stacks/stacks-project

Fetching contributors…
Cannot retrieve contributors at this time
7424 lines (6781 sloc) 274 KB
 \input{preamble} % OK, start here. % \begin{document} \title{Formal Deformation Theory} \maketitle \phantomsection \label{section-phantom} \tableofcontents \section{Introduction} \label{section-introduction} \noindent This chapter develops formal deformation theory in a form applicable later in the Stacks project, closely following Rim \cite[Exposee VI]{SGA7-I} and Schlessinger \cite{Sch}. We strongly encourage the reader new to this topic to read the paper by Schlessinger first, as it is sufficiently general for most applications, and Schlessinger's results are indeed used in most papers that use this kind of formal deformation theory. \medskip\noindent Let $\Lambda$ be a complete Noetherian local ring with residue field $k$, and let $\mathcal{C}_\Lambda$ denote the category of Artinian local $\Lambda$-algebras with residue field $k$. Given a functor $F : \mathcal{C}_\Lambda \to \textit{Sets}$ such that $F(k)$ is a one element set, Schlessinger's paper introduced conditions (H1)-(H4) such that: \begin{enumerate} \item $F$ has a hull'' if and only if (H1)-(H3) hold. \item $F$ is prorepresentable if and only (H1)-(H4) hold. \end{enumerate} The purpose of this chapter is to generalize these results in two ways exactly as is done in Rim's paper: \begin{enumerate} \item[(A)] The functor $F$ is replaced by a category $\mathcal{F}$ cofibered in groupoids over $\mathcal{C}_\Lambda$, see Section \ref{section-CLambda}. \item[(B)] We let $\Lambda$ be a Noetherian ring and $\Lambda \to k$ a finite ring map to a field. The category $\mathcal{C}_\Lambda$ is the category of Artinian local $\Lambda$-algebras $A$ endowed with a given identification $A/\mathfrak m_A = k$. \end{enumerate} The analogue of the condition that $F(k)$ is a one element set is that $\mathcal{F}(k)$ is the trivial groupoid. If $\mathcal{F}$ satisfies this condition then we say it is a {\it predeformation category}, but in general we do not make this assumption. Rim's paper \cite[Exposee VI]{SGA7-I} is the original source for the results in this document. We also mention the useful paper \cite{Vistoli}, which discusses deformation theory with groupoids but in less generality than we do here. \medskip\noindent An important role is played by the completion'' $\widehat{\mathcal{C}}_\Lambda$ of the category $\mathcal{C}_\Lambda$. An object of $\widehat{\mathcal{C}}_\Lambda$ is a Noetherian complete local $\Lambda$-algebra $R$ whose residue field is identified with $k$, see Section \ref{section-category-completion-CLambda}. On the one hand $\mathcal{C}_\Lambda \subset \widehat{\mathcal{C}}_\Lambda$ is a strictly full subcategory and on the other hand $\widehat{\mathcal{C}}_\Lambda$ is a full subcategory of the category of pro-objects of $\mathcal{C}_\Lambda$. A functor $\mathcal{C}_\Lambda \to \textit{Sets}$ is {\it prorepresentable} if it is isomorphic to the restriction of a representable functor $\underline{R} = \Mor_{\widehat{\mathcal{C}}_\Lambda}(R, -)$ to $\mathcal{C}_\Lambda$ where $R \in \Ob(\widehat{\mathcal{C}}_\Lambda)$. \medskip\noindent {\it Categories cofibred in groupoids} are dual to categories fibred in groupoids; we introduce them in Section \ref{section-preliminary}. A {\it smooth} morphism of categories cofibred in groupoids over $\mathcal{C}_\Lambda$ is one that satisfies the infinitesimal lifting criterion for objects, see Section \ref{section-smooth-morphisms}. This is analogous to the definition of a formally smooth ring map, see Algebra, Definition \ref{algebra-definition-formally-smooth} and is exactly dual to the notion in Criteria for Representability, Section \ref{criteria-section-formally-smooth}. This is an important notion as we eventually want to prove that certain kinds of categories cofibred in groupoids have a smooth prorepresentable presentation, much like the characterization of algebraic stacks in Algebraic Stacks, Sections \ref{algebraic-section-stack-to-presentation} and \ref{algebraic-section-smooth-groupoid-gives-algebraic-stack}. A {\it versal formal object} of a category $\mathcal{F}$ cofibred in groupoids over $\mathcal{C}_\Lambda$ is an object $\xi \in \widehat{\mathcal{F}}(R)$ of the completion such that the associated morphism $\underline{\xi} : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$ is smooth. \medskip\noindent In Section \ref{section-schlessinger-conditions}, we define conditions (S1) and (S2) on $\mathcal{F}$ generalizing Schlessinger's (H1) and (H2). The analogue of Schlessinger's (H3)---the condition that $\mathcal{F}$ has finite dimensional tangent space---is not given a name. A key step in the development of the theory is the existence of versal formal objects for predeformation categories satisfying (S1), (S2) and (H3), see Lemma \ref{lemma-versal-object-existence}. Schlessinger's notion of a {\it hull} for a functor $F : \mathcal{C}_\Lambda \to \textit{Sets}$ is, in our terminology, a versal formal object $\xi \in \widehat{F}(R)$ such that the induced map of tangent spaces $d\underline{\xi} : T\underline{R}|_{\mathcal{C}_\Lambda} \to TF$ is an isomorphism. In the literature a hull is often called a miniversal'' object. We do not do so, and here is why. It can happen that a functor has a versal formal object without having a hull. Moreover, we show in Section \ref{section-minimal-versal} that if a predeformation category has a versal formal object, then it always has a {\it minimal} one (as defined in Definition \ref{definition-minimal-versal}) which is unique up to isomorphism, see Lemma \ref{lemma-minimal-versal}. But it can happen that the minimal versal formal object does not induce an isomorphism on tangent spaces! (See Examples \ref{example-do-not-get-S2} and \ref{example-smooth-continued}.) \medskip\noindent Keeping in mind the differences pointed out above, Theorem \ref{theorem-miniversal-object-existence} is the direct generalization of (1) above: it recovers Schlessinger's result in the case that $\mathcal{F}$ is a functor and it characterizes minimal versal formal objects, in the presence of conditions (S1) and (S2), in terms of the map $d\underline{\xi} : T\underline{R}|_{\mathcal{C}_\Lambda} \to TF$ on tangent spaces. \medskip\noindent In Section \ref{section-RS-condition}, we define Rim's condition (RS) on $\mathcal{F}$ generalizing Schlessinger's (H4). A {\it deformation category} is defined as a predeformation category satisfying (RS). The analogue to prorepresentable functors are the categories cofibred in groupoids over $\mathcal{C}_\Lambda$ which have a {\it presentation by a smooth prorepresentable groupoid in functors} on $\mathcal{C}_\Lambda$, see Definitions \ref{definition-groupoid-in-functors}, \ref{definition-prorepresentable-groupoid-in-functors}, and \ref{definition-smooth-groupoid-in-functors}. This notion of a presentation takes into account the groupoid structure of the fibers of $\mathcal{F}$. In Theorem \ref{theorem-presentation-deformation-groupoid} we prove that $\mathcal{F}$ has a presentation by a smooth prorepresentable groupoid in functors if and only if $\mathcal{F}$ has a finite dimensional tangent space and finite dimensional infinitesimal automorphism space. This is the generalization of (2) above: it reduces to Schlessinger's result in the case that $\mathcal{F}$ is a functor. There is a final Section \ref{section-minimality} where we discuss how to use minimal versal formal objects to produce a (unique up to isomorphism) minimal presentation by a smooth prorepresentable groupoid in functors. \medskip\noindent We also find the following conceptual explanation for Schlessinger's conditions. If a predeformation category $\mathcal{F}$ satisfies (RS), then the associated functor of isomorphism classes $\overline{\mathcal{F}}: \mathcal{C}_\Lambda \to \textit{Sets}$ satisfies (H1) and (H2) (Lemmas \ref{lemma-RS-implies-S1-S2} and \ref{lemma-S1-S2-associated-functor}). Conversely, if a functor $F : \mathcal{C}_\Lambda \to \textit{Sets}$ arises naturally as the functor of isomorphism classes of a category $\mathcal{F}$ cofibered in groupoids, then it seems to happen in practice that an argument showing $F$ satisfies (H1) and (H2) will also show $\mathcal{F}$ satisfies (RS) (see Artin's Axioms, Section \ref{artin-section-examples} for examples). Moreover, if $\mathcal{F}$ satisfies (RS), then condition (H4) for $\overline{\mathcal{F}}$ has a simple interpretation in terms of extending automorphisms of objects of $\mathcal{F}$ (Lemma \ref{lemma-RS-associated-functor}). These observations suggest that (RS) should be regarded as the fundamental deformation theoretic glueing condition. \section{Notation and Conventions} \label{section-notations-conventions} \noindent A ring is commutative with $1$. The maximal ideal of a local ring $A$ is denoted by $\mathfrak{m}_A$. The set of positive integers is denoted by $\mathbf{N} = \{1, 2, 3, \ldots\}$. If $U$ is an object of a category $\mathcal{C}$, we denote by $\underline{U}$ the functor $\Mor_\mathcal{C}(U, -): \mathcal{C} \to \textit{Sets}$, see Remarks \ref{remarks-cofibered-groupoids} (\ref{item-definition-yoneda}). Warning: this may conflict with the notation in other chapters where we sometimes use $\underline{U}$ to denote $h_U(-) = \Mor_\mathcal{C}(-, U)$. \medskip\noindent Throughout this chapter $\Lambda$ is a Noetherian ring and $\Lambda \to k$ is a finite ring map from $\Lambda$ to a field. The kernel of this map is denoted $\mathfrak m_\Lambda$ and the image $k' \subset k$. It turns out that $\mathfrak m_\Lambda$ is a maximal ideal, $k' = \Lambda/\mathfrak m_\Lambda$ is a field, and the extension $k' \subset k$ is finite. See discussion surrounding (\ref{equation-k-prime}). \section{The base category} \label{section-CLambda} \noindent Motivation. An important application of formal deformation theory is to criteria for representability by algebraic spaces. Suppose given a locally Noetherian base $S$ and a functor $F : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$. Let $k$ be a finite type field over $S$, i.e., we are given a finite type morphism $\Spec(k) \to S$. One of Artin's criteria is that for any element $x \in F(\Spec(k))$ the predeformation functor associated to the triple $(S, k, x)$ should be prorepresentable. By Morphisms, Lemma \ref{morphisms-lemma-point-finite-type} the condition that $k$ is of finite type over $S$ means that there exists an affine open $\Spec(\Lambda) \subset S$ such that $k$ is a finite $\Lambda$-algebra. This motivates why we work throughout this chapter with a base category as follows. \begin{definition} \label{definition-CLambda} Let $\Lambda$ be a Noetherian ring and let $\Lambda \to k$ be a finite ring map where $k$ is a field. We define {\it $\mathcal{C}_\Lambda$} to be the category with \begin{enumerate} \item objects are pairs $(A, \varphi)$ where $A$ is an Artinian local $\Lambda$-algebra and where $\varphi : A/\mathfrak m_A \to k$ is a $\Lambda$-algebra isomorphism, and \item morphisms $f : (B, \psi) \to (A, \varphi)$ are local $\Lambda$-algebra homomorphisms such that $\varphi \circ (f \bmod \mathfrak m) = \psi$. \end{enumerate} We say we are in the {\it classical case} if $\Lambda$ is a Noetherian complete local ring and $k$ is its residue field. \end{definition} \noindent Note that if $\Lambda \to k$ is surjective and if $A$ is an Artinian local $\Lambda$-algebra, then the identification $\varphi$, if it exists, is unique. Moreover, in this case any $\Lambda$-algebra map $A \to B$ is going to be compatible with the identifications. Hence in this case $\mathcal{C}_\Lambda$ is just the category of local Artinian $\Lambda$-algebras whose residue field is'' $k$. By abuse of notation we also denote objects of $\mathcal{C}_\Lambda$ simply $A$ in the general case. Moreover, we will often write $A/\mathfrak m = k$, i.e., we will pretend all rings in $\mathcal{C}_\Lambda$ have residue field $k$ (since all ring maps in $\mathcal{C}_\Lambda$ are compatible with the given identifications this should never cause any problems). Throughout the rest of this chapter the base ring $\Lambda$ and the field $k$ are fixed. The category $\mathcal{C}_\Lambda$ will be the base category for the cofibered categories considered below. \begin{definition} \label{definition-small-extension} Let $f: B \to A$ be a ring map in $\mathcal{C}_\Lambda$. We say $f$ is a {\it small extension} if it is surjective and $\Ker(f)$ is a nonzero principal ideal which is annihilated by $\mathfrak{m}_B$. \end{definition} \noindent By the following lemma we can often reduce arguments involving surjective ring maps in $\mathcal{C}_\Lambda$ to the case of small extensions. \begin{lemma} \label{lemma-factor-small-extension} Let $f: B \to A$ be a surjective ring map in $\mathcal{C}_\Lambda$. Then $f$ can be factored as a composition of small extensions. \end{lemma} \begin{proof} Let $I$ be the kernel of $f$. The maximal ideal $\mathfrak{m}_B$ is nilpotent since $B$ is Artinian, say $\mathfrak{m}_B^n = 0$. Hence we get a factorization $$B = B/I\mathfrak{m}_B^{n-1} \to B/I\mathfrak{m}_B^{n-2} \to \ldots \to B/I \cong A$$ of $f$ into a composition of surjective maps whose kernels are annihilated by the maximal ideal. Thus it suffices to prove the lemma when $f$ itself is such a map, i.e.\ when $I$ is annihilated by $\mathfrak{m}_B$. In this case $I$ is a $k$-vector space, which has finite dimension, see Algebra, Lemma \ref{algebra-lemma-artinian-finite-length}. Take a basis $x_1, \ldots, x_n$ of $I$ as a $k$-vector space to get a factorization $$B \to B/(x_1) \to \ldots \to B/(x_1, \ldots, x_n) \cong A$$ of $f$ into a composition of small extensions. \end{proof} \noindent The next lemma says that we can compute the length of a module over a local $\Lambda$-algebra with residue field $k$ in terms of the length over $\Lambda$. To explain the notation in the statement, let $k' \subset k$ be the image of our fixed finite ring map $\Lambda \to k$. Note that $k/k'$ is a finite extension of rings. Hence $k'$ is a field and $k'/k$ is a finite extension, see Algebra, Lemma \ref{algebra-lemma-integral-under-field}. Moreover, as $\Lambda \to k'$ is surjective we see that its kernel is a maximal ideal $\mathfrak m_\Lambda$. Thus \begin{equation} \label{equation-k-prime} [k : k'] = [k : \Lambda/\mathfrak m_\Lambda] < \infty \end{equation} and in the classical case we have $k = k'$. The notation $k' = \Lambda/\mathfrak m_\Lambda$ will be fixed throughout this chapter. \begin{lemma} \label{lemma-length} Let $A$ be a local $\Lambda$-algebra with residue field $k$. Let $M$ be an $A$-module. Then $[k : k'] \text{length}_A(M) = \text{length}_\Lambda(M)$. In the classical case we have $\text{length}_A(M) = \text{length}_\Lambda(M)$. \end{lemma} \begin{proof} If $M$ is a simple $A$-module then $M \cong k$ as an $A$-module, see Algebra, Lemma \ref{algebra-lemma-characterize-length-1}. In this case $\text{length}_A(M) = 1$ and $\text{length}_\Lambda(M) = [k' : k]$, see Algebra, Lemma \ref{algebra-lemma-dimension-is-length}. If $\text{length}_A(M)$ is finite, then the result follows on choosing a filtration of $M$ by $A$-submodules with simple quotients using additivity, see Algebra, Lemma \ref{algebra-lemma-length-additive}. If $\text{length}_A(M)$ is infinite, the result follows from the obvious inequality $\text{length}_A(M) \leq \text{length}_\Lambda(M)$. \end{proof} \begin{lemma} \label{lemma-surjective} Let $A \to B$ be a ring map in $\mathcal{C}_\Lambda$. The following are equivalent \begin{enumerate} \item $f$ is surjective, \item $\mathfrak m_A/\mathfrak m_A^2 \to \mathfrak m_B/\mathfrak m_B^2$ is surjective, and \item $\mathfrak m_A/(\mathfrak m_\Lambda A + \mathfrak m_A^2) \to \mathfrak m_B/(\mathfrak m_\Lambda B + \mathfrak m_B^2)$ is surjective. \end{enumerate} \end{lemma} \begin{proof} For any ring map $f : A \to B$ in $\mathcal{C}_\Lambda$ we have $f(\mathfrak m_A) \subset \mathfrak m_B$ for example because $\mathfrak m_A$, $\mathfrak m_B$ is the set of nilpotent elements of $A$, $B$. Suppose $f$ is surjective. Let $y \in \mathfrak m_B$. Choose $x \in A$ with $f(x) = y$. Since $f$ induces an isomorphism $A/\mathfrak m_A \to B/\mathfrak m_B$ we see that $x \in \mathfrak m_A$. Hence the induced map $\mathfrak m_A/\mathfrak m_A^2 \to \mathfrak m_B/\mathfrak m_B^2$ is surjective. In this way we see that (1) implies (2). \medskip\noindent It is clear that (2) implies (3). The map $A \to B$ gives rise to a canonical commutative diagram $$\xymatrix{ \mathfrak m_\Lambda/\mathfrak m_\Lambda^2 \otimes_{k'} k \ar[r] \ar[d] & \mathfrak m_A/\mathfrak m_A^2 \ar[r] \ar[d] & \mathfrak m_A/(\mathfrak m_\Lambda A + \mathfrak m_A^2) \ar[r] \ar[d] & 0 \\ \mathfrak m_\Lambda/\mathfrak m_\Lambda^2 \otimes_{k'} k \ar[r] & \mathfrak m_B/\mathfrak m_B^2 \ar[r] & \mathfrak m_B/(\mathfrak m_\Lambda B + \mathfrak m_B^2) \ar[r] & 0 }$$ with exact rows. Hence if (3) holds, then so does (2). \medskip\noindent Assume (2). To show that $A \to B$ is surjective it suffices by Nakayama's lemma (Algebra, Lemma \ref{algebra-lemma-NAK}) to show that $A/\mathfrak m_A \to B/\mathfrak m_AB$ is surjective. (Note that $\mathfrak m_A$ is a nilpotent ideal.) As $k = A/\mathfrak m_A = B/\mathfrak m_B$ it suffices to show that $\mathfrak m_AB \to \mathfrak m_B$ is surjective. Applying Nakayama's lemma once more we see that it suffices to see that $\mathfrak m_AB/\mathfrak m_A\mathfrak m_B \to \mathfrak m_B/\mathfrak m_B^2$ is surjective which is what we assumed. \end{proof} \noindent If $A \to B$ is a ring map in $\mathcal{C}_\Lambda$, then the map $\mathfrak m_A/(\mathfrak m_\Lambda A + \mathfrak m_A^2) \to \mathfrak m_B/(\mathfrak m_\Lambda B + \mathfrak m_B^2)$ is the map on relative cotangent spaces. Here is a formal definition. \begin{definition} \label{definition-tangent-space-ring} Let $R \to S$ be a local homomorphism of local rings. The {\it relative cotangent space}\footnote{Caution: We will see later that in our general setting the tangent space of an object $A \in \mathcal{C}_\Lambda$ over $\Lambda$ should not be defined simply as the $k$-linear dual of the relative cotangent space. In fact, the correct definition of the relative cotangent space is $\Omega_{S/R} \otimes_S S/\mathfrak m_S$.} of $R$ over $S$ is the $S/\mathfrak m_S$-vector space $\mathfrak m_S/(\mathfrak m_R S + \mathfrak m_S^2)$. \end{definition} \noindent If $f_1: A_1 \to A$ and $f_2: A_2 \to A$ are two ring maps, then the fiber product $A_1 \times_A A_2$ is the subring of $A_1 \times A_2$ consisting of elements whose two projections to $A$ are equal. Throughout this chapter we will be considering conditions involving such a fiber product when $f_1$ and $f_2$ are in $\mathcal{C}_\Lambda$. It isn't always the case that the fibre product is an object of $\mathcal{C}_\Lambda$. \begin{example} \label{example-fibre-product} Let $p$ be a prime number and let $n \in \mathbf{N}$. Let $\Lambda = \mathbf{F}_p(t_1, t_2, \ldots, t_n)$ and let $k = \mathbf{F}_p(x_1, \ldots, x_n)$ with map $\Lambda \to k$ given by $t_i \mapsto x_i^p$. Let $A = k[\epsilon] = k[x]/(x^2)$. Then $A$ is an object of $\mathcal{C}_\Lambda$. Suppose that $D : k \to k$ is a derivation of $k$ over $\Lambda$, for example $D = \partial/\partial x_i$. Then the map $$f_D : k \longrightarrow k[\epsilon], \quad a \mapsto a + D(a)\epsilon$$ is a morphism of $\mathcal{C}_\Lambda$. Set $A_1 = A_2 = k$ and set $f_1 = f_{\partial/\partial x_1}$ and $f_2(a) = a$. Then $A_1 \times_A A_2 = \{a \in k \mid \partial/\partial x_1(a) = 0\}$ which does not surject onto $k$. Hence the fibre product isn't an object of $\mathcal{C}_\Lambda$. \end{example} \noindent It turns out that this problem can only occur if the residue field extension $k' \subset k$ (\ref{equation-k-prime}) is inseparable and neither $f_1$ nor $f_2$ is surjective. \begin{lemma} \label{lemma-fiber-product-CLambda} Let $f_1 : A_1 \to A$ and $f_2 : A_2 \to A$ be ring maps in $\mathcal{C}_\Lambda$. Then: \begin{enumerate} \item If $f_1$ or $f_2$ is surjective, then $A_1 \times_A A_2$ is in $\mathcal{C}_\Lambda$. \item If $f_2$ is a small extension, then so is $A_1 \times_A A_2 \to A_1$. \item If the field extension $k' \subset k$ is separable, then $A_1 \times_A A_2$ is in $\mathcal{C}_\Lambda$. \end{enumerate} \end{lemma} \begin{proof} The ring $A_1 \times_A A_2$ is a $\Lambda$-algebra via the map $\Lambda \to A_1 \times_A A_2$ induced by the maps $\Lambda \to A_1$ and $\Lambda \to A_2$. It is a local ring with unique maximal ideal $$\mathfrak m_{A_1} \times_{\mathfrak m_A} \mathfrak m_{A_2} = \Ker(A_1 \times_A A_2 \longrightarrow k)$$ A ring is Artinian if and only if it has finite length as a module over itself, see Algebra, Lemma \ref{algebra-lemma-artinian-finite-length}. Since $A_1$ and $A_2$ are Artinian, Lemma \ref{lemma-length} implies $\text{length}_\Lambda(A_1)$ and $\text{length}_\Lambda(A_2)$, and hence $\text{length}_\Lambda(A_1 \times A_2)$, are all finite. As $A_1 \times_A A_2 \subset A_1 \times A_2$ is a $\Lambda$-submodule, this implies $\text{length}_{A_1 \times_A A_2}(A_1 \times_A A_2) \leq \text{length}_\Lambda(A_1 \times_A A_2)$ is finite. So $A_1 \times_A A_2$ is Artinian. Thus the only thing that is keeping $A_1 \times_A A_2$ from being an object of $\mathcal{C}_\Lambda$ is the possibility that its residue field maps to a proper subfield of $k$ via the map $A_1 \times_A A_2 \to A \to A/\mathfrak m_A = k$ above. \medskip\noindent Proof of (1). If $f_2$ is surjective, then the projection $A_1 \times_A A_2 \to A_1$ is surjective. Hence the composition $A_1 \times_A A_2 \to A_1 \to A_1/\mathfrak m_{A_1} = k$ is surjective and we conclude that $A_1 \times_A A_2$ is an object of $\mathcal{C}_\Lambda$. \medskip\noindent Proof of (2). If $f_2$ is a small extension then $A_2 \to A$ and $A_1 \times_A A_2 \to A_1$ are both surjective with the same kernel. Hence the kernel of $A_1 \times_A A_2 \to A_1$ is a $1$-dimensional $k$-vector space and we see that $A_1 \times_A A_2 \to A_1$ is a small extension. \medskip\noindent Proof of (3). Choose $\overline{x} \in k$ such that $k = k'(\overline{x})$ (see Fields, Lemma \ref{fields-lemma-primitive-element}). Let $P'(T) \in k'[T]$ be the minimal polynomial of $\overline{x}$ over $k'$. Since $k/k'$ is separable we see that $\text{d}P/\text{d}T(\overline{x}) \not = 0$. Choose a monic $P \in \Lambda[T]$ which maps to $P'$ under the surjective map $\Lambda[T] \to k'[T]$. Because $A, A_1, A_2$ are henselian, see Algebra, Lemma \ref{algebra-lemma-local-dimension-zero-henselian}, we can find $x, x_1, x_2 \in A, A_1, A_2$ with $P(x) = 0, P(x_1) = 0, P(x_2) = 0$ and such that the image of $x, x_1, x_2$ in $k$ is $\overline{x}$. Then $(x_1, x_2) \in A_1 \times_A A_2$ because $x_1, x_2$ map to $x \in A$ by uniqueness, see Algebra, Lemma \ref{algebra-lemma-uniqueness}. Hence the residue field of $A_1 \times_A A_2$ contains a generator of $k$ over $k'$ and we win. \end{proof} \noindent Next we define essential surjections in $\mathcal{C}_\Lambda$. A necessary and sufficient condition for a surjection in $\mathcal{C}_\Lambda$ to be essential is given in Lemma \ref{lemma-essential-surjection}. \begin{definition} \label{definition-essential-surjection} Let $f: B \to A$ be a ring map in $\mathcal{C}_\Lambda$. We say $f$ is an {\it essential surjection} if it has the following properties: \begin{enumerate} \item $f$ is surjective. \item If $g: C \to B$ is a ring map in $\mathcal{C}_\Lambda$ such that $f \circ g$ is surjective, then $g$ is surjective. \end{enumerate} \end{definition} \noindent Using Lemma \ref{lemma-surjective}, we can characterize essential surjections in $\mathcal{C}_\Lambda$ as follows. \begin{lemma} \label{lemma-essential-surjection-mod-squares} Let $f: B \to A$ be a ring map in $\mathcal{C}_\Lambda$. The following are equivalent \begin{enumerate} \item $f$ is an essential surjection, \item the map $B/\mathfrak m_B^2 \to A/\mathfrak m_A^2$ is an essential surjection, and \item the map $B/(\mathfrak m_\Lambda B + \mathfrak m_B^2) \to A/(\mathfrak m_\Lambda A + \mathfrak m_A^2)$ is an essential surjection. \end{enumerate} \end{lemma} \begin{proof} Assume (3). Let $C \to B$ be a ring map in $\mathcal{C}_\Lambda$ such that $C \to A$ is surjective. Then $C \to A/(\mathfrak m_\Lambda A + \mathfrak m_A^2)$ is surjective too. We conclude that $C \to B/(\mathfrak m_\Lambda B + \mathfrak m_B^2)$ is surjective by our assumption. Hence $C \to B$ is surjective by applying Lemma \ref{lemma-surjective} (2 times). \medskip\noindent Assume (1). Let $C \to B/(\mathfrak m_\Lambda B + \mathfrak m_B^2)$ be a morphism of $\mathcal{C}_\Lambda$ such that $C \to A/(\mathfrak m_\Lambda A + \mathfrak m_A^2)$ is surjective. Set $C' = C \times_{B/(\mathfrak m_\Lambda B + \mathfrak m_B^2)} B$ which is an object of $\mathcal{C}_\Lambda$ by Lemma \ref{lemma-fiber-product-CLambda}. Note that $C' \to A/(\mathfrak m_\Lambda A + \mathfrak m_A^2)$ is still surjective, hence $C' \to A$ is surjective by Lemma \ref{lemma-surjective}. Thus $C' \to B$ is surjective by our assumption. This implies that $C' \to B/(\mathfrak m_\Lambda B + \mathfrak m_B^2)$ is surjective, which implies by the construction of $C'$ that $C \to B/(\mathfrak m_\Lambda B + \mathfrak m_B^2)$ is surjective. \medskip\noindent In the first paragraph we proved (3) $\Rightarrow$ (1) and in the second paragraph we proved (1) $\Rightarrow$ (3). The equivalence of (2) and (3) is a special case of the equivalence of (1) and (3), hence we are done. \end{proof} \noindent To analyze essential surjections in $\mathcal{C}_\Lambda$ a bit more we introduce some notation. Suppose that $A$ is an object of $\mathcal{C}_\Lambda$. There is a canonical exact sequence \begin{equation} \label{equation-sequence} \mathfrak m_A/\mathfrak m_A^2 \xrightarrow{\text{d}_A} \Omega_{A/\Lambda} \otimes_A k \to \Omega_{k/\Lambda} \to 0 \end{equation} see Algebra, Lemma \ref{algebra-lemma-differential-seq}. Note that $\Omega_{k/\Lambda} = \Omega_{k/k'}$ with $k'$ as in (\ref{equation-k-prime}). Let $H_1(L_{k/\Lambda})$ be the first homology module of the naive cotangent complex of $k$ over $\Lambda$, see Algebra, Definition \ref{algebra-definition-naive-cotangent-complex}. Then we can extend (\ref{equation-sequence}) to the exact sequence \begin{equation} \label{equation-sequence-extended} H_1(L_{k/\Lambda}) \to \mathfrak m_A/\mathfrak m_A^2 \xrightarrow{\text{d}_A} \Omega_{A/\Lambda} \otimes_A k \to \Omega_{k/\Lambda} \to 0, \end{equation} see Algebra, Lemma \ref{algebra-lemma-exact-sequence-NL}. If $B \to A$ is a ring map in $\mathcal{C}_\Lambda$ then we obtain a commutative diagram \begin{equation} \label{equation-sequence-functorial} \vcenter{ \xymatrix{ H_1(L_{k/\Lambda}) \ar[r] \ar@{=}[d] & \mathfrak m_B/\mathfrak m_B^2 \ar[r]_{\text{d}_B} \ar[d] & \Omega_{B/\Lambda} \otimes_B k \ar[r] \ar[d] & \Omega_{k/\Lambda} \ar[r] \ar@{=}[d] & 0 \\ H_1(L_{k/\Lambda}) \ar[r] & \mathfrak m_A/\mathfrak m_A^2 \ar[r]^{\text{d}_A} & \Omega_{A/\Lambda} \otimes_A k \ar[r] & \Omega_{k/\Lambda} \ar[r] & 0 } } \end{equation} with exact rows. \begin{lemma} \label{lemma-H1-separable-case} There is a canonical map $$\mathfrak m_\Lambda/\mathfrak m_\Lambda^2 \longrightarrow H_1(L_{k/\Lambda}).$$ If $k' \subset k$ is separable (for example if the characteristic of $k$ is zero), then this map induces an isomorphism $\mathfrak m_\Lambda/\mathfrak m_\Lambda^2 \otimes_{k'} k = H_1(L_{k/\Lambda})$. If $k = k'$ (for example in the classical case), then $\mathfrak m_\Lambda/\mathfrak m_\Lambda^2 = H_1(L_{k/\Lambda})$. The composition $$\mathfrak m_\Lambda/\mathfrak m_\Lambda^2 \longrightarrow H_1(L_{k/\Lambda}) \longrightarrow \mathfrak m_A/\mathfrak m_A^2$$ comes from the canonical map $\mathfrak m_\Lambda \to \mathfrak m_A$. \end{lemma} \begin{proof} Note that $H_1(L_{k'/\Lambda}) = \mathfrak m_\Lambda/\mathfrak m_\Lambda^2$ as $\Lambda \to k'$ is surjective with kernel $\mathfrak m_\Lambda$. The map arises from functoriality of the naive cotangent complex. If $k' \subset k$ is separable, then $k' \to k$ is an \'etale ring map, see Algebra, Lemma \ref{algebra-lemma-etale-over-field}. Thus its naive cotangent complex has trivial homology groups, see Algebra, Definition \ref{algebra-definition-etale}. Then Algebra, Lemma \ref{algebra-lemma-exact-sequence-NL} applied to the ring maps $\Lambda \to k' \to k$ implies that $\mathfrak m_\Lambda/\mathfrak m_\Lambda^2 \otimes_{k'} k = H_1(L_{k/\Lambda})$. We omit the proof of the final statement. \end{proof} \begin{lemma} \label{lemma-essential-surjection} Let $f: B \to A$ be a ring map in $\mathcal{C}_\Lambda$. Notation as in (\ref{equation-sequence-functorial}). \begin{enumerate} \item The equivalent conditions of Lemma \ref{lemma-essential-surjection-mod-squares} characterizing when $f$ is surjective are also equivalent to \begin{enumerate} \item $\Im(\text{d}_B) \to \Im(\text{d}_A)$ is surjective, and \item the map $\Omega_{B/\Lambda} \otimes_B k \to \Omega_{A/\Lambda} \otimes_A k$ is surjective. \end{enumerate} \item The following are equivalent \begin{enumerate} \item $f$ is an essential surjection, \item the map $\Im(\text{d}_B) \to \Im(\text{d}_A)$ is an isomorphism, and \item the map $\Omega_{B/\Lambda} \otimes_B k \to \Omega_{A/\Lambda} \otimes_A k$ is an isomorphism. \end{enumerate} \item If $k/k'$ is separable, then $f$ is an essential surjection if and only if the map $\mathfrak m_B/(\mathfrak m_\Lambda B + \mathfrak m_B^2) \to \mathfrak m_A/(\mathfrak m_\Lambda A + \mathfrak m_A^2)$ is an isomorphism. \item If $f$ is a small extension, then $f$ is not essential if and only if $f$ has a section $s: A \to B$ in $\mathcal{C}_\Lambda$ with $f \circ s = \text{id}_A$. \end{enumerate} \end{lemma} \begin{proof} Proof of (1). It follows from (\ref{equation-sequence-functorial}) that (1)(a) and (1)(b) are equivalent. Also, if $A \to B$ is surjective, then (1)(a) and (1)(b) hold. Assume (1)(a). Since the kernel of $\text{d}_A$ is the image of $H_1(L_{k/\Lambda})$ which also maps to $\mathfrak m_B/\mathfrak m_B^2$ we conclude that $\mathfrak m_B/\mathfrak m_B^2 \to \mathfrak m_A/\mathfrak m_A^2$ is surjective. Hence $B \to A$ is surjective by Lemma \ref{lemma-surjective}. This finishes the proof of (1). \medskip\noindent Proof of (2). The equivalence of (2)(b) and (2)(c) is immediate from (\ref{equation-sequence-functorial}). \medskip\noindent Assume (2)(b). Let $g : C \to B$ be a ring map in $\mathcal{C}_\Lambda$ such that $f \circ g$ is surjective. We conclude that $\mathfrak m_C/\mathfrak m_C^2 \to \mathfrak m_A/\mathfrak m_A^2$ is surjective by Lemma \ref{lemma-surjective}. Hence $\Im(\text{d}_C) \to \Im(\text{d}_A)$ is surjective and by the assumption we see that $\Im(\text{d}_C) \to \Im(\text{d}_B)$ is surjective. It follows that $C \to B$ is surjective by (1). \medskip\noindent Assume (2)(a). Then $f$ is surjective and we see that $\Omega_{B/\Lambda} \otimes_B k \to \Omega_{A/\Lambda} \otimes_A k$ is surjective. Let $K$ be the kernel. Note that $K = \text{d}_B(\Ker(\mathfrak m_B/\mathfrak m_B^2 \to \mathfrak m_A/\mathfrak m_A^2))$ by (\ref{equation-sequence-functorial}). Choose a splitting $$\Omega_{B/\Lambda} \otimes_B k = \Omega_{A/\Lambda} \otimes_A k \oplus K$$ of $k$-vector space. The map $\text{d} : B \to \Omega_{B/\Lambda}$ induces via the projection onto $K$ a map $D : B \to K$. Set $C = \{b \in B \mid D(b) = 0\}$. The Leibniz rule shows that this is a $\Lambda$-subalgebra of $B$. Let $\overline{x} \in k$. Choose $x \in B$ mapping to $\overline{x}$. If $D(x) \not = 0$, then we can find an element $y \in \mathfrak m_B$ such that $D(y) = D(x)$. Hence $x - y \in C$ is an element which maps to $\overline{x}$. Thus $C \to k$ is surjective and $C$ is an object of $\mathcal{C}_\Lambda$. Similarly, pick $\omega \in \Im(\text{d}_A)$. We can find $x \in \mathfrak m_B$ such that $\text{d}_B(x)$ maps to $\omega$ by (1). If $D(x) \not = 0$, then we can find an element $y \in \mathfrak m_B$ which maps to zero in $\mathfrak m_A/\mathfrak m_A^2$ such that $D(y) = D(x)$. Hence $z = x - y$ is an element of $\mathfrak m_C$ whose image $\text{d}_C(z) \in \Omega_{C/k} \otimes_C k$ maps to $\omega$. Hence $\Im(\text{d}_C) \to \Im(\text{d}_A)$ is surjective. We conclude that $C \to A$ is surjective by (1). Hence $C \to B$ is surjective by assumption. Hence $D = 0$, i.e., $K = 0$, i.e., (2)(c) holds. This finishes the proof of (2). \medskip\noindent Proof of (3). If $k'/k$ is separable, then $H_1(L_{k/\Lambda}) = \mathfrak m_\Lambda/\mathfrak m_\Lambda^2 \otimes_{k'} k$, see Lemma \ref{lemma-H1-separable-case}. Hence $\Im(\text{d}_A) = \mathfrak m_A/(\mathfrak m_\Lambda A + \mathfrak m_A^2)$ and similarly for $B$. Thus (3) follows from (2). \medskip\noindent Proof of (4). A section $s$ of $f$ is not surjective (by definition a small extension has nontrivial kernel), hence $f$ is not essentially surjective. Conversely, assume $f$ is a small surjection but not an essential surjection. Choose a ring map $C \to B$ in $\mathcal{C}_\Lambda$ which is not surjective, such that $C \to A$ is surjective. Let $C' \subset B$ be the image of $C \to B$. Then $C' \not = B$ but $C'$ surjects onto $A$. Since $f : B \to A$ is a small extension, $\text{length}_C(B) = \text{length}_C(A) + 1$. Thus $\text{length}_C(C') \leq \text{length}_C(A)$ since $C'$ is a proper subring of $B$. But $C' \to A$ is surjective, so in fact we must have $\text{length}_C(C') = \text{length}_C(A)$ and $C' \to A$ is an isomorphism which gives us our section. \end{proof} \begin{example} \label{example-essential-surjection} Let $\Lambda = k[[x]]$ be the power series ring in $1$ variable over $k$. Set $A = k$ and $B = \Lambda/(x^2)$. Then $B \to A$ is an essential surjection by Lemma \ref{lemma-essential-surjection} because it is a small extension and the map $B \to A$ does not have a right inverse (in the category $\mathcal{C}_\Lambda$). But the map $$k \cong \mathfrak m_B/\mathfrak m_B^2 \longrightarrow \mathfrak m_A/\mathfrak m_A^2 = 0$$ is not an isomorphism. Thus in Lemma \ref{lemma-essential-surjection} (3) it is necessary to consider the map of relative cotangent spaces $\mathfrak m_B/(\mathfrak m_\Lambda B + \mathfrak m_B^2) \to \mathfrak m_A/(\mathfrak m_\Lambda A + \mathfrak m_A^2)$. \end{example} \section{The completed base category} \label{section-category-completion-CLambda} \noindent The following completion'' of the category $\mathcal{C}_\Lambda$ will serve as the base category of the completion of a category cofibered in groupoids over $\mathcal{C}_\Lambda$ (Section \ref{section-formal-objects}). \begin{definition} \label{definition-completion-CLambda} Let $\Lambda$ be a Noetherian ring and let $\Lambda \to k$ be a finite ring map where $k$ is a field. We define {\it $\widehat{\mathcal{C}}_\Lambda$} to be the category with \begin{enumerate} \item objects are pairs $(R, \varphi)$ where $R$ is a Noetherian complete local $\Lambda$-algebra and where $\varphi : R/\mathfrak m_R \to k$ is a $\Lambda$-algebra isomorphism, and \item morphisms $f : (S, \psi) \to (R, \varphi)$ are local $\Lambda$-algebra homomorphisms such that $\varphi \circ (f \bmod \mathfrak m) = \psi$. \end{enumerate} \end{definition} \noindent As in the discussion following Definition \ref{definition-CLambda} we will usually denote an object of $\widehat{\mathcal{C}}_\Lambda$ simply $R$, with the identification $R/\mathfrak m_R = k$ understood. In this section we discuss some basic properties of objects and morphisms of the category $\widehat{\mathcal{C}}_\Lambda$ paralleling our discussion of the category $\mathcal{C}_\Lambda$ in the previous section. \medskip\noindent Our first observation is that any object $A \in \mathcal{C}_\Lambda$ is an object of $\widehat{\mathcal{C}}_\Lambda$ as an Artinian local ring is always Noetherian and complete with respect to its maximal ideal (which is after all a nilpotent ideal). Moreover, it is clear from the definitions that $\mathcal{C}_\Lambda \subset \widehat{\mathcal{C}}_\Lambda$ is the strictly full subcategory consisting of all Artinian rings. As it turns out, conversely every object of $\widehat{\mathcal{C}}_\Lambda$ is a limit of objects of $\mathcal{C}_\Lambda$. \medskip\noindent Suppose that $R$ is an object of $\widehat{\mathcal{C}}_\Lambda$. Consider the rings $R_n = R/\mathfrak m_R^n$ for $n \in \mathbf{N}$. These are Noetherian local rings with a unique nilpotent prime ideal, hence Artinian, see Algebra, Proposition \ref{algebra-proposition-dimension-zero-ring}. The ring maps $$\ldots \to R_{n + 1} \to R_n \to \ldots \to R_2 \to R_1 = k$$ are all surjective. Completeness of $R$ by definition means that $R = \lim R_n$. If $f : R \to S$ is a ring map in $\widehat{\mathcal{C}}_\Lambda$ then we obtain a system of ring maps $f_n : R_n \to S_n$ whose limit is the given map. \begin{lemma} \label{lemma-surjective-cotangent-space} Let $f: R \to S$ be a ring map in $\widehat{\mathcal{C}}_\Lambda$. The following are equivalent \begin{enumerate} \item $f$ is surjective, \item the map $\mathfrak m_R/\mathfrak m_R^2 \to \mathfrak m_S/\mathfrak m_S^2$ is surjective, and \item the map $\mathfrak m_R/(\mathfrak m_\Lambda R + \mathfrak m_R^2) \to \mathfrak m_S/(\mathfrak m_\Lambda S + \mathfrak m_S^2)$ is surjective. \end{enumerate} \end{lemma} \begin{proof} Note that for $n \geq 2$ we have the equality of relative cotangent spaces $$\mathfrak m_R/(\mathfrak m_\Lambda R + \mathfrak m_R^2) = \mathfrak m_{R_n}/(\mathfrak m_\Lambda R_n + \mathfrak m_{R_n}^2)$$ and similarly for $S$. Hence by Lemma \ref{lemma-surjective} we see that $R_n \to S_n$ is surjective for all $n$. Now let $K_n$ be the kernel of $R_n \to S_n$. Then the sequences $$0 \to K_n \to R_n \to S_n \to 0$$ form an exact sequence of directed inverse systems. The system $(K_n)$ is Mittag-Leffler since each $K_n$ is Artinian. Hence by Algebra, Lemma \ref{algebra-lemma-ML-exact-sequence} taking limits preserves exactness. So $\lim R_n \to \lim S_n$ is surjective, i.e., $f$ is surjective. \end{proof} \begin{lemma} \label{lemma-CLambdahat-pushouts} The category $\widehat{\mathcal{C}}_\Lambda$ admits pushouts. \end{lemma} \begin{proof} Let $R \to S_1$ and $R \to S_2$ be morphisms of $\widehat{\mathcal{C}}_\Lambda$. Consider the ring $C = S_1 \otimes_R S_2$. This ring has a finitely generated maximal ideal $\mathfrak m = \mathfrak m_{S_1} \otimes S_2 + S_1 \otimes \mathfrak m_{S_2}$ with residue field $k$. Set $C^\wedge$ equal to the completion of $C$ with respect to $\mathfrak m$. Then $C^\wedge$ is a Noetherian ring complete with respect to the maximal ideal $\mathfrak m^\wedge = \mathfrak mC^\wedge$ whose residue field is identified with $k$, see Algebra, Lemma \ref{algebra-lemma-completion-Noetherian}. Hence $C^\wedge$ is an object of $\widehat{\mathcal{C}}_\Lambda$. Then $S_1 \to C^\wedge$ and $S_2 \to C^\wedge$ turn $C^\wedge$ into a pushout over $R$ in $\widehat{\mathcal{C}}_\Lambda$ (details omitted). \end{proof} \noindent We will not need the following lemma. \begin{lemma} \label{lemma-CLambdahat-coproducts} The category $\widehat{\mathcal{C}}_\Lambda$ admits coproducts of pairs of objects. \end{lemma} \begin{proof} Let $R$ and $S$ be objects of $\widehat{\mathcal{C}}_\Lambda$. Consider the ring $C = R \otimes_\Lambda S$. There is a canonical surjective map $C \to R \otimes_\Lambda S \to k \otimes_\Lambda k \to k$ where the last map is the multiplication map. The kernel of $C \to k$ is a maximal ideal $\mathfrak m$. Note that $\mathfrak m$ is generated by $\mathfrak m_R C$, $\mathfrak m_S C$ and finitely many elements of $C$ which map to generators of the kernel of $k \otimes_\Lambda k \to k$. Hence $\mathfrak m$ is a finitely generated ideal. Set $C^\wedge$ equal to the completion of $C$ with respect to $\mathfrak m$. Then $C^\wedge$ is a Noetherian ring complete with respect to the maximal ideal $\mathfrak m^\wedge = \mathfrak mC^\wedge$ with residue field $k$, see Algebra, Lemma \ref{algebra-lemma-completion-Noetherian}. Hence $C^\wedge$ is an object of $\widehat{\mathcal{C}}_\Lambda$. Then $R \to C^\wedge$ and $S \to C^\wedge$ turn $C^\wedge$ into a coproduct in $\widehat{\mathcal{C}}_\Lambda$ (details omitted). \end{proof} \noindent An empty coproduct in a category is an initial object of the category. In the classical case $\widehat{\mathcal{C}}_\Lambda$ has an initial object, namely $\Lambda$ itself. More generally, if $k' = k$, then the completion $\Lambda^\wedge$ of $\Lambda$ with respect to $\mathfrak m_\Lambda$ is an initial object. More generally still, if $k' \subset k$ is separable, then $\widehat{\mathcal{C}}_\Lambda$ has an initial object too. Namely, choose a monic polynomial $P \in \Lambda[T]$ such that $k \cong k'[T]/(P')$ where $p' \in k'[T]$ is the image of $P$. Then $R = \Lambda^\wedge[T]/(P)$ is an initial object, see proof of Lemma \ref{lemma-fiber-product-CLambda}. \medskip\noindent If $R$ is an initial object as above, then we have $\mathcal{C}_\Lambda = \mathcal{C}_R$ and $\widehat{\mathcal{C}}_\Lambda = \widehat{\mathcal{C}}_R$ which effectively brings the whole discussion in this chapter back to the classical case. But, if $k' \subset k$ is inseparable, then an initial object does not exist. \begin{lemma} \label{lemma-derivations-finite} Let $S$ be an object of $\widehat{\mathcal{C}}_\Lambda$. Then $\dim_k \text{Der}_\Lambda(S, k) < \infty$. \end{lemma} \begin{proof} Let $x_1, \ldots, x_n \in \mathfrak m_S$ map to a $k$-basis for the relative cotangent space $\mathfrak m_S/(\mathfrak m_\Lambda S + \mathfrak m_S^2)$. Choose $y_1, \ldots, y_m \in S$ whose images in $k$ generate $k$ over $k'$. We claim that $\dim_k \text{Der}_\Lambda(S, k) \leq n + m$. To see this it suffices to prove that if $D(x_i) = 0$ and $D(y_j) = 0$, then $D = 0$. Let $a \in S$. We can find a polynomial $P = \sum \lambda_J y^J$ with $\lambda_J \in \Lambda$ whose image in $k$ is the same as the image of $a$ in $k$. Then we see that $D(a - P) = D(a) - D(P) = D(a)$ by our assumption that $D(y_j) = 0$ for all $j$. Thus we may assume $a \in \mathfrak m_S$. Write $a = \sum a_i x_i$ with $a_i \in S$. By the Leibniz rule $$D(a) = \sum x_iD(a_i) + \sum a_iD(x_i) = \sum x_iD(a_i)$$ as we assumed $D(x_i) = 0$. We have $\sum x_iD(a_i) = 0$ as multiplication by $x_i$ is zero on $k$. \end{proof} \begin{lemma} \label{lemma-derivations-surjective} Let $f : R \to S$ be a morphism of $\widehat{\mathcal{C}}_\Lambda$. If $\text{Der}_\Lambda(S, k) \to \text{Der}_\Lambda(R, k)$ is injective, then $f$ is surjective. \end{lemma} \begin{proof} If $f$ is not surjective, then $\mathfrak m_S/(\mathfrak m_R S + \mathfrak m_S^2)$ is nonzero by Lemma \ref{lemma-surjective-cotangent-space}. Then also $Q = S/(f(R) + \mathfrak m_R S + \mathfrak m_S^2)$ is nonzero. Note that $Q$ is a $k = R/\mathfrak m_R$-vector space via $f$. We turn $Q$ into an $S$-module via $S \to k$. The quotient map $D : S \to Q$ is an $R$-derivation: if $a_1, a_2 \in S$, we can write $a_1 = f(b_1) + a_1'$ and $a_2 = f(b_2) + a_2'$ for some $b_1, b_2 \in R$ and $a_1', a_2' \in \mathfrak m_S$. Then $b_i$ and $a_i$ have the same image in $k$ for $i = 1, 2$ and \begin{align*} a_1a_2 & = (f(b_1) + a_1')(f(b_2) + a_2') \\ & = f(b_1)a_2' + f(b_2)a_1' \\ & = f(b_1)(f(b_2) + a_2') + f(b_2)(f(b_1) + a_1') \\ & = f(b_1)a_2 + f(b_2)a_1 \end{align*} in $Q$ which proves the Leibniz rule. Hence $D : S \to Q$ is a $\Lambda$-derivation which is zero on composing with $R \to S$. Since $Q \not = 0$ there also exist derivations $D : S \to k$ which are zero on composing with $R \to S$, i.e., $\text{Der}_\Lambda(S, k) \to \text{Der}_\Lambda(R, k)$ is not injective. \end{proof} \begin{lemma} \label{lemma-m-adic-topology} Let $R$ be an object of $\widehat{\mathcal{C}}_\Lambda$. Let $(J_n)$ be a decreasing sequence of ideals such that $\mathfrak m_R^n \subset J_n$. Set $J = \bigcap J_n$. Then the sequence $(J_n/J)$ defines the $\mathfrak m_{R/J}$-adic topology on $R/J$. \end{lemma} \begin{proof} It is clear that $\mathfrak m_{R/J}^n \subset J_n/J$. Thus it suffices to show that for every $n$ there exists an $N$ such that $J_N/J \subset \mathfrak m_{R/J}^n$. This is equivalent to $J_N \subset \mathfrak m_R^n + J$. For each $n$ the ring $R/\mathfrak m_R^n$ is Artinian, hence there exists a $N_n$ such that $$J_{N_n} + \mathfrak m_R^n = J_{N_n + 1} + \mathfrak m_R^n = \ldots$$ Set $E_n = (J_{N_n} + \mathfrak m_R^n)/\mathfrak m_R^n$. Set $E = \lim E_n \subset \lim R/\mathfrak m_R^n = R$. Note that $E \subset J$ as for any $f \in E$ and any $m$ we have $f \in J_m + \mathfrak m_R^n$ for all $n \gg 0$, so $f \in J_m$ by Artin-Rees, see Algebra, Lemma \ref{algebra-lemma-intersect-powers-ideal-module-zero}. Since the transition maps $E_n \to E_{n - 1}$ are all surjective, we see that $J$ surjects onto $E_n$. Hence for $N = N_n$ works. \end{proof} \begin{lemma} \label{lemma-limit-artinian} Let $\ldots \to A_3 \to A_2 \to A_1$ be a sequence of surjective ring maps in $\mathcal{C}_\Lambda$. If $\dim_k (\mathfrak m_{A_n}/\mathfrak m_{A_n}^2)$ is bounded, then $S = \lim A_n$ is an object in $\widehat{\mathcal{C}}_\Lambda$ and the ideals $I_n = \Ker(S \to A_n)$ define the $\mathfrak m_S$-adic topology on $S$. \end{lemma} \begin{proof} We will use freely that the maps $S \to A_n$ are surjective for all $n$. Note that the maps $\mathfrak m_{A_{n + 1}}/\mathfrak m_{A_{n + 1}}^2 \to \mathfrak m_{A_n}/\mathfrak m_{A_n}^2$ are surjective, see Lemma \ref{lemma-surjective-cotangent-space}. Hence for $n$ sufficiently large the dimension $\dim_k (\mathfrak m_{A_n}/\mathfrak m_{A_n}^2)$ stabilizes to an integer, say $r$. Thus we can find $x_1, \ldots, x_r \in \mathfrak m_S$ whose images in $A_n$ generate $\mathfrak m_{A_n}$. Moreover, pick $y_1, \ldots, y_t \in S$ whose images in $k$ generate $k$ over $\Lambda$. Then we get a ring map $P = \Lambda[z_1, \ldots, z_{r + t}] \to S$, $z_i \mapsto x_i$ and $z_{r + j} \mapsto y_j$ such that the composition $P \to S \to A_n$ is surjective for all $n$. Let $\mathfrak m \subset P$ be the kernel of $P \to k$. Let $R = P^\wedge$ be the $\mathfrak m$-adic completion of $P$; this is an object of $\widehat{\mathcal{C}}_\Lambda$. Since we still have the compatible system of (surjective) maps $R \to A_n$ we get a map $R \to S$. Set $J_n = \Ker(R \to A_n)$. Set $J = \bigcap J_n$. By Lemma \ref{lemma-m-adic-topology} we see that $R/J = \lim R/J_n = \lim A_n = S$ and that the ideals $J_n/J = I_n$ define the $\mathfrak m$-adic topology. (Note that for each $n$ we have $\mathfrak m_R^{N_n} \subset J_n$ for some $N_n$ and not necessarily $N_n = n$, so a renumbering of the ideals $J_n$ may be necessary before applying the lemma.) \end{proof} \begin{lemma} \label{lemma-power-series} Let $R', R \in \Ob(\widehat{\mathcal{C}}_\Lambda)$. Suppose that $R = R' \oplus I$ for some ideal $I$ of $R$. Let $x_1, \ldots, x_r \in I$ map to a basis of $I/\mathfrak m_R I$. Set $S = R'[[X_1, \ldots, X_r]]$ and consider the $R'$-algebra map $S \to R$ mapping $X_i$ to $x_i$. Assume that for every $n \gg 0$ the map $S/\mathfrak m_S^n \to R/\mathfrak m_R^n$ has a left inverse in $\mathcal{C}_\Lambda$. Then $S \to R$ is an isomorphism. \end{lemma} \begin{proof} As $R = R' \oplus I$ we have $$\mathfrak m_R/\mathfrak m_R^2 = \mathfrak m_{R'}/\mathfrak m_{R'}^2 \oplus I/\mathfrak m_RI$$ and similarly $$\mathfrak m_R/\mathfrak m_R^2 = \mathfrak m_{R'}/\mathfrak m_{R'}^2 \oplus \bigoplus kX_i$$ Hence for $n > 1$ the map $S/\mathfrak m_S^n \to R/\mathfrak m_R^n$ induces an isomorphism on cotangent spaces. Thus a left inverse $h_n : R/\mathfrak m_R^n \to S/\mathfrak m_S^n$ is surjective by Lemma \ref{lemma-surjective-cotangent-space}. Since $h_n$ is injective as a left inverse it is an isomorphism. Thus the canonical surjections $S/\mathfrak m_S^n \to R/\mathfrak m_R^n$ are all isomorphisms and we win. \end{proof} \section{Categories cofibered in groupoids} \label{section-preliminary} \noindent In developing the theory we work with categories {\it cofibered} in groupoids. We assume as known the definition and basic properties of categories {\it fibered} in groupoids, see Categories, Section \ref{categories-section-fibred-groupoids}. \begin{definition} \label{definition-category-cofibred-groupoids} Let $\mathcal{C}$ be a category. A {\it category cofibered in groupoids over $\mathcal{C}$} is a category $\mathcal{F}$ equipped with a functor $p: \mathcal{F} \to \mathcal{C}$ such that $\mathcal{F}^{opp}$ is a category fibered in groupoids over $\mathcal{C}^{opp}$ via $p^{opp}: \mathcal{F}^{opp} \to \mathcal{C}^{opp}$. \end{definition} \noindent Explicitly, $p: \mathcal{F} \to \mathcal{C}$ is cofibered in groupoids if the following two conditions hold: \begin{enumerate} \item For every morphism $f: U \to V$ in $\mathcal{C}$ and every object $x$ lying over $U$, there is a morphism $x \to y$ of $\mathcal{F}$ lying over $f$. \item For every pair of morphisms $a: x \to y$ and $b: x \to z$ of $\mathcal{F}$ and any morphism $f: p(y) \to p(z)$ such that $p(b) = f \circ p(a)$, there exists a unique morphism $c: y \to z$ of $\mathcal F$ lying over $f$ such that $b = c \circ a$. \end{enumerate} \begin{remarks} \label{remarks-cofibered-groupoids} Everything about categories fibered in groupoids translates directly to the cofibered setting. The following remarks are meant to fix notation. Let $\mathcal{C}$ be a category. \begin{enumerate} \item We often omit the functor $p: \mathcal{F} \to \mathcal{C}$ from the notation. \item The fiber category over an object $U$ in $\mathcal{C}$ is denoted by $\mathcal{F}(U)$. Its objects are those of $\mathcal{F}$ lying over $U$ and its morphisms are those of $\mathcal{F}$ lying over $\text{id}_U$. If $x, y$ are objects of $\mathcal{F}(U)$, we sometimes write $\Mor_U(x, y)$ for $\Mor_{\mathcal{F}(U)}(x, y)$. \item The fibre categories $\mathcal{F}(U)$ are groupoids, see Categories, Lemma \ref{categories-lemma-fibred-groupoids}. Hence the morphisms in $\mathcal{F}(U)$ are all isomorphisms. We sometimes write $\text{Aut}_U(x)$ for $\Mor_{\mathcal{F}(U)}(x, x)$. \item \label{item-pushforward} Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}$, let $f: U \to V$ be a morphism in $\mathcal{C}$, and let $x \in \Ob(\mathcal{F}(U))$. A {\it pushforward} of $x$ along $f$ is a morphism $x \to y$ of $\mathcal{F}$ lying over $f$. A pushforward is unique up to unique isomorphism (see the discussion following Categories, Definition \ref{categories-definition-cartesian-over-C}). We sometimes write $x \to f_*x$ for the'' pushforward of $x$ along $f$. \item A {\it choice of pushforwards for $\mathcal{F}$} is the choice of a pushforward of $x$ along $f$ for every pair $(x, f)$ as above. We can make such a choice of pushforwards for $\mathcal{F}$ by the axiom of choice. \item Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}$. Given a choice of pushforwards for $\mathcal{F}$, there is an associated pseudo-functor $\mathcal{C} \to \textit{Groupoids}$. We will never use this construction so we give no details. \item \label{item-cofibered-morphism} A morphism of categories cofibered in groupoids over $\mathcal{C}$ is a functor commuting with the projections to $\mathcal{C}$. If $\mathcal{F}$ and $\mathcal{F}'$ are categories cofibered in groupoids over $\mathcal{C}$, we denote the morphisms from $\mathcal{F}$ to $\mathcal{F}'$ by $\Mor_\mathcal{C}(\mathcal{F}, \mathcal{F}')$. \item \label{item-definition-cofibered-groupoids-2-category} Categories cofibered in groupoids form a $(2, 1)$-category $\text{Cof}(\mathcal{C})$. Its 1-morphisms are the morphisms described in (\ref{item-cofibered-morphism}). If $p : \mathcal{F} \to C$ and $p': \mathcal{F}' \to \mathcal{C}$ are categories cofibered in groupoids and $\varphi, \psi : \mathcal{F} \to \mathcal{F}'$ are $1$-morphisms, then a 2-morphism $t : \varphi \to \psi$ is a morphism of functors such that $p'(t_x) = \text{id}_{p(x)}$ for all $x \in \Ob(\mathcal{F})$. \item \label{item-construction-associated-cofibered-groupoid} Let $F : \mathcal{C} \to \textit{Groupoids}$ be a functor. There is a category cofibered in groupoids $\mathcal{F} \to \mathcal{C}$ associated to $F$ as follows. An object of $\mathcal{F}$ is a pair $(U, x)$ where $U \in \Ob(\mathcal{C})$ and $x \in \Ob(F(U))$. A morphism $(U, x) \to (V, y)$ is a pair $(f, a)$ where $f \in \Mor_\mathcal{C}(U, V)$ and $a \in \Mor_{F(V)}(F(f)(x), y)$. The functor $\mathcal{F} \to \mathcal{C}$ sends $(U, x)$ to $U$. See Categories, Section \ref{categories-section-presheaves-groupoids}. \item \label{item-associated-functor-isomorphism-classes} Let $\mathcal{F}$ be cofibered in groupoids over $\mathcal{C}$. For $U \in \Ob(\mathcal{C})$ set $\overline{\mathcal{F}}(U)$ equal to the set of isomorphisms classes of the category $\mathcal{F}(U)$. If $f : U \to V$ is a morphism of $\mathcal{C}$, then we obtain a map of sets $\overline{\mathcal{F}}(U) \to \overline{\mathcal{F}}(V)$ by mapping the isomorphism class of $x$ to the isomorphism class of a pushforward $f_*x$ of $x$ see (\ref{item-pushforward}). Then $\overline{\mathcal{F}} : \mathcal{C} \to \textit{Sets}$ is a functor. Similarly, if $\varphi : \mathcal{F} \to \mathcal{G}$ is a morphism of cofibered categories, we denote by $\overline{\varphi}: \overline{\mathcal{F}} \to \overline{\mathcal{G}}$ the associated morphism of functors. \item \label{item-convention-cofibered-sets} Let $F: \mathcal{C} \to \textit{Sets}$ be a functor. We can think of a set as a discrete category, i.e., as a groupoid with only identity morphisms. Then the construction (\ref{item-construction-associated-cofibered-groupoid}) associates to $F$ a category cofibered in sets. This defines a fully faithful embedding of the category of functors $\mathcal{C} \to \textit{Sets}$ to the category of categories cofibered in groupoids over $\mathcal{C}$. We identify the category of functors with its image under this embedding. Hence if $F : \mathcal{C} \to \textit{Sets}$ is a functor, we denote the associated category cofibered in sets also by $F$; and if $\varphi : F \to G$ is a morphism of functors, we denote still by $\varphi$ the corresponding morphism of categories cofibered in sets, and vice-versa. See Categories, Section \ref{categories-section-fibred-in-sets}. \item \label{item-definition-yoneda} Let $U$ be an object of $\mathcal{C}$. We write $\underline{U}$ for the functor $\Mor_\mathcal{C}(U, -): \mathcal{C} \to \textit{Sets}$. This defines a fully faithful embedding of $\mathcal C^{opp}$ into the category of functors $\mathcal{C} \to \textit{Sets}$. Hence, if $f : U \to V$ is a morphism, we are justified in denoting still by $f$ the induced morphism $\underline{V} \to \underline{U}$, and vice-versa. \item \label{item-fibre-product} Fiber products of categories cofibered in groupoids: If $\mathcal{F} \to \mathcal{H}$ and $\mathcal{G} \to \mathcal{H}$ are morphisms of categories cofibered in groupoids over $\mathcal{C}_\Lambda$, then a construction of their 2-fiber product is given by the construction for their 2-fiber product as categories over $\mathcal{C}_\Lambda$, as described in Categories, Lemma \ref{categories-lemma-2-product-categories-over-C}. \item \label{item-definition-restricting-base-category} Restricting the base category: Let $p : \mathcal{F} \to \mathcal{C}$ be a category cofibered in groupoids, and let $\mathcal{C}'$ be a full subcategory of $\mathcal{C}$. The restriction $\mathcal{F}|_{\mathcal{C}'}$ is the full subcategory of $\mathcal{F}$ whose objects lie over objects of $\mathcal{C}'$. It is a category cofibered in groupoids via the functor $p|_{\mathcal{C}'}: \mathcal{F}|_{\mathcal{C}'} \to \mathcal{C}'$. \end{enumerate} \end{remarks} \section{Prorepresentable functors and predeformation categories} \label{section-cofibered-groupoids} \noindent Our basic goal is to understand categories cofibered in groupoids over $\mathcal{C}_\Lambda$ and $\widehat{\mathcal{C}}_\Lambda$. Since $\mathcal{C}_\Lambda$ is a full subcategory of $\widehat{\mathcal{C}}_\Lambda$ we can restrict categories cofibred in groupoids over $\widehat{\mathcal{C}}_\Lambda$ to $\mathcal{C}_\Lambda$, see Remarks \ref{remarks-cofibered-groupoids} (\ref{item-definition-restricting-base-category}). In particular we can do this with functors, in particular with representable functors. The functors on $\mathcal{C}_\Lambda$ one obtains in this way are called prorepresentable functors. \begin{definition} \label{definition-prorepresentable} Let $F : \mathcal{C}_\Lambda \to \textit{Sets}$ be a functor. We say $F$ is {\it prorepresentable} if there exists an isomorphism $F \cong \underline{R}|_{\mathcal{C}_\Lambda}$ of functors for some $R \in \Ob(\widehat{\mathcal{C}}_\Lambda)$. \end{definition} \noindent Note that if $F : \mathcal{C}_\Lambda \to \textit{Sets}$ is prorepresentable by $R \in \Ob(\widehat{\mathcal{C}}_\Lambda)$, then $$F(k) = \Mor_{\widehat{\mathcal{C}}_\Lambda}(R, k) = \{*\}$$ is a singleton. The categories cofibered in groupoids over $\mathcal{C}_\Lambda$ that are arise in deformation theory will often satisfy an analogous condition. \begin{definition} \label{definition-predeformation-category} A {\it predeformation category} $\mathcal{F}$ is a category cofibered in groupoids over $\mathcal{C}_\Lambda$ such that $\mathcal{F}(k)$ is equivalent to a category with a single object and a single morphism, i.e., $\mathcal{F}(k)$ contains at least one object and there is a unique morphism between any two objects. A {\it morphism of predeformation categories} is a morphism of categories cofibered in groupoids over $\mathcal{C}_\Lambda$. \end{definition} \noindent A feature of a predeformation category is the following. Let $x_0 \in \Ob(\mathcal{F}(k))$. Then every object of $\mathcal{F}$ comes equipped with a unique morphism to $x_0$. Namely, if $x$ is an object of $\mathcal{F}$ over $A$, then we can choose a pushforward $x \to q_*x$ where $q : A \to k$ is the quotient map. There is a unique isomorphism $q_*x \to x_0$ and the composition $x \to q_*x \to x_0$ is the desired morphism. \begin{remark} \label{remark-predeformation-functor} We say that a functor $F: \mathcal{C}_\Lambda \to \textit{Sets}$ is a {\it predeformation functor} if the associated cofibered set is a predeformation category, i.e.\ if $F(k)$ is a one element set. Thus if $\mathcal{F}$ is a predeformation category, then $\overline{\mathcal{F}}$ is a predeformation functor. \end{remark} \begin{remark} \label{remark-localize-cofibered-groupoid} Let $p : \mathcal{F} \to \mathcal{C}_\Lambda$ be a category cofibered in groupoids, and let $x \in \Ob(\mathcal{F}(k))$. We denote by $\mathcal{F}_x$ the category of objects over $x$. An object of $\mathcal{F}_x$ is an arrow $y \to x$. A morphism $(y \to x) \to (z \to x)$ in $\mathcal{F}_x$ is a commutative diagram $$\xymatrix{ y \ar[rr] \ar[dr] & & z \ar[dl] \\ & x & }$$ There is a forgetful functor $\mathcal{F}_x \to \mathcal{F}$. We define the functor $p_x : \mathcal{F}_x \to \mathcal{C}_\Lambda$ as the composition $\mathcal{F}_x \to \mathcal{F} \xrightarrow{p} \mathcal{C}_\Lambda$. Then $p_x : \mathcal{F}_x \to \mathcal{C}_\Lambda$ is a predeformation category (proof omitted). In this way we can pass from an arbitrary category cofibered in groupoids over $\mathcal{C}_\Lambda$ to a predeformation category at any $x \in \Ob(\mathcal{F}(k))$. \end{remark} \section{Formal objects and completion categories} \label{section-formal-objects} \noindent In this section we discuss how to go between categories cofibred in groupoids over $\mathcal{C}_\Lambda$ to categories cofibred in groupoids over $\widehat{\mathcal{C}}_\Lambda$ and vice versa. \begin{definition} \label{definition-formal-objects} Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda$. The {\it category $\widehat{\mathcal{F}}$ of formal objects of $\mathcal{F}$} is the category with the following objects and morphisms. \begin{enumerate} \item A {\it formal object $\xi = (R, \xi_n, f_n)$ of $\mathcal{F}$} consists of an object $R$ of $\widehat{\mathcal{C}}_\Lambda$, and a collection indexed by $n \in \mathbf{N}$ of objects $\xi_n$ of $\mathcal{F}(R/\mathfrak m_R^n)$ and morphisms $f_n : \xi_{n + 1} \to \xi_n$ lying over the projection $R/\mathfrak m_R^{n + 1} \to R/\mathfrak m_R^n$. \item Let $\xi = (R, \xi_n, f_n)$ and $\eta = (S, \eta_n, g_n)$ be formal objects of $\mathcal{F}$. A {\it morphism $a : \xi \to \eta$ of formal objects} consists of a map $a_0 : R \to S$ in $\widehat{\mathcal{C}}_\Lambda$ and a collection $a_n : \xi_n \to \eta_n$ of morphisms of $\mathcal{F}$ lying over $R/\mathfrak m_R^n \to S/\mathfrak m_S^n$, such that for every $n$ the diagram $$\xymatrix{ \xi_{n + 1} \ar[r]_{f_n} \ar[d]_{a_{n + 1}} & \xi_n \ar[d]^{a_n} \\ \eta_{n + 1} \ar[r]^{g_n} & \eta_n }$$ commutes. \end{enumerate} \end{definition} \noindent The category of formal objects comes with a functor $\widehat{p}: \widehat{\mathcal{F}} \to \widehat{\mathcal{C}}_\Lambda$ which sends an object $(R, \xi_n, f_n)$ to $R$ and a morphism $(R, \xi_n, f_n) \to (S, \eta_n, g_n)$ to the map $R \to S$. \begin{lemma} \label{lemma-completion-cofibred} Let $p : \mathcal{F} \to \mathcal{C}_\Lambda$ be a category cofibered in groupoids. Then $\widehat{p} : \widehat{\mathcal{F}} \to \widehat{\mathcal{C}}_\Lambda$ is a category cofibered in groupoids. \end{lemma} \begin{proof} Let $R \to S$ be a ring map in $\widehat{\mathcal{C}}_\Lambda$. Let $(R, \xi_n, f_n)$ be an object of $\widehat{\mathcal{F}}$. For each $n$ choose a pushforward $\xi_n \to \eta_n$ of $\xi_n$ along $R/\mathfrak m_R^n \to S/\mathfrak m_S^n$. For each $n$ there exists a unique morphism $g_n : \eta_{n + 1} \to \eta_n$ in $\mathcal{F}$ lying over $S/\mathfrak m_S^{n + 1} \to S/\mathfrak m_S^n$ such that $$\xymatrix{ \xi_{n + 1} \ar[d] \ar[r]_{f_n} & \xi_n \ar[d] \\ \eta_{n + 1} \ar[r]^{g_n} & \eta_n }$$ commutes (by the first axiom of a category cofibred in groupoids). Hence we obtain a morphism $(R, \xi_n, f_n) \to (S, \eta_n, g_n)$ lying over $R \to S$, i.e., the first axiom of a category cofibred in groupoids holds for $\widehat{\mathcal{F}}$. To see the second axiom suppose that we have morphisms $a : (R, \xi_n, f_n) \to (S, \eta_n, g_n)$ and $b : (R, \xi_n, f_n) \to (T, \theta_n, h_n)$ in $\widehat{\mathcal{F}}$ and a morphism $c_0 : S \to T$ in $\widehat{\mathcal{C}}_\Lambda$ such that $c_0 \circ a_0 = b_0$. By the second axiom of a category cofibred in groupoids for $\mathcal{F}$ we obtain unique maps $c_n : \eta_n \to \theta_n$ lying over $S/\mathfrak m_S^n \to T/\mathfrak m_T^n$ such that $c_n \circ a_n = b_n$. Setting $c = (c_n)_{n \geq 0}$ gives the desired morphism $c : (S, \eta_n, g_n) \to (T, \theta_n, h_n)$ in $\widehat{\mathcal{F}}$ (we omit the verification that $h_n \circ c_{n + 1} = c_n \circ g_n$). \end{proof} \begin{definition} \label{definition-completion} Let $p : \mathcal{F} \to \mathcal{C}_\Lambda$ be a category cofibered in groupoids. The category cofibered in groupoids $\widehat{p} : \widehat{\mathcal F} \to \widehat{\mathcal{C}}_\Lambda$ is called the {\it completion of $\mathcal{F}$}. \end{definition} \noindent If $\mathcal{F}$ is a category cofibered in groupoids over $\mathcal C_\Lambda$, we have defined $\widehat{\mathcal{F}}(R)$ for $R \in \Ob(\widehat{\mathcal{C}}_\Lambda)$ in terms of the filtration of $R$ by powers of its maximal ideal. But suppose $\mathcal{I} = (I_n)$ is a filtration of $R$ by ideals inducing the $\mathfrak{m}_R$-adic topology. We define $\widehat{\mathcal{F}}_\mathcal{I}(R)$ to be the category with the following objects and morphisms: \begin{enumerate} \item An object is a collection $(\xi_n, f_n)_{n \in \mathbf{N}}$ of objects $\xi_n$ of $\mathcal{F}(R/I_n)$ and morphisms $f_n : \xi_{n + 1} \to \xi_n$ lying over the projections $R/I_{n + 1} \to R/I_n$. \item A morphism $a : (\xi_n, f_n) \to (\eta_n, g_n)$ consists of a collection $a_n : \xi_n \to \eta_n$ of morphisms in $\mathcal{F}(R/I_n)$, such that for every $n$ the diagram $$\xymatrix{ \xi_{n + 1} \ar[r]^{f_n} \ar[d]_{a_{n + 1}} & \xi_n \ar[d]^{a_n} \\ \eta_{n + 1} \ar[r]^{g_n} & \eta_n }$$ commutes. \end{enumerate} \begin{lemma} \label{lemma-formal-objects-different-filtration} In the situation above, $\widehat{\mathcal{F}}_\mathcal{I}(R)$ is equivalent to the category $\widehat{\mathcal{F}}(R)$. \end{lemma} \begin{proof} An equivalence $\widehat{\mathcal{F}}_\mathcal{I}(R) \to \widehat{\mathcal{F}}(R)$ can be defined as follows. For each $n$, let $m(n)$ be the least $m$ that $I_m \subset \mathfrak m_R^n$. Given an object $(\xi_n, f_n)$ of $\widehat{\mathcal{F}}_\mathcal{I}(R)$, let $\eta_n$ be the pushforward of $\xi_{m(n)}$ along $R/I_{m(n)} \to R/\mathfrak m_R^n$. Let $g_n : \eta_{n + 1} \to \eta_n$ be the unique morphism of $\mathcal{F}$ lying over $R/\mathfrak m_R^{n + 1} \to R/\mathfrak m_R^n$ such that $$\xymatrix{ \xi_{m(n + 1)} \ar[rrr]_{f_{m(n)} \circ \ldots \circ f_{m(n + 1) - 1}} \ar[d] & & & \xi_{m(n)} \ar[d] \\ \eta_{n + 1} \ar[rrr]^{g_n} & & & \eta_n }$$ commutes (existence and uniqueness is guaranteed by the axioms of a cofibred category). The functor $\widehat{\mathcal{F}}_\mathcal{I}(R) \to \widehat{\mathcal{F}}(R)$ sends $(\xi_n, f_n)$ to $(R, \eta_n, g_n)$. We omit the verification that this is indeed an equivalence of categories. \end{proof} \begin{remark} \label{remark-different-sequence-ideals} Let $p : \mathcal{F} \to \mathcal{C}_\Lambda$ be a category cofibered in groupoids. Suppose that for each $R \in \Ob(\widehat{\mathcal{C}}_\Lambda)$ we are given a filtration $\mathcal{I}_R$ of $R$ by ideals. If $\mathcal{I}_R$ induces the $\mathfrak m_R$-adic topology on $R$ for all $R$, then one can define a category $\widehat{\mathcal{F}}_\mathcal{I}$ by mimicking the definition of $\widehat{\mathcal{F}}$. This category comes equipped with a morphism $\widehat{p}_\mathcal{I} : \widehat{\mathcal{F}}_\mathcal{I} \to \widehat{\mathcal{C}}_\Lambda$ making it into a category cofibered in groupoids such that $\widehat{\mathcal{F}}_\mathcal{I}(R)$ is isomorphic to $\widehat{\mathcal{F}}_{\mathcal{I}_R}(R)$ as defined above. The categories cofibered in groupoids $\widehat{\mathcal{F}}_\mathcal{I}$ and $\widehat{\mathcal{F}}$ are equivalent, by using over an object $R \in \Ob(\widehat{\mathcal{C}}_\Lambda)$ the equivalence of Lemma \ref{lemma-formal-objects-different-filtration}. \end{remark} \begin{remark} \label{remark-completion-functor} Let $F: \mathcal{C}_\Lambda \to \textit{Sets}$ be a functor. Identifying functors with cofibered sets, the completion of $F$ is the functor $\widehat{F} : \widehat{\mathcal{C}}_\Lambda \to \textit{Sets}$ given by $\widehat{F}(S) = \lim F(S/\mathfrak{m}_S^{n})$. This agrees with the definition in Schlessinger's paper \cite{Sch}. \end{remark} \begin{remark} \label{remark-restrict-completion} Let $\mathcal{F}$ be a category cofibred in groupoids over $\mathcal{C}_\Lambda$. We claim that there is a canonical equivalence $$can : \widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda} \longrightarrow \mathcal{F}.$$ Namely, let $A \in \Ob(\mathcal{C}_\Lambda)$ and let $(A, \xi_n, f_n)$ be an object of $\widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda}(A)$. Since $A$ is Artinian there is a minimal $m \in \mathbf{N}$ such that $\mathfrak m_A^m = 0$. Then $can$ sends $(A, \xi_n, f_n)$ to $\xi_m$. This functor is an equivalence of categories cofibered in groupoids by Categories, Lemma \ref{categories-lemma-equivalence-fibred-categories} because it is an equivalence on all fibre categories by Lemma \ref{lemma-formal-objects-different-filtration} and the fact that the $\mathfrak m_A$-adic topology on a local Artinian ring $A$ comes from the zero ideal. We will frequently identify $\mathcal{F}$ with a full subcategory of $\widehat{\mathcal{F}}$ via a quasi-inverse to the functor $can$. \end{remark} \begin{remark} \label{remark-completion-morphism} Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of categories cofibered in groupoids over $\mathcal{C}_\Lambda$. Then there is an induced morphism $\widehat{\varphi}: \widehat{\mathcal{F}} \to \widehat{\mathcal{G}}$ of categories cofibered in groupoids over $\widehat{\mathcal{C}}_\Lambda$. It sends an object $\xi = (R, \xi_n, f_n)$ of $\widehat{\mathcal{F}}$ to $(R, \varphi(\xi_n), \varphi(f_n))$, and it sends a morphism $(a_0 : R \to S, a_n : \xi_n \to \eta_n)$ between objects $\xi$ and $\eta$ of $\widehat{\mathcal{F}}$ to $(a_0 : R \to S, \varphi(a_n) : \varphi(\xi_n) \to \varphi(\eta_n))$. Finally, if $t : \varphi \to \varphi'$ is a $2$-morphism between $1$-morphisms $\varphi, \varphi': \mathcal{F} \to \mathcal{G}$ of categories cofibred in groupoids, then we obtain a $2$-morphism $\widehat{t} : \widehat{\varphi} \to \widehat{\varphi}'$. Namely, for $\xi = (R, \xi_n, f_n)$ as above we set $\widehat{t}_\xi = (t_{\varphi(\xi_n)})$. Hence completion defines a functor between $2$-categories $$\widehat{~} : \text{Cof}(\mathcal{C}_\Lambda) \longrightarrow \text{Cof}(\widehat{\mathcal{C}}_\Lambda)$$ from the $2$-category of categories cofibred in groupoids over $\mathcal{C}_\Lambda$ to the $2$-category of categories cofibred in groupoids over $\widehat{\mathcal{C}}_\Lambda$. \end{remark} \begin{remark} \label{remark-completion-restriction-adjoint} We claim the completion functor of Remark \ref{remark-completion-morphism} and the restriction functor $|_{\mathcal{C}_\Lambda} : \text{Cof}(\widehat{\mathcal{C}}_\Lambda) \to \text{Cof}(\mathcal{C}_\Lambda)$ of Remarks \ref{remarks-cofibered-groupoids} (\ref{item-definition-restricting-base-category}) are 2-adjoint'' in the following precise sense. Let $\mathcal{F} \in \Ob(\text{Cof}(\mathcal{C}_\Lambda))$ and let $\mathcal{G} \in \Ob(\text{Cof}(\widehat{\mathcal{C}}_\Lambda))$. Then there is an equivalence of categories $$\Phi : \Mor_{\mathcal{C}_\Lambda}( \mathcal{G}|_{\mathcal{C}_\Lambda}, \mathcal{F}) \longrightarrow \Mor_{\widehat{\mathcal{C}}_\Lambda}(\mathcal{G}, \widehat{\mathcal{F}})$$ To describe this equivalence, we define canonical morphisms $\mathcal{G} \to \widehat{\mathcal{G}|_{\mathcal{C}_\Lambda}}$ and $\widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$ as follows \begin{enumerate} \item Let $R \in \Ob(\widehat{\mathcal{C}}_\Lambda))$ and let $\xi$ be an object of the fiber category $\mathcal{G}(R)$. Choose a pushforward $\xi \to \xi_n$ of $\xi$ to $R/\mathfrak m_R^n$ for each $n \in \mathbf{N}$, and let $f_n : \xi_{n + 1} \to \xi_n$ be the induced morphism. Then $\mathcal{G} \to \widehat{\mathcal{G}|_{\mathcal{C}_\Lambda}}$ sends $\xi$ to $(R, \xi_n, f_n)$. \item This is the equivalence $can : \widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$ of Remark \ref{remark-restrict-completion}. \end{enumerate} Having said this, the equivalence $\Phi : \Mor_{\mathcal{C}_\Lambda}( \mathcal{G}|_{\mathcal{C}_\Lambda}, \mathcal{F}) \to \Mor_{\widehat{\mathcal{C}}_\Lambda}(\mathcal{G}, \widehat{\mathcal{F}})$ sends a morphism $\varphi : \mathcal{G}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$ to $$\mathcal{G} \to \widehat{\mathcal{G}|_{\mathcal{C}_\Lambda}} \xrightarrow{\widehat{\varphi}} \widehat{\mathcal{F}}$$ There is a quasi-inverse $\Psi : \Mor_{\widehat{\mathcal{C}}_\Lambda}( \mathcal{G}, \widehat{\mathcal{F}}) \to \Mor_{\mathcal{C}_\Lambda}( \mathcal{G}|_{\mathcal{C}_\Lambda}, \mathcal{F})$ to $\Phi$ which sends $\psi : \mathcal{G} \to \widehat{\mathcal{F}}$ to $$\mathcal{G}|_{\mathcal{C}_\Lambda} \xrightarrow{\psi|_{\mathcal{C}_\Lambda}} \widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda} \to \mathcal{F}.$$ We omit the verification that $\Phi$ and $\Psi$ are quasi-inverse. We also do not address functoriality of $\Phi$ (because it would lead into 3-category territory which we want to avoid at all cost). \end{remark} \begin{remark} \label{remark-completion-restriction-cofset-adjoint} For a category $\mathcal{C}$ we denote by $\text{CofSet}(\mathcal{C})$ the category of cofibered sets over $\mathcal{C}$. It is a $1$-category isomorphic the category of functors $\mathcal{C} \to \textit{Sets}$. See Remarks \ref{remarks-cofibered-groupoids} (\ref{item-convention-cofibered-sets}). The completion and restriction functors restrict to functors $\widehat{~} : \text{CofSet}(\mathcal{C}_\Lambda) \to \text{CofSet}(\widehat{\mathcal{C}}_\Lambda)$ and $|_{\mathcal{C}_\Lambda} : \text{CofSet}(\widehat{\mathcal{C}}_\Lambda) \to \text{CofSet}(\mathcal{C}_\Lambda)$ which we denote by the same symbols. As functors on the categories of cofibered sets, completion and restriction are adjoints in the usual 1-categorical sense: the same construction as in Remark \ref{remark-completion-restriction-adjoint} defines a functorial bijection $$\Mor_{\mathcal{C}_\Lambda}(G|_{\mathcal{C}_\Lambda}, F) \longrightarrow \Mor_{\widehat{\mathcal{C}}_\Lambda}(G, \widehat{F})$$ for $F \in \Ob(\text{CofSet}(\mathcal{C}_\Lambda))$ and $G \in \Ob(\text{CofSet}(\widehat{\mathcal{C}}_\Lambda))$. Again the map $\widehat{F}|_{\mathcal{C}_\Lambda} \to F$ is an isomorphism. \end{remark} \begin{remark} \label{remark-restrict-complete-continuous-functor} Let $G : \widehat{\mathcal{C}}_\Lambda \to \textit{Sets}$ be a functor that commutes with limits. Then the map $G \to \widehat{G|_{\mathcal{C}_\Lambda}}$ described in Remark \ref{remark-completion-restriction-adjoint} is an isomorphism. Indeed, if $S$ is an object of $\widehat{\mathcal{C}}_\Lambda$, then we have canonical bijections $$\widehat{G|_{\mathcal{C}_\Lambda}}(S) = \lim_n G(S/\mathfrak{m}_S^n) = G(\lim_n S/\mathfrak{m}_S^n) = G(S).$$ In particular, if $R$ is an object of $\widehat{\mathcal{C}}_\Lambda$ then $\underline{R} = \widehat{\underline{R}|_{\mathcal{C}_\Lambda}}$ because the representable functor $\underline{R}$ commutes with limits by definition of limits. \end{remark} \begin{remark} \label{remark-formal-objects-yoneda} Let $R$ be an object of $\widehat{\mathcal{C}}_\Lambda$. It defines a functor $\underline{R}: \widehat{\mathcal{C}}_\Lambda \to \textit{Sets}$ as described in Remarks \ref{remarks-cofibered-groupoids} (\ref{item-definition-yoneda}). As usual we identify this functor with the associated cofibered set. If $\mathcal{F}$ is a cofibered category over $\mathcal{C}_\Lambda$, then there is an equivalence of categories \begin{equation} \label{equation-formal-objects-maps} \Mor_{\mathcal{C}_\Lambda}( \underline{R}|_{\mathcal{C}_\Lambda}, \mathcal{F}) \longrightarrow \widehat{\mathcal{F}}(R). \end{equation} It is given by the composition $$\Mor_{\mathcal{C}_\Lambda}( \underline{R}|_{\mathcal{C}_\Lambda}, \mathcal{F}) \xrightarrow{\Phi} \Mor_{\widehat{\mathcal{C}}_\Lambda}( \underline{R}, \widehat{\mathcal{F}}) \xrightarrow{\sim} \widehat{\mathcal{F}}(R)$$ where $\Phi$ is as in Remark \ref{remark-completion-restriction-adjoint} and the second equivalence comes from the 2-Yoneda lemma (the cofibered analogue of Categories, Lemma \ref{categories-lemma-yoneda-2category}). Explicitly, the equivalence sends a morphism $\varphi : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$ to the formal object $(R, \varphi(R \to R/\mathfrak{m}_R^n), \varphi(f_n))$ in $\widehat{\mathcal{F}}(R)$, where $f_n : R/\mathfrak m_R^{n + 1} \to R/\mathfrak m_R^n$ is the projection. \medskip\noindent Assume a choice of pushforwards for $\mathcal{F}$ has been made. Given any $\xi \in \Ob(\widehat{\mathcal{F}}(R))$ we construct an explicit $\underline{\xi} : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$ which maps to $\xi$ under (\ref{equation-formal-objects-maps}). Namely, say $\xi = (R, \xi_n, f_n)$. An object $\alpha$ in $\underline{R}|_{\mathcal{C}_\Lambda}$ is the same thing as a morphism $\alpha : R \to A$ of $\widehat{\mathcal{C}}_\Lambda$ with $A$ Artinian. Let $m \in \mathbf{N}$ be minimal such that $\mathfrak m_A^m = 0$. Then $\alpha$ factors through a unique $\alpha_m : R/\mathfrak m_R^m \to A$ and we can set $\underline{\xi}(\alpha) = \alpha_{m, *}\xi_m$. We omit the description of $\underline{\xi}$ on morphisms and we omit the proof that $\underline{\xi}$ maps to $\xi$ via (\ref{equation-formal-objects-maps}). \medskip\noindent Assume a choice of pushforwards for $\widehat{\mathcal{F}}$ has been made. In this case the proof of Categories, Lemma \ref{categories-lemma-yoneda-2category} gives an explicit quasi-inverse $$\iota : \widehat{\mathcal{F}}(R) \longrightarrow \Mor_{\widehat{\mathcal{C}}_\Lambda}( \underline{R}, \widehat{\mathcal{F}})$$ to the 2-Yoneda equivalence which takes $\xi$ to the morphism $\iota(\xi) : \underline{R} \to \widehat{\mathcal{F}}$ sending $f \in \underline{R}(S) = \Mor_{\mathcal{C}_\Lambda}(R, S)$ to $f_*\xi$. A quasi-inverse to (\ref{equation-formal-objects-maps}) is then $$\widehat{\mathcal{F}}(R) \xrightarrow{\iota} \Mor_{\widehat{\mathcal{C}}_\Lambda}( \underline{R}, \widehat{\mathcal{F}}) \xrightarrow{\Psi} \Mor_{\mathcal{C}_\Lambda}( \underline{R}|_{\mathcal{C}_\Lambda}, \mathcal{F})$$ where $\Psi$ is as in Remark \ref{remark-completion-restriction-adjoint}. Given $\xi \in \Ob(\widehat{\mathcal{F}}(R))$ we have $\Psi(\iota(\xi)) \cong \underline{\xi}$ where $\underline{\xi}$ is as in the previous paragraph, because both are mapped to $\xi$ under the equivalence of categories (\ref{equation-formal-objects-maps}). Using $\underline{R} = \widehat{\underline{R}|_{\mathcal{C}_\Lambda}}$ (see Remark \ref{remark-restrict-complete-continuous-functor}) and unwinding the definitions of $\Phi$ and $\Psi$ we conclude that $\iota(\xi)$ is isomorphic to the completion of $\underline{\xi}$. \end{remark} \begin{remark} \label{remark-formal-objects-yoneda-map} Let $\mathcal{F}$ be a category cofibred in groupoids over $\mathcal{C}_\Lambda$. Let $\xi = (R, \xi_n, f_n)$ and $\eta = (S, \eta_n, g_n)$ be formal objects of $\mathcal{F}$. Let $a = (a_n) : \xi \to \eta$ be a morphism of formal objects, i.e., a morphism of $\widehat{\mathcal{F}}$. Let $f = \widehat{p}(a) = a_0 : R \to S$ be the projection of $a$ in $\widehat{\mathcal{C}}_\Lambda$. Then we obtain a $2$-commutative diagram $$\xymatrix{ \underline{R}|_{\mathcal{C}_\Lambda} \ar[rd]_{\underline{\xi}} & & \underline{S}|_{\mathcal{C}_\Lambda} \ar[ll]^f \ar[ld]^{\underline{\eta}} \\ & \mathcal{F} }$$ where $\underline{\xi}$ and $\underline{\eta}$ are the morphisms constructed in Remark \ref{remark-formal-objects-yoneda}. To see this let $\alpha : S \to A$ be an object of $\underline{S}|_{\mathcal{C}_\Lambda}$ (see loc.\ cit.). Let $m \in \mathbf{N}$ be minimal such that $\mathfrak m_A^m = 0$. We get a commutative diagram $$\xymatrix{ R \ar[d]^f \ar[r] & R/\mathfrak m_R^m \ar[d]_{f_m} \ar[rd]^{\beta_m} \\ S \ar[r] & S/\mathfrak m_S^m \ar[r]^{\alpha_m} & A }$$ such that the bottom arrows compose to give $\alpha$. Then $\underline{\eta}(\alpha) = \alpha_{m, *}\eta_m$ and $\underline{\xi}(\alpha \circ f) = \beta_{m, *}\xi_m$. The morphism $a_m : \xi_m \to \eta_m$ lies over $f_m$ hence we obtain a canonical morphism $$\underline{\xi}(\alpha \circ f) = \beta_{m, *}\xi_m \longrightarrow \underline{\eta}(\alpha) = \alpha_{m, *}\eta_m$$ lying over $\text{id}_A$ such that $$\xymatrix{ \xi_m \ar[r] \ar[d]^{a_m} & \beta_{m, *}\xi_m \ar[d] \\ \eta_m \ar[r] & \alpha_{m, *}\eta_m }$$ commutes by the axioms of a category cofibred in groupoids. This defines a transformation of functors $\underline{\xi} \circ f \to \underline{\eta}$ which witnesses the 2-commutativity of the first diagram of this remark. \end{remark} \begin{remark} \label{remark-spell-out-formal-object} According to Remark \ref{remark-formal-objects-yoneda}, giving a formal object $\xi$ of $\mathcal{F}$ is equivalent to giving a prorepresentable functor $U : \mathcal{C}_\Lambda \to \textit{Sets}$ and a morphism $U \to \mathcal{F}$. \end{remark} \section{Smooth morphisms} \label{section-smooth-morphisms} \noindent In this section we discuss smooth morphisms of categories cofibered in groupoids over $\mathcal{C}_\Lambda$. \begin{definition} \label{definition-smooth-morphism} Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of categories cofibered in groupoids over $\mathcal{C}_\Lambda$. We say $\varphi$ is {\it smooth} if it satisfies the following condition: Let $B \to A$ be a surjective ring map in $\mathcal{C}_\Lambda$. Let $y \in \Ob(\mathcal{G}(B)), x \in \Ob(\mathcal{F}(A))$, and $y \to \varphi(x)$ be a morphism lying over $B \to A$. Then there exists $x' \in \Ob(\mathcal{F}(B))$, a morphism $x' \to x$ lying over $B \to A$, and a morphism $\varphi(x') \to y$ lying over $\text{id}: B \to B$, such that the diagram $$\xymatrix{ \varphi(x') \ar[r] \ar[dr] & y \ar[d] \\ & \varphi(x) }$$ commutes. \end{definition} \begin{lemma} \label{lemma-smoothness-small-extensions} Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of categories cofibered in groupoids over $\mathcal{C}_\Lambda$. Then $\varphi$ is smooth if the condition in Definition \ref{definition-smooth-morphism} is assumed to hold only for small extensions $B \to A$. \end{lemma} \begin{proof} Let $B \to A$ be a surjective ring map in $\mathcal{C}_\Lambda$. Let $y \in \Ob(\mathcal{G}(B))$, $x \in \Ob(\mathcal{F}(A))$, and $y \to \varphi(x)$ be a morphism lying over $B \to A$. By Lemma \ref{lemma-factor-small-extension} we can factor $B \to A$ into small extensions $B = B_n \to B_{n-1} \to \ldots \to B_0 = A$. We argue by induction on $n$. If $n = 1$ the result is true by assumption. If $n > 1$, then denote $f : B = B_n \to B_{n - 1}$ and denote $g : B_{n - 1} \to B_0 = A$. Choose a pushforward $y \to f_* y$ of $y$ along $f$, so that the morphism $y \to \varphi(x)$ factors as $y \to f_* y \to \varphi(x)$. By the induction hypothesis we can find $x_{n - 1} \to x$ lying over $g : B_{n - 1} \to A$ and $a : \varphi(x_{n - 1}) \to f_*y$ lying over $\text{id} : B_{n - 1} \to B_{n - 1}$ such that $$\xymatrix{ \varphi(x_{n - 1}) \ar[r]_-a \ar[dr] & f_*y \ar[d] \\ & \varphi(x) }$$ commutes. We can apply the assumption to the composition $y \to \varphi(x_{n - 1})$ of $y \to f_*y$ with $a^{-1} : f_*y \to \varphi(x_{n - 1})$. We obtain $x_n \to x_{n - 1}$ lying over $B_n \to B_{n - 1}$ and $\varphi(x_n) \to y$ lying over $\text{id} : B_n \to B_n$ so that the diagram $$\xymatrix{ \varphi(x_n) \ar[r] \ar[d] & y \ar[d] \\ \varphi(x_{n - 1}) \ar[r]^-a \ar[dr] & f_*y \ar[d] \\ & \varphi(x) }$$ commutes. Then the composition $x_n \to x_{n - 1} \to x$ and $\varphi(x_n) \to y$ are the morphisms required by the definition of smoothness. \end{proof} \begin{remark} \label{remark-smoothness-2-categorical} Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of categories cofibered in groupoids over $\mathcal{C}_\Lambda$. Let $B \to A$ be a ring map in $\mathcal{C}_\Lambda$. Choices of pushforwards along $B \to A$ for objects in the fiber categories $\mathcal{F}(B)$ and $\mathcal{G}(B)$ determine functors $\mathcal{F}(B) \to \mathcal{F}(A)$ and $\mathcal{G}(B) \to \mathcal{G}(A)$ fitting into a $2$-commutative diagram $$\xymatrix{ \mathcal{F}(B) \ar[r]^{\varphi} \ar[d] & \mathcal{G}(B) \ar[d] \\ \mathcal{F}(A) \ar[r]^{\varphi} & \mathcal{G}(A) . }$$ Hence there is an induced functor $\mathcal{F}(B) \to \mathcal{F}(A) \times_{\mathcal{G}(A)} \mathcal{G}(B)$. Unwinding the definitions shows that $\varphi : \mathcal{F} \to \mathcal{G}$ is smooth if and only if this induced functor is essentially surjective whenever $B \to A$ is surjective (or equivalently, by Lemma \ref{lemma-smoothness-small-extensions}, whenever $B \to A$ is a small extension). \end{remark} \begin{remark} \label{remark-compare-smooth-schlessinger} The characterization of smooth morphisms in Remark \ref{remark-smoothness-2-categorical} is analogous to Schlessinger's notion of a smooth morphism of functors, cf.\ \cite[Definition 2.2.]{Sch}. In fact, when $\mathcal{F}$ and $\mathcal{G}$ are cofibered in sets then our notion is equivalent to Schlessinger's. Namely, in this case let $F, G : \mathcal{C}_\Lambda \to \textit{Sets}$ be the corresponding functors, see Remarks \ref{remarks-cofibered-groupoids} (\ref{item-convention-cofibered-sets}). Then $F \to G$ is smooth if and only if for every surjection of rings $B \to A$ in $\mathcal{C}_\Lambda$ the map $F(B) \to F(A) \times_{G(A)} G(B)$ is surjective. \end{remark} \begin{remark} \label{remark-smooth-to-iso-classes} Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda$. Then the morphism $\mathcal{F} \to \overline{\mathcal{F}}$ is smooth. Namely, suppose that $f : B \to A$ is a ring map in $\mathcal{C}_\Lambda$. Let $x \in \Ob(\mathcal{F}(A))$ and let $\overline{y} \in \overline{\mathcal{F}}(B)$ be the isomorphism class of $y \in \Ob(\mathcal{F}(B))$ such that $\overline{f_*y} = \overline{x}$. Then we simply take $x' = y$, the implied morphism $x' = y \to x$ over $B \to A$, and the equality $\overline{x'} = \overline{y}$ as the solution to the problem posed in Definition \ref{definition-smooth-morphism}. \end{remark} \noindent If $R \to S$ is a ring map $\widehat{\mathcal{C}}_\Lambda$, then there is an induced morphism $\underline{S} \to \underline{R}$ between the functors $\underline{S}, \underline{R}: \widehat{\mathcal{C}}_\Lambda \to \textit{Sets}$. In this situation, smoothness of the restriction $\underline{S}|_{\mathcal{C}_\Lambda} \to \underline{R}|_{\mathcal{C}_\Lambda}$ is a familiar notion: \begin{lemma} \label{lemma-smooth-morphism-power-series} Let $R \to S$ be a ring map in $\widehat{\mathcal{C}}_\Lambda$. Then the induced morphism $\underline{S}|_{\mathcal{C}_\Lambda} \to \underline{R}|_{\mathcal{C}_\Lambda}$ is smooth if and only if $S$ is a power series ring over $R$. \end{lemma} \begin{proof} Assume $S$ is a power series ring over $R$. Say $S = R[[x_1, \ldots, x_n]]$. Smoothness of $\underline{S}|_{\mathcal{C}_\Lambda} \to \underline{R}|_{\mathcal{C}_\Lambda}$ means the following (see Remark \ref{remark-compare-smooth-schlessinger}): Given a surjective ring map $B \to A$ in $\mathcal{C}_\Lambda$, a ring map $R \to B$, a ring map $S \to A$ such that the solid diagram $$\xymatrix{ S \ar[r] \ar@{..>}[rd] & A \\ R \ar[u] \ar[r] & B \ar[u] }$$ is commutative then a dotted arrow exists making the diagram commute. (Note the similarity with Algebra, Definition \ref{algebra-definition-formally-smooth}.) To construct the dotted arrow choose elements $b_i \in B$ whose images in $A$ are equal to the images of $x_i$ in $A$. Note that $b_i \in \mathfrak m_B$ as $x_i$ maps to an element of $\mathfrak m_A$. Hence there is a unique $R$-algebra map $R[[x_1, \ldots, x_n]] \to B$ which maps $x_i$ to $b_i$ and which can serve as our dotted arrow. \medskip\noindent Conversely, assume $\underline{S}|_{\mathcal{C}_\Lambda} \to \underline{R}|_{\mathcal{C}_\Lambda}$ is smooth. Let $x_1, \ldots, x_n \in S$ be elements whose images form a basis in the relative cotangent space $\mathfrak m_S/(\mathfrak m_R S + \mathfrak m_S^2)$ of $S$ over $R$. Set $T = R[[X_1, \ldots, X_n]]$. Note that both $$S/(\mathfrak m_R S + \mathfrak m_S^2) \cong R/\mathfrak m_R[x_1, \ldots, x_n]/(x_ix_j)$$ and $$T/(\mathfrak m_R T + \mathfrak m_T^2) \cong R/\mathfrak m_R[X_1, \ldots, X_n]/(X_iX_j).$$ Let $S/(\mathfrak m_R S + \mathfrak m_S^2) \to T/(\mathfrak m_R T + \mathfrak m_T^2)$ be the local $R$-algebra isomorphism given by mapping the class of $x_i$ to the class of $X_i$. Let $f_1 : S \to T/(\mathfrak m_R T + \mathfrak m_T^2)$ be the composition $S \to S/(\mathfrak m_R S + \mathfrak m_S^2) \to T/(\mathfrak m_R T + \mathfrak m_T^2)$. The assumption that $\underline{S}|_{\mathcal{C}_\Lambda} \to \underline{R}|_{\mathcal{C}_\Lambda}$ is smooth means we can lift $f_1$ to a map $f_2 : S \to T/\mathfrak{m}_T^2$, then to a map $f_3 : S \to T/\mathfrak{m}_T^3$, and so on, for all $n \geq 1$. Thus we get an induced map $f : S \to T = \lim T/\mathfrak m_T^n$ of local $R$-algebras. By our choice of $f_1$, the map $f$ induces an isomorphism $\mathfrak m_S/(\mathfrak m_R S + \mathfrak m_S^2) \to \mathfrak m_T/(\mathfrak m_R T + \mathfrak m_T^2)$ of relative cotangent spaces. Hence $f$ is surjective by Lemma \ref{lemma-surjective-cotangent-space} (where we think of $f$ as a map in $\widehat{\mathcal{C}}_R$). Choose preimages $y_i \in S$ of $X_i \in T$ under $f$. As $T$ is a power series ring over $R$ there exists a local $R$-algebra homomorphism $s : T \to S$ mapping $X_i$ to $y_i$. By construction $f \circ s = \text{id}$. Then $s$ is injective. But $s$ induces an isomorphism on relative cotangent spaces since $f$ does, so it is also surjective by Lemma \ref{lemma-surjective-cotangent-space} again. Hence $s$ and $f$ are isomorphisms. \end{proof} \noindent Smooth morphisms satisfy the following functorial properties. \begin{lemma} \label{lemma-smooth-properties} Let $\varphi : \mathcal{F} \to \mathcal{G}$ and $\psi : \mathcal{G} \to \mathcal{H}$ be morphisms of categories cofibered in groupoids over $\mathcal{C}_\Lambda$. \begin{enumerate} \item If $\varphi$ and $\psi$ are smooth, then $\psi \circ \varphi$ is smooth. \item If $\varphi$ is essentially surjective and $\psi \circ \varphi$ is smooth, then $\psi$ is smooth. \item If $\mathcal{G}' \to \mathcal{G}$ is a morphism of categories cofibered in groupoids and $\varphi$ is smooth, then $\mathcal{F} \times_\mathcal{G} \mathcal{G}' \to \mathcal{G}'$ is smooth. \end{enumerate} \end{lemma} \begin{proof} Statements (1) and (2) follow immediately from the definitions. Proof of (3) omitted. Hints: use the formulation of smoothness given in Remark \ref{remark-smoothness-2-categorical} and use that $\mathcal{F} \times_\mathcal{G} \mathcal{G}'$ is the $2$-fibre product, see Remarks \ref{remarks-cofibered-groupoids} (\ref{item-fibre-product}). \end{proof} \begin{lemma} \label{lemma-smooth-morphism-essentially-surjective} Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a smooth morphism of categories cofibered in groupoids over $\mathcal{C}_\Lambda$. Assume $\varphi : \mathcal{F}(k) \to \mathcal{G}(k)$ is essentially surjective. Then $\varphi : \mathcal{F} \to \mathcal{G}$ and $\widehat{\varphi} : \widehat{\mathcal{F}} \to \widehat{\mathcal{G}}$ are essentially surjective. \end{lemma} \begin{proof} Let $y$ be an object of $\mathcal{G}$ lying over $A \in \Ob(\mathcal{C}_\Lambda)$. Let $y \to y_0$ be a pushforward of $y$ along $A \to k$. By the assumption on essential surjectivity of $\varphi : \mathcal{F}(k) \to \mathcal{G}(k)$ there exist an object $x_0$ of $\mathcal{F}$ lying over $k$ and an isomorphism $y_0 \to \varphi(x_0)$. Smoothness of $\varphi$ implies there exists an object $x$ of $\mathcal{F}$ over $A$ whose image $\varphi(x)$ is isomorphic to $y$. Thus $\varphi : \mathcal{F} \to \mathcal{G}$ is essentially surjective. \medskip\noindent Let $\eta = (R, \eta_n, g_n)$ be an object of $\widehat{\mathcal{G}}$. We construct an object $\xi$ of $\widehat{\mathcal{F}}$ with an isomorphism $\eta \to \varphi(\xi)$. By the assumption on essential surjectivity of $\varphi : \mathcal{F}(k) \to \mathcal{G}(k)$, there exists a morphism $\eta_1 \to \varphi(\xi_1)$ in $\mathcal{G}(k)$ for some $\xi_1 \in \Ob(\mathcal{F}(k))$. The morphism $\eta_2 \xrightarrow{g_1} \eta_1 \to \varphi(\xi_1)$ lies over the surjective ring map $R/\mathfrak m_R^2 \to k$, hence by smoothness of $\varphi$ there exists $\xi_2 \in \Ob(\mathcal{F}(R/\mathfrak m_R^2))$, a morphism $f_1: \xi_2 \to \xi_1$ lying over $R/\mathfrak m_R^2 \to k$, and a morphism $\eta_2 \to \varphi(\xi_2)$ such that $$\xymatrix{ \varphi(\xi_2) \ar[r]^{\varphi(f_1)} & \varphi(\xi_{1}) \\ \eta_2 \ar[u] \ar[r]^{g_1} & \eta_1 \ar[u] \\ }$$ commutes. Continuing in this way we construct an object $\xi = (R, \xi_n, f_n)$ of $\widehat{\mathcal{F}}$ and a morphism $\eta \to \varphi(\xi) = (R, \varphi(\xi_n), \varphi(f_n))$ in $\widehat{\mathcal{G}}(R)$. \end{proof} \begin{remark} \label{remark-cofibered-groupoid-projection-smooth} Let $p : \mathcal{F} \to \mathcal{C}_\Lambda$ be a category cofibered in groupoids. We can consider $\mathcal{C}_\Lambda$ as the trivial category cofibered in groupoids over $\mathcal{C}_\Lambda$, and then $p$ is a morphism of categories cofibered in groupoids over $\mathcal{C}_\Lambda$. We say $\mathcal{F}$ is {\it smooth} if its structure morphism $p$ is smooth. This is the absolute'' notion of smoothness for a category cofibered in groupoids over $\mathcal{C}_\Lambda$. \end{remark} \begin{example} \label{example-smooth} Let $R \in \Ob(\widehat{\mathcal{C}}_\Lambda)$. When is $\underline{R}|_{\mathcal{C}_\Lambda}$ smooth? In the classical case this means that $R$ is a power series ring over $\Lambda$, see Lemma \ref{lemma-smooth-morphism-power-series}. (Strictly speaking this uses that $\underline{\Lambda}|_{\mathcal{C}_\Lambda} = \mathcal{C}_\Lambda$ because $\Lambda$ is an initial object of $\widehat{\mathcal{C}}_\Lambda$ in the classical case.) In the general case we can construct examples as follows. Pick an integer $n \geq 0$ and a maximal ideal $\mathfrak m \subset \Lambda[x_1, \ldots, x_n]$ lying over $\mathfrak m_\Lambda$ so that $$k' = \Lambda/\mathfrak m_\Lambda \longrightarrow \Lambda[x_1, \ldots, x_n]/\mathfrak m$$ is isomorphic to $k' \to k$. Fix such an identification $k = \Lambda[x_1, \ldots, x_n]/\mathfrak m$. Set $R = \Lambda[x_1, \ldots, x_n]^\wedge$ equal to the $\mathfrak m$-adic completion of $\Lambda[x_1, \ldots, x_n]$. Then $R$ is an object of $\widehat{\mathcal{C}}_\Lambda$. Namely, it is a complete local Noetherian ring (see Algebra, Lemma \ref{algebra-lemma-completion-Noetherian-Noetherian}) and its residue field is identified with $k$. We claim that $\underline{R}|_{\mathcal{C}_\Lambda}$ is smooth. To see this we have to show: Given a surjection $B \to A$ in $\mathcal{C}_\Lambda$ and a map $R \to A$ there exists a lift of this map to $B$. This is clear as we can first lift the composition $\Lambda[x_1, \ldots, x_n] \to R \to A$ to a map $\Lambda[x_1, \ldots, x_n] \to B$ and then observe that this latter map factors through the completion $R$ as $B$ is complete (being Artinian). In fact, it turns out that whenever $\underline{R}|_{\mathcal{C}_\Lambda}$ is smooth, then $R$ is isomorphic to a completion of a smooth algebra over $\Lambda$, but we won't use this. \end{example} \begin{example} \label{example-smooth-explicit} Here is a more explicit example of an $R$ as in Example \ref{example-smooth}. Let $p$ be a prime number and let $n \in \mathbf{N}$. Let $\Lambda = \mathbf{F}_p(t_1, t_2, \ldots, t_n)$ and let $k = \mathbf{F}_p(x_1, \ldots, x_n)$ with map $\Lambda \to k$ given by $t_i \mapsto x_i^p$. Then we can take $$R = \Lambda[x_1, \ldots, x_n]^\wedge_{(x_1^p - t_1, \ldots, x_n^p - t_n)}$$ We cannot do better'' in this example, i.e., we cannot approximate $\mathcal{C}_\Lambda$ by a smaller smooth object of $\widehat{\mathcal{C}}_\Lambda$ (one can argue that the dimension of $R$ has to be at least $n$ since the map $\Omega_{R/\Lambda} \otimes_R k \to \Omega_{k/\Lambda}$ is surjective). We will discuss this phenomenon later in more detail. \end{example} \begin{remark} \label{remark-smooth-on-top} Suppose $\mathcal{F}$ is a predeformation category admitting a smooth morphism $\varphi : \mathcal U \to \mathcal{F}$ from a predeformation category $\mathcal{U}$. Then by Lemma \ref{lemma-smooth-morphism-essentially-surjective} $\varphi$ is essentially surjective, so by Lemma \ref{lemma-smooth-properties} $p: \mathcal{F} \to \mathcal{C}_\Lambda$ is smooth if and only if the composition $\mathcal U \xrightarrow{\varphi} \mathcal{F} \xrightarrow{p} \mathcal{C}_\Lambda$ is smooth, i.e.\ $\mathcal{F}$ is smooth if and only if $\mathcal{U}$ is smooth. \end{remark} \noindent Later we are interested in producing smooth morphisms from prorepresentable functors to predeformation categories $\mathcal{F}$. By the discussion in Remark \ref{remark-formal-objects-yoneda} these morphisms correspond to certain formal objects of $\mathcal{F}$ More precisely, these are the so-called versal formal objects of $\mathcal{F}$. \begin{definition} \label{definition-versal} Let $\mathcal{F}$ be a category cofibered in groupoids. Let $\xi$ be a formal object of $\mathcal{F}$ lying over $R \in \Ob(\widehat{\mathcal C}_\Lambda)$. We say $\xi$ is {\it versal} if the corresponding morphism $\underline{\xi}: \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$ of Remark \ref{remark-formal-objects-yoneda} is smooth. \end{definition} \begin{remark} \label{remark-versal-object} Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal C_\Lambda$, and let $\xi$ be a formal object of $\mathcal{F}$. It follows from the definition of smoothness that versality of $\xi$ is equivalent to the following condition: If $$\xymatrix{ & y \ar[d] \\ \xi \ar[r] & x }$$ is a diagram in $\widehat{\mathcal{F}}$ such that $y \to x$ lies over a surjective map $B \to A$ of Artinian rings (we may assume it is a small extension), then there exists a morphism $\xi \to y$ such that $$\xymatrix{ & y \ar[d] \\ \xi \ar[r] \ar[ur] & x }$$ commutes. In particular, the condition that $\xi$ be versal does not depend on the choices of pushforwards made in the construction of $\underline{\xi} : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$ in Remark \ref{remark-formal-objects-yoneda}. \end{remark} \begin{lemma} \label{lemma-versal-object-quasi-initial} Let $\mathcal{F}$ be a predeformation category. Let $\xi$ be a versal formal object of $\mathcal{F}$. For any formal object $\eta$ of $\widehat{\mathcal{F}}$, there exists a morphism $\xi \to \eta$. \end{lemma} \begin{proof} By assumption the morphism $\underline{\xi} : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$ is smooth. Then $\iota(\xi) : \underline{R} \to \widehat{\mathcal{F}}$ is the completion of $\underline{\xi}$, see Remark \ref{remark-formal-objects-yoneda}. By Lemma \ref{lemma-smooth-morphism-essentially-surjective} there exists an object $f$ of $\underline{R}$ such that $\iota(\xi)(f) = \eta$. Then $f$ is a ring map $f : R \to S$ in $\widehat{\mathcal{C}}_\Lambda$. And $\iota(\xi)(f) = \eta$ means that $f_*\xi \cong \eta$ which means exactly that there is a morphism $\xi \to \eta$ lying over $f$. \end{proof} \section{Schlessinger's conditions} \label{section-schlessinger-conditions} \noindent In the following we often consider fibre products $A_1 \times_A A_2$ of rings in the category $\mathcal{C}_\Lambda$. We have seen in Example \ref{example-fibre-product} that such a fibre product may not always be an object of $\mathcal{C}_\Lambda$. However, in virtually all cases below one of the two maps $A_i \to A$ is surjective and $A_1 \times_A A_2$ will be an object of $\mathcal{C}_\Lambda$ by Lemma \ref{lemma-fiber-product-CLambda}. We will use this result without further mention. \medskip\noindent We denote by $k[\epsilon]$ the ring of dual numbers over $k$. More generally, for a $k$-vector space $V$, we denote by $k[V]$ the $k$-algebra whose underlying vector space is $k \oplus V$ and whose multiplication is given by $(a, v) \cdot (a', v') = (aa', av' + a'v)$. When $V = k$, $k[V]$ is the ring of dual numbers over $k$. For any finite dimensional $k$-vector space $V$ the ring $k[V]$ is in $\mathcal{C}_\Lambda$. \begin{definition} \label{definition-S1-S2} Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal C_\Lambda$. We define {\it conditions (S1) and (S2)} on $\mathcal{F}$ as follows: \begin{enumerate} \item[(S1)] Every diagram in $\mathcal{F}$ $$\vcenter{ \xymatrix{ & x_2 \ar[d] \\ x_1 \ar[r] & x } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ & A_2 \ar[d] \\ A_1 \ar[r] & A } }$$ in $\mathcal{C}_\Lambda$ with $A_2 \to A$ surjective can be completed to a commutative diagram $$\vcenter{ \xymatrix{ y \ar[r] \ar[d] & x_2 \ar[d] \\ x_1 \ar[r] & x } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ A_1 \times_A A_2 \ar[r] \ar[d] & A_2 \ar[d] \\ A_1 \ar[r] & A. } }$$ \item[(S2)] The condition of (S1) holds for diagrams in $\mathcal{F}$ lying over a diagram in $\mathcal{C}_\Lambda$ of the form $$\xymatrix{ & k[\epsilon] \ar[d] \\ A \ar[r] & k. }$$ Moreover, if we have two commutative diagrams in $\mathcal{F}$ $$\vcenter{ \xymatrix{ y \ar[r]_c \ar[d]_a & x_\epsilon \ar[d]^e \\ x \ar[r]^d & x_0 } } \quad\text{and}\quad \vcenter{ \xymatrix{ y' \ar[r]_{c'} \ar[d]_{a'} & x_\epsilon \ar[d]^e \\ x \ar[r]^d & x_0 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ A \times_k k[\epsilon] \ar[r] \ar[d] & k[\epsilon] \ar[d] \\ A \ar[r] & k } }$$ then there exists a morphism $b : y \to y'$ in $\mathcal{F}(A \times_k k[\epsilon])$ such that $a = a' \circ b$. \end{enumerate} \end{definition} \noindent We can partly explain the meaning of conditions (S1) and (S2) in terms of fibre categories. Suppose that $f_1 : A_1 \to A$ and $f_2 : A_2 \to A$ are ring maps in $\mathcal{C}_\Lambda$ with $f_2$ surjective. Denote $p_i : A_1 \times_A A_2 \to A_i$ the projection maps. Assume a choice of pushforwards for $\mathcal{F}$ has been made. Then the commutative diagram of rings translates into a $2$-commutative diagram $$\xymatrix{ \mathcal{F}(A_1 \times_A A_2) \ar[r]_-{p_{2, *}} \ar[d]_{p_{1, *}} & \mathcal{F}(A_2) \ar[d]^{f_{2, *}} \\ \mathcal{F}(A_1) \ar[r]^{f_{1, *}} & \mathcal{F}(A) }$$ of fibre categories whence a functor \begin{equation} \label{equation-compare} \mathcal{F}(A_1 \times_A A_2) \to \mathcal{F}(A_1) \times_{\mathcal{F}(A)} \mathcal{F}(A_2) \end{equation} into the $2$-fibre product of categories. Condition (S1) requires that this functor be essentially surjective. The first part of condition (S2) requires that this functor be a essentially surjective if $f_2$ equals the map $k[\epsilon] \to k$. Moreover in this case, the second part of (S2) implies that two objects which become isomorphic in the target are isomorphic in the source (but it is {\it not} equivalent to this statement). The advantage of stating the conditions as in the definition is that no choices have to be made. \begin{lemma} \label{lemma-S1-small-extensions} Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal C_\Lambda$. Then $\mathcal{F}$ satisfies (S1) if the condition of (S1) is assumed to hold only when $A_2 \to A$ is a small extension. \end{lemma} \begin{proof} Proof omitted. Hints: apply Lemma \ref{lemma-factor-small-extension} and use induction similar to the proof of Lemma \ref{lemma-smoothness-small-extensions}. \end{proof} \begin{remark} \label{remark-compare-S1-S2-schlessinger} When $\mathcal{F}$ is cofibered in sets, conditions (S1) and (S2) are exactly conditions (H1) and (H2) from Schlessinger's paper \cite{Sch}. Namely, for a functor $F: \mathcal{C}_\Lambda \to \textit{Sets}$, conditions (S1) and (S2) state: \begin{enumerate} \item [(S1)] If $A_1 \to A$ and $A_2 \to A$ are maps in $\mathcal{C}_\Lambda$ with $A_2 \to A$ surjective, then the induced map $F(A_1 \times_A A_2) \to F(A_1) \times_{F(A)} F(A_2)$ is surjective. \item [(S2)] If $A \to k$ is a map in $\mathcal{C}_\Lambda$, then the induced map $F(A \times_k k[\epsilon]) \to F(A) \times_{F(k)} F(k[\epsilon])$ is bijective. \end{enumerate} The injectivity of the map $F(A \times_k k[\epsilon]) \to F(A) \times_{F(k)} F(k[\epsilon])$ comes from the second part of condition (S2) and the fact that morphisms are identities. \end{remark} \begin{lemma} \label{lemma-S2-extensions} Let $\mathcal{F}$ be a category cofibred in groupoids over $\mathcal{C}_\Lambda$. If $\mathcal{F}$ satisfies (S2), then the condition of (S2) also holds when $k[\epsilon]$ is replaced by $k[V]$ for any finite dimensional $k$-vector space $V$. \end{lemma} \begin{proof} In the case that $\mathcal{F}$ is cofibred in sets, i.e., corresponds to a functor $F : \mathcal{C}_\Lambda \to \textit{Sets}$ this follows from the description of (S2) for $F$ in Remark \ref{remark-compare-S1-S2-schlessinger} and the fact that $k[V] \cong k[\epsilon] \times_k \ldots \times_k k[\epsilon]$ with $\dim_k V$ factors. The case of functors is what we will use in the rest of this chapter. \medskip\noindent We prove the general case by induction on $\dim(V)$. If $\dim(V) = 1$, then $k[V] \cong k[\epsilon]$ and the result holds by assumption. If $\dim(V) > 1$ we write $V = V' \oplus k\epsilon$. Pick a diagram $$\vcenter{ \xymatrix{ & x_V \ar[d] \\ x \ar[r] & x_0 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ & k[V] \ar[d] \\ A \ar[r] & k } }$$ Choose a morphism $x_V \to x_{V'}$ lying over $k[V] \to k[V']$ and a morphism $x_V \to x_\epsilon$ lying over $k[V] \to k[\epsilon]$. Note that the morphism $x_V \to x_0$ factors as $x_V \to x_{V'} \to x_0$ and as $x_V \to x_\epsilon \to x_0$. By induction hypothesis we can find a diagram $$\vcenter{ \xymatrix{ y' \ar[d] \ar[r] & x_{V'} \ar[d] \\ x \ar[r] & x_0 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ A \times_k k[V'] \ar[d] \ar[r] & k[V'] \ar[d] \\ A \ar[r] & k } }$$ This gives us a commutative diagram $$\vcenter{ \xymatrix{ & x_\epsilon \ar[d] \\ y' \ar[r] & x_0 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ & k[\epsilon] \ar[d] \\ A \times_k k[V'] \ar[r] & k } }$$ Hence by (S2) we get a commutative diagram $$\vcenter{ \xymatrix{ y \ar[d] \ar[r] & x_\epsilon \ar[d] \\ y' \ar[r] & x_0 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ (A \times_k k[V']) \times_k k[\epsilon] \ar[d] \ar[r] & k[\epsilon] \ar[d] \\ A \times_k k[V'] \ar[r] & k } }$$ Note that $(A \times_k k[V']) \times_k k[\epsilon] = A \times_k k[V' \oplus k\epsilon] = A \times_k k[V]$. We claim that $y$ fits into the correct commutative diagram. To see this we let $y \to y_V$ be a morphism lying over $A \times_k k[V] \to k[V]$. We can factor the morphisms $y \to y' \to x_{V'}$ and $y \to x_\epsilon$ through the morphism $y \to y_V$ (by the axioms of categories cofibred in groupoids). Hence we see that both $y_V$ and $x_V$ fit into commutative diagrams $$\vcenter{ \xymatrix{ y_V \ar[r] \ar[d] & x_\epsilon \ar[d] \\ x_{V'} \ar[r] & x_0 } } \quad\text{and}\quad \vcenter{ \xymatrix{ x_V \ar[r] \ar[d] & x_\epsilon \ar[d] \\ x_{V'} \ar[r] & x_0 } }$$ and hence by the second part of (S2) there exists an isomorphism $y_V \to x_V$ compatible with $y_V \to x_{V'}$ and $x_V \to x_{V'}$ and in particular compatible with the maps to $x_0$. The composition $y \to y_V \to x_V$ then fits into the required commutative diagram $$\vcenter{ \xymatrix{ y \ar[r] \ar[d] & x_V \ar[d] \\ x \ar[r] & x_0 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ A \times_k k[V] \ar[d] \ar[r] & k[V] \ar[d] \\ A \ar[r] & k } }$$ In this way we see that the first part of $(S2)$ holds with $k[\epsilon]$ replaced by $k[V]$. \medskip\noindent To prove the second part suppose given two commutative diagrams $$\vcenter{ \xymatrix{ y \ar[r] \ar[d] & x_V \ar[d] \\ x \ar[r] & x_0 } } \quad\text{and}\quad \vcenter{ \xymatrix{ y' \ar[r] \ar[d] & x_V \ar[d] \\ x \ar[r] & x_0 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ A \times_k k[V] \ar[d] \ar[r] & k[V] \ar[d] \\ A \ar[r] & k } }$$ We will use the morphisms $x_V \to x_{V'} \to x_0$ and $x_V \to x_\epsilon \to x_0$ introduced in the first paragraph of the proof. Choose morphisms $y \to y_{V'}$ and $y' \to y'_{V'}$ lying over $A \times_k k[V] \to A \times_k k[V']$. The axioms of a cofibred category imply we can find commutative diagrams $$\vcenter{ \xymatrix{ y_{V'} \ar[r] \ar[d] & x_{V'} \ar[d] \\ x \ar[r] & x_0 } } \quad\text{and}\quad \vcenter{ \xymatrix{ y'_{V'} \ar[r] \ar[d] & x_{V'} \ar[d] \\ x \ar[r] & x_0 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ A \times_k k[V'] \ar[d] \ar[r] & k[V'] \ar[d] \\ A \ar[r] & k } }$$ By induction hypothesis we obtain an isomorphism $b : y_{V'} \to y'_{V'}$ compatible with the morphisms $y_{V'} \to x$ and $y'_{V'} \to x$, in particular compatible with the morphisms to $x_0$. Then we have commutative diagrams $$\vcenter{ \xymatrix{ y \ar[r] \ar[d] & x_\epsilon \ar[d] \\ y'_{V'} \ar[r] & x_0 } } \quad\text{and}\quad \vcenter{ \xymatrix{ y' \ar[r] \ar[d] & x_\epsilon \ar[d] \\ y'_{V'} \ar[r] & x_0 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ A \times_k k[\epsilon] \ar[d] \ar[r] & k[\epsilon] \ar[d] \\ A \ar[r] & k } }$$ where the morphism $y \to y'_{V'}$ is the composition $y \to y_{V'} \xrightarrow{b} y'_{V'}$ and where the morphisms $y \to x_\epsilon$ and $y' \to x_\epsilon$ are the compositions of the maps $y \to x_V$ and $y' \to x_V$ with the morphism $x_V \to x_\epsilon$. Then the second part of (S2) guarantees the existence of an isomorphism $y \to y'$ compatible with the maps to $y'_{V'}$, in particular compatible with the maps to $x$ (because $b$ was compatible with the maps to $x$). \end{proof} \begin{lemma} \label{lemma-S1-S2-associated-functor} Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda$. \begin{enumerate} \item If $\mathcal{F}$ satisfies (S1), then so does $\overline{\mathcal{F}}$. \item If $\mathcal{F}$ satisfies (S2), then so does $\overline{\mathcal{F}}$ provided at least one of the following conditions is satisfied \begin{enumerate} \item $\mathcal{F}$ is a predeformation category, \item the category $\mathcal{F}(k)$ is a set or a setoid, or \item for any morphism $x_\epsilon \to x_0$ of $\mathcal{F}$ lying over $k[\epsilon] \to k$ the pushforward map $\text{Aut}_{k[\epsilon]}(x_\epsilon) \to \text{Aut}_k(x_0)$ is surjective. \end{enumerate} \end{enumerate} \end{lemma} \begin{proof} Assume $\mathcal{F}$ has (S1). Suppose we have ring maps $f_i : A_i \to A$ in $\mathcal{C}_\Lambda$ with $f_2$ surjective. Let $x_i \in \mathcal{F}(A_i)$ such that the pushforwards $f_{1, *}(x_1)$ and $f_{2, *}(x_2)$ are isomorphic. Then we can denote $x$ an object of $\mathcal{F}$ over $A$ isomorphic to both of these and we obtain a diagram as in (S1). Hence we find an object $y$ of $\mathcal{F}$ over $A_1 \times_A A_2$ whose pushforward to $A_1$, resp.\ $A_2$ is isomorphic to $x_1$, resp.\ $x_2$. In this way we see that (S1) holds for $\overline{\mathcal{F}}$. \medskip\noindent Assume $\mathcal{F}$ has (S2). The first part of (S2) for $\overline{\mathcal{F}}$ follows as in the argument above. The second part of (S2) for $\overline{\mathcal{F}}$ signifies that the map $$\overline{\mathcal{F}}(A \times_k k[\epsilon]) \to \overline{\mathcal{F}}(A) \times_{\overline{\mathcal{F}}(k)} \overline{\mathcal{F}}(k[\epsilon])$$ is injective for any ring $A$ in $\mathcal{C}_\Lambda$. Suppose that $y, y' \in \mathcal{F}(A \times_k k[\epsilon])$. Using the axioms of cofibred categories we can choose commutative diagrams $$\vcenter{ \xymatrix{ y \ar[r]_c \ar[d]_a & x_\epsilon \ar[d]^e \\ x \ar[r]^d & x_0 } } \quad\text{and}\quad \vcenter{ \xymatrix{ y' \ar[r]_{c'} \ar[d]_{a'} & x'_\epsilon \ar[d]^{e'} \\ x' \ar[r]^{d'} & x'_0 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ A \times_k k[\epsilon] \ar[d] \ar[r] & k[\epsilon] \ar[d] \\ A \ar[r] & k } }$$ Assume that there exist isomorphisms $\alpha : x \to x'$ in $\mathcal{F}(A)$ and $\beta : x_\epsilon \to x'_\epsilon$ in $\mathcal{F}(k[\epsilon])$. This also means there exists an isomorphism $\gamma : x_0 \to x'_0$ compatible with $\alpha$. To prove (S2) for $\overline{\mathcal{F}}$ we have to show that there exists an isomorphism $y \to y'$ in $\mathcal{F}(A \times_k k[\epsilon])$. By (S2) for $\mathcal{F}$ such a morphism will exist if we can choose the isomorphisms $\alpha$ and $\beta$ and $\gamma$ such that $$\xymatrix{ x \ar[d]^\alpha \ar[r] & x_0 \ar[d]^\gamma & x_\epsilon \ar[d]^\beta \ar[l]^e \\ x' \ar[r] & x'_0 & x'_\epsilon \ar[l]_{e'} }$$ is commutative (because then we can replace $x$ by $x'$ and $x_\epsilon$ by $x'_\epsilon$ in the previous displayed diagram). The left hand square commutes by our choice of $\gamma$. We can factor $e' \circ \beta$ as $\gamma' \circ e$ for some second map $\gamma' : x_0 \to x'_0$. Now the question is whether we can arrange it so that $\gamma = \gamma'$? This is clear if $\mathcal{F}(k)$ is a set, or a setoid. Moreover, if $\text{Aut}_{k[\epsilon]}(x_\epsilon) \to \text{Aut}_k(x_0)$ is surjective, then we can adjust the choice of $\beta$ by precomposing with an automorphism of $x_\epsilon$ whose image is $\gamma^{-1} \circ \gamma'$ to make things work. \end{proof} \begin{lemma} \label{lemma-S1-S2-localize} Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda$. Let $x_0 \in \Ob(\mathcal{F}(k))$. Let $\mathcal{F}_{x_0}$ be the category cofibred in groupoids over $\mathcal{C}_\Lambda$ constructed in Remark \ref{remark-localize-cofibered-groupoid}. \begin{enumerate} \item If $\mathcal{F}$ satisfies (S1), then so does $\mathcal{F}_{x_0}$. \item If $\mathcal{F}$ satisfies (S2), then so does $\mathcal{F}_{x_0}$. \end{enumerate} \end{lemma} \begin{proof} Any diagram as in Definition \ref{definition-S1-S2} in $\mathcal{F}_{x_0}$ gives rise to a diagram in $\mathcal{F}$ and the output of condition (S1) or (S2) for this diagram in $\mathcal{F}$ can be viewed as an output for $\mathcal{F}_{x_0}$ as well. \end{proof} \begin{lemma} \label{lemma-lifting-section} Let $p: \mathcal{F} \to \mathcal{C}_\Lambda$ be a category cofibered in groupoids. Consider a diagram of $\mathcal{F}$ $$\vcenter{ \xymatrix{ y \ar[r] \ar[d]_a & x_\epsilon \ar[d]_e \\ x \ar[r]^d & x_0 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ A \times_k k[\epsilon] \ar[r] \ar[d] & k[\epsilon] \ar[d] \\ A \ar[r] & k. } }$$ in $\mathcal{C}_\Lambda$. Assume $\mathcal{F}$ satisfies (S2). Then there exists a morphism $s : x \to y$ with $a \circ s = \text{id}_x$ if and only if there exists a morphism $s_\epsilon : x \to x_\epsilon$ with $e \circ s_\epsilon = d$. \end{lemma} \begin{proof} The only if'' direction is clear. Conversely, assume there exists a morphism $s_\epsilon : x \to x_\epsilon$ with $e \circ s_\epsilon = d$. Note that $p(s_\epsilon) : A \to k[\epsilon]$ is a ring map compatible with the map $A \to k$. Hence we obtain $$\sigma = (\text{id}_A, p(s_\epsilon)) : A \to A \times_k k[\epsilon].$$ Choose a pushforward $x \to \sigma_*x$. By construction we can factor $s_\epsilon$ as $x \to \sigma_*x \to x_\epsilon$. Moreover, as $\sigma$ is a section of $A \times_k k[\epsilon] \to A$, we get a morphism $\sigma_*x \to x$ such that $x \to \sigma_*x \to x$ is $\text{id}_x$. Because $e \circ s_\epsilon = d$ we find that the diagram $$\xymatrix{ \sigma_*x \ar[r] \ar[d] & x_\epsilon \ar[d]_e \\ x \ar[r]^d & x_0 }$$ is commutative. Hence by (S2) we obtain a morphism $\sigma_*x \to y$ such that $\sigma_*x \to y \to x$ is the given map $\sigma_*x \to x$. The solution to the problem is now to take $a : x \to y$ equal to the composition $x \to \sigma_*x \to y$. \end{proof} \begin{lemma} \label{lemma-lifting-along-small-extension} Consider a commutative diagram in a predeformation category $\mathcal{F}$ $$\vcenter{ \xymatrix{ y \ar[r] \ar[d] & x_2 \ar[d]^{a_2} \\ x_1 \ar[r]^{a_1} & x } } \quad\text{lying over} \vcenter{ \xymatrix{ A_1 \times_A A_2 \ar[r] \ar[d] & A_2 \ar[d]^{f_2} \\ A_1 \ar[r]^{f_1} & A } }$$ in $\mathcal{C}_\Lambda$ where $f_2 : A_2 \to A$ is a small extension. Assume there is a map $h : A_1 \to A_2$ such that $f_2 = f_1 \circ h$. Let $I = \Ker(f_2)$. Consider the ring map $$g : A_1 \times_A A_2 \longrightarrow k[I] = k \oplus I, \quad (u, v) \longmapsto \overline{u} \oplus (v - h(u))$$ Choose a pushforward $y \to g_*y$. Assume $\mathcal{F}$ satisfies (S2). If there exists a morphism $x_1 \to g_*y$, then there exists a morphism $b: x_1 \to x_2$ such that $a_1 = a_2 \circ b$. \end{lemma} \begin{proof} Note that $\text{id}_{A_1} \times g : A_1 \times_A A_2 \to A_1 \times_k k[I]$ is an isomorphism and that $k[I] \cong k[\epsilon]$. Hence we have a diagram $$\vcenter{ \xymatrix{ y \ar[r] \ar[d] & g_*y \ar[d] \\ x_1 \ar[r] & x_0 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ A_1 \times_k k[\epsilon] \ar[r] \ar[d] & k[\epsilon] \ar[d] \\ A_1 \ar[r] & k. } }$$ where $x_0$ is an object of $\mathcal{F}$ lying over $k$ (every object of $\mathcal{F}$ has a unique morphism to $x_0$, see discussion following Definition \ref{definition-predeformation-category}). If we have a morphism $x_1 \to g_*y$ then Lemma \ref{lemma-lifting-section} provides us with a section $s : x_1 \to y$ of the map $y \to x_1$. Composing this with the map $y \to x_2$ we obtain $b : x_1 \to x_2$ which has the property that $a_1 = a_2 \circ b$ because the diagram of the lemma commutes and because $s$ is a section. \end{proof} \section{Tangent spaces of functors} \label{section-tangent-spaces-functors} \noindent Let $R$ be a ring. We write $\text{Mod}_R$ for the category of $R$-modules and $\text{Mod}^{fg}_R$ for the category of finitely generated $R$-modules. \begin{definition} \label{definition-linear} Let $L: \text{Mod}^{fg}_R \to \text{Mod}_R$, resp.\ $L: \text{Mod}_R \to \text{Mod}_R$ be a functor. We say that $L$ is {\it $R$-linear} if for every pair of objects $M, N$ of $\text{Mod}^{fg}_R$, resp.\ $\text{Mod}_R$ the map $$L : \Hom_R(M, N) \longrightarrow \Hom_R(L(M), L(N))$$ is a map of $R$-modules. \end{definition} \begin{remark} \label{remark-linear-enriched-over-modules} One can define the notion of an $R$-linearity for any functor between categories enriched over $\text{Mod}_R$. We made the definition specifically for functors $L: \text{Mod}^{fg}_R \to \text{Mod}_R$ and $L: \text{Mod}_R \to \text{Mod}_R$ because these are the cases that we have needed so far. \end{remark} \begin{remark} \label{remark-linear-functor} If $L: \text{Mod}^{fg}_R \to \text{Mod}_R$ is an $R$-linear functor, then $L$ preserves finite products and sends the zero module to the zero module, see Homology, Lemma \ref{homology-lemma-additive-additive}. On the other hand, if a functor $\text{Mod}^{fg}_R \to \textit{Sets}$ preserves finite products and sends the zero module to a one element set, then it has a unique lift to a $R$-linear functor, see Lemma \ref{lemma-linear-functor}. \end{remark} \begin{lemma} \label{lemma-linear-functor} Let $L: \text{Mod}^{fg}_R \to \textit{Sets}$, resp.\ $L: \text{Mod}_R \to \textit{Sets}$ be a functor. Suppose $L(0)$ is a one element set and $L$ preserves finite products. Then there exists a unique $R$-linear functor $\widetilde{L} : \text{Mod}^{fg}_R \to \text{Mod}_R$, resp.\ $\widetilde{L} : \text{Mod}^{fg}_R \to \text{Mod}_R$, such that $$\vcenter{ \xymatrix{ & \text{Mod}_R \ar[dr]^{\text{forget}} & \\ \text{Mod}^{fg}_R \ar[ur]^{\widetilde{L}} \ar[rr]^{L} & & \textit{Sets} } } \quad\text{resp.}\quad \vcenter{ \xymatrix{ & \text{Mod}_R \ar[dr]^{\text{forget}} & \\ \text{Mod}_R \ar[ur]^{\widetilde{L}} \ar[rr]^{L} & & \textit{Sets} } }$$ commutes. \end{lemma} \begin{proof} We only prove this in case $L: \text{Mod}^{fg}_R \to \textit{Sets}$. Let $M$ be a finitely generated $R$-module. We define $\widetilde{L}(M)$ to be the set $L(M)$ with the following $R$-module structure. \medskip\noindent Multiplication: If $r \in R$, multiplication by $r$ on $L(M)$ is defined to be the map $L(M) \to L(M)$ induced by the multiplication map $r \cdot : M \to M$. \medskip\noindent Addition: The sum map $M \times M \to M: (m_1, m_2) \mapsto m_1 + m_2$ induces a map $L(M \times M) \to L(M)$. By assumption $L(M \times M)$ is canonically isomorphic to $L(M) \times L(M)$. Addition on $L(M)$ is defined by the map $L(M) \times L(M) \cong L(M \times M) \to L(M)$. \medskip\noindent Zero: There is a unique map $0 \to M$. The zero element of $L(M)$ is the image of $L(0) \to L(M)$. \medskip\noindent We omit the verification that this defines an $R$-module $\widetilde{L}(M)$, the unique such that is $R$-linearly functorial in $M$. \end{proof} \begin{lemma} \label{lemma-morphism-linear-functors} Let $L_1, L_2: \text{Mod}^{fg}_R \to \textit{Sets}$ be functors that take $0$ to a one element set and preserve finite products. Let $t : L_1 \to L_2$ be a morphism of functors. Then $t$ induces a morphism $\widetilde{t} : \widetilde{L}_1 \to \widetilde{L}_2$ between the functors guaranteed by Lemma \ref{lemma-linear-functor}, which is given simply by $\widetilde{t}_M = t_M: \widetilde{L}_1(M) \to \widetilde{L}_2(M)$ for each $M \in \Ob(\text{Mod}^{fg}_R)$. In other words, $t_M: \widetilde{L}_1(M) \to \widetilde{L}_2(M)$ is a map of $R$-modules. \end{lemma} \begin{proof} Omitted. \end{proof} \noindent In the case $R = K$ is a field, a $K$-linear functor $L : \text{Mod}^{fg}_K \to \text{Mod}_K$ is determined by its value $L(K)$. \begin{lemma} \label{lemma-linear-functor-over-field} Let $K$ be a field. Let $L: \text{Mod}^{fg}_K \to \text{Mod}_K$ be a $K$-linear functor. Then $L$ is isomorphic to the functor $L(K) \otimes_K - : \text{Mod}^{fg}_K \to \text{Mod}_K$. \end{lemma} \begin{proof} For $V \in \Ob(\text{Mod}^{fg}_K)$, the isomorphism $L(K) \otimes_K V \to L(V)$ is given on pure tensors by $x \otimes v \mapsto L(f_v)(x)$, where $f_v: K \to V$ is the $K$-linear map sending $1 \mapsto v$. When $V = K$, this is the isomorphism $L(K) \otimes_K K \to L(K)$ given by multiplication by $K$. For general $V$, it is an isomorphism by the case $V = K$ and the fact that $L$ commutes with finite products (Remark \ref{remark-linear-functor}). \end{proof} \noindent For a ring $R$ and an $R$-module $M$, let $R[M]$ be the $R$-algebra whose underlying $R$-module is $R \oplus M$ and whose multiplication is given by $(r, m) \cdot (r', m') = (rr', rm' + r'm)$. When $M = R$ this is the ring of dual numbers over $R$, which we denote by $R[\epsilon]$. \medskip\noindent Now let $S$ be a ring and assume $R$ is an $S$-algebra. Then the assignment $M \mapsto R[M]$ determines a functor $\text{Mod}_R \to S\text{-Alg}/R$, where $S\text{-Alg}/R$ denotes the category of $S$-algebras over $R$. Note that $S\text{-Alg}/R$ admits finite products: if $A_1 \to R$ and $A_2 \to R$ are two objects, then $A_1 \times_R A_2$ is a product. \begin{lemma} \label{lemma-preserves-products} Let $R$ be an $S$-algebra. Then the functor $\text{Mod}_R \to S\text{-Alg}/R$ described above preserves finite products. \end{lemma} \begin{proof} This is merely the statement that if $M$ and $N$ are $R$-modules, then the map $R[M \times N] \to R[M] \times_R R[N]$ is an isomorphism in $S\text{-Alg}/R$. \end{proof} \begin{lemma} \label{lemma-tangent-space-functor} Let $R$ be an $S$-algebra, and let $\mathcal{C}$ be a strictly full subcategory of $S\text{-Alg}/R$ containing $R[M]$ for all $M \in \Ob(\text{Mod}^{fg}_R)$. Let $F: \mathcal{C} \to \textit{Sets}$ be a functor. Suppose that $F(R)$ is a one element set and that for any $M, N \in \Ob(\text{Mod}^{fg}_R)$, the induced map $$F(R[M] \times_R R[N]) \to F(R[M]) \times F(R[N])$$ is a bijection. Then $F(R[M])$ has a natural $R$-module structure for any $M \in \Ob(\text{Mod}^{fg}_R)$. \end{lemma} \begin{proof} Note that $R \cong R[0]$ and $R[M] \times_R R[N] \cong R[M \times N]$ hence $R$ and $R[M] \times_R R[N]$ are objects of $\mathcal{C}$ by our assumptions on $\mathcal{C}$. Thus the conditions on $F$ make sense. The functor $\text{Mod}_R \to S\text{-Alg}/R$ of Lemma \ref{lemma-preserves-products} restricts to a functor $\text{Mod}^{fg}_R \to \mathcal{C}$ by the assumption on $\mathcal{C}$. Let $L$ be the composition $\text{Mod}^{fg}_R \to \mathcal{C} \to \textit{Sets}$, i.e., $L(M) = F(R[M])$. Then $L$ preserves finite products by Lemma \ref{lemma-preserves-products} and the assumption on $F$. Hence Lemma \ref{lemma-linear-functor} shows that $L(M) = F(R[M])$ has a natural $R$-module structure for any $M \in \Ob(\text{Mod}^{fg}_R)$. \end{proof} \begin{definition} \label{definition-tangent-space-over-R} Let $\mathcal{C}$ be a category as in Lemma \ref{lemma-tangent-space-functor}. Let $F : \mathcal{C} \to \textit{Sets}$ be a functor such that $F(R)$ is a one element set. The {\it tangent space $TF$ of $F$} is $F(R[\epsilon])$. \end{definition} \noindent When $F : \mathcal{C} \to \textit{Sets}$ satisfies the hypotheses of Lemma \ref{lemma-tangent-space-functor}, the tangent space $TF$ has a natural $R$-module structure. \begin{example} \label{example-tangent-space-functor} Since $\mathcal{C}_\Lambda$ contains all $k[V]$ for finite dimensional vector spaces $V$ we see that Definition \ref{definition-tangent-space-over-R} applies with $S = \Lambda$, $R = k$, $\mathcal{C} = \mathcal{C}_\Lambda$, and $F : \mathcal{C}_\Lambda \to \textit{Sets}$ a predeformation functor. The tangent space is $TF = F(k[\epsilon])$. \end{example} \begin{example} \label{example-tangent-space-prorepresentable-functor} Let us work out the tangent space of Example \ref{example-tangent-space-functor} when