# stacks/stacks-project

Update section on rank and determinant in modules

 @@ -2741,8 +2741,8 @@ \section{Flat modules} \begin{lemma} \label{lemma-flat-locally-finite-presentation} Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be locally of finite presentation and flat. Then $\mathcal{F}$ is Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be a flat $\mathcal{O}_X$-module of finite presentation. Then $\mathcal{F}$ is locally a direct summand of a finite free $\mathcal{O}_X$-module. \end{lemma} @@ -3943,16 +3943,16 @@ \section{Rank and determinant} \noindent Let $(X, \mathcal{O}_X)$ be a ringed space. Consider the category $\mathcal{C}$ of finite locally free $\mathcal{O}_X$-modules. $\textit{Vect}(X)$ of finite locally free $\mathcal{O}_X$-modules. This is an exact category (see Injectives, Remark \ref{injectives-remark-embed-exact-category}) whose admissible epimorphisms are surjections and whose admissible monomorphisms are kernels of surjections. Moreover, there is a set of isomorphism classes of objects of $\mathcal{C}$ (proof omitted). Thus we can form the Grothendieck $K$-group $K(\mathcal{C})$, which is often denoted $K_0^{naive}(X)$. Explicitly, in this case $K_0^{naive}(X)$ is the abelian group generated by $[\mathcal{E}]$ for $\mathcal{E}$ of objects of $\textit{Vect}(X)$ (proof omitted). Thus we can form the zeroth Grothendieck $K$-group $K_0(\textit{Vect}(X))$. Explicitly, in this case $K_0(\textit{Vect}(X))$ is the abelian group generated by $[\mathcal{E}]$ for $\mathcal{E}$ a finite locally free $\mathcal{O}_X$-module, subject to the relations $$[\mathcal{E}'] = [\mathcal{E}] + [\mathcal{E}''] @@ -3962,44 +3962,46 @@ \section{Rank and determinant} of finite locally free \mathcal{O}_X-modules. \medskip\noindent Ranks. Given a finite locally free \mathcal{O}_X-module \mathcal{E}, {\bf Ranks.} Assume all stalks \mathcal{O}_{X, x} are nonzero rings. Given a finite locally free \mathcal{O}_X-module \mathcal{E}, the {\it rank} is a locally constant function$$ r = r_\mathcal{E} : X \longrightarrow \mathbf{Z}_{\geq 0},\quad \text{rank}_\mathcal{E} : X \longrightarrow \mathbf{Z}_{\geq 0},\quad x \longmapsto \text{rank}_{\mathcal{O}_{X, x}} \mathcal{E}_x $$This makes sense as \mathcal{E}_x \cong \mathcal{O}_{X, x}^{\oplus r(x)} and this uniquely determines r(x). By definition of locally free modules the function r is locally constant. If See Lemma \ref{lemma-rank}. By definition of locally free modules the function \text{rank}_\mathcal{E} is locally constant. If 0 \to \mathcal{E}' \to \mathcal{E} \to \mathcal{E}'' \to 0 is a short exact sequence of finite locally free \mathcal{O}_X-modules, then r_\mathcal{E} = r_{\mathcal{E}'} + r_{\mathcal{E}''}, then \text{rank}_\mathcal{E} = \text{rank}_{\mathcal{E}'} + \text{rank}_{\mathcal{E}''}, Thus the rank defines a homomorphism$$ K_0^{naive}(X) \longrightarrow \text{Map}_{cont}(X, \mathbf{Z}),\quad K_0(\textit{Vec}(X)) \longrightarrow \text{Map}_{cont}(X, \mathbf{Z}),\quad [\mathcal{E}] \longmapsto r_\mathcal{E} $$\medskip\noindent Determinants. Given a finite locally free {\bf Determinants.} Given a finite locally free \mathcal{O}_X-module \mathcal{E} we obtain a disjoint union decomposition$$ X = X_0 \amalg X_1 \amalg X_2 \amalg \ldots $$with X_i open and closed, such that \mathcal{E} is finite locally free of rank i on X_i (this is exactly the same as saying the r_\mathcal{E} is locally constant). In this case we define \text{rank}_\mathcal{E} is locally constant). In this case we define \det(\mathcal{E}) as the invertible sheaf on X which is equal to \wedge^i(\mathcal{E}|_{X_i}) on X_i for all i \geq 0. Since the decomposition above is disjoint, there are no glueing conditions to check. By Lemma \ref{lemma-det-ses} below this defines a homomorphism$$ \det : K_0^{naive}(X) \longrightarrow \Pic(X),\quad \det : K_0(\textit{Vect}(X)) \longrightarrow \Pic(X),\quad [\mathcal{E}] \longmapsto \det(\mathcal{E})  of abelian groups. of abelian groups. The elements of $\Pic(X)$ we get in this manner are locally free of rank $1$ (see below the lemma for a generalization). \begin{lemma} \label{lemma-det-ses} @@ -4040,6 +4042,37 @@ \section{Rank and determinant} $s''_i$ to a section of $\mathcal{E}$. We omit the details. \end{proof} \noindent {\bf Determinants, reprise.} Let $(X, \mathcal{O}_X)$ be a ringed space. Instead of looking at finite locally free $\mathcal{O}_X$-modules we could look at those $\mathcal{O}_X$-modules $\mathcal{F}$ which are locally on $X$ a direct summand of a finite free $\mathcal{O}_X$-module. This is the same thing as asking $\mathcal{F}$ to be a flat $\mathcal{O}_X$-module of finite presentation, see Lemma \ref{lemma-flat-locally-finite-presentation}. For such a module in general the rank function is undefined; for example $X$ could be a point and $\Gamma(X, \mathcal{O}_X)$ could be the product $A \times B$ of two nonzero rings and $\mathcal{F}$ could correspond to $A \times 0$. On the other hand, for $\mathcal{F}$ flat and of finite presentation we can still define $\det(\mathcal{F})$ and this will be an invertible $\mathcal{O}_X$-module in the sense of Definition \ref{definition-invertible} (not necessarily locally free of rank $1$). This is done using the construction of More on Algebra, Section \ref{more-algebra-section-determinants} on the values of $\mathcal{F}$ on sufficiently small opens of $X$. If we ever need this we will precisely state and prove the relevant lemmas here.