# stacks/stacks-project

Another sanity check for localized chern classes

 @@ -5215,14 +5215,28 @@ \section{Bivariant intersection theory} on $A^*(X)$ is commutative, but we will see that chern classes live in its center. \begin{remark} \label{remark-restriction-bivariant} Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X \to Y$ and $Y' \to Y$ be morphisms of schemes locally of finite type over $S$. Let $X' = Y' \times_Y X$. Then there is an obvious restriction map $$A^p(X \to Y) \longrightarrow A^p(X' \to Y'),\quad c \longmapsto c'$$ obtained by viewing a scheme $Y''$ locally of finite type over $Y'$ as a scheme locally of finite type over $Y$ and settting $c' \cap \alpha'' = c \cap \alpha''$ for any $\alpha'' \in A_k(Y'')$. This restriction operation is compatible with compositions in an obvious manner. \end{remark} \begin{remark} \label{remark-pullback-cohomology} Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $f : X \to Y$ be a morphism of schemes locally of finite type over $S$. Then there is a canonical $\mathbf{Z}$-algebra map $A^*(Y) \to A^*(X)$. Namely, given $c \in A^p(Y)$ and $X' \to X$, then we can let $f^*c$ be defined by the map $c \cap - : A_k(X') \to A_{k - p}(X')$ which is given by thinking of $X'$ as a scheme over $Y$. Let $f : Y' \to Y$ be a morphism of schemes locally of finite type over $S$. As a special case of Remark \ref{remark-restriction-bivariant} there is a canonical $\mathbf{Z}$-algebra map $f^* : A^*(Y) \to A^*(Y')$. \end{remark} \begin{lemma} @@ -7433,28 +7447,28 @@ \section{Preparation for localized chern classes} $c' = (E' \to Z)_* \circ c'_2 \circ c'_1$ with $c'_1 \in A^0(W'_\infty \to X)$ and $c'_2 \in A^p(E' \to W'_\infty)$. By Lemma \ref{lemma-gysin-at-infty-independent} we have $g_{\infty, *} \circ c'_1 = c_1$. Above $E' \subset W'_\infty$ denotes the inverse image of $Z$ in $W'_\infty$, hence $E'$ is also $g_{\infty, *} \circ c'_1 = c_1$. Here $E' \subset W'_\infty$ denotes the inverse image of $Z$ in $W'_\infty$. Hence $E'$ is also the inverse image of $E$ in $W'_\infty$ by $g_\infty$. Since moreover $Q' = g^*Q$ we find that $c'_2$ is simply the restriction of $c_2$ to schemes lying over $W'_\infty$. Thus we obtain restriction of $c_2$ to schemes lying over $W'_\infty$, see Remark \ref{remark-restriction-bivariant}. Thus we obtain \begin{align*} c' \cap \alpha & = (E' \to Z)_*(c'_2 \cap c'_1 \cap \alpha) \\ & = (E \to Z)_*(E' \to E)_*(c'_2 \cap c'_1 \cap \alpha) \\ (E \to Z)_*(E' \to E)_*(c_2 \cap c'_1 \cap \alpha) \\ & = (E \to Z)_*(c_2 \cap g_{\infty, *}(c'_1 \cap \alpha)) \\ & = (E \to Z)_*(c_2 \cap c_1 \cap \alpha) \\ & = c \cap \alpha \end{align*} In the third equality we used the relationship between $c'_2$ and $c_2$ and the fact that $c_2$ commutes with proper pushforward as it is a bivariant class. This proves the independence. In the third equality we used that $c_2$ commutes with proper pushforward as it is a bivariant class. \end{proof} @@ -7532,10 +7546,21 @@ \section{Localized chern classes} \begin{lemma} \label{lemma-independent-loc-chern} The localized classes constructed above are independent of the choices. The localized classes constructed above are independent of choices. \end{lemma} \begin{proof} \begin{proof}[First proof] The only choice we made above was the choice of the bounded complex $\mathcal{E}^\bullet$ of finite locally free $\mathcal{O}_X$-modules representing $E$. However, in More on Flatness, Lemma \ref{flat-lemma-complex-and-divisor-derived} we have seen that the blowing up $b : W \to \mathbf{P}^1_X$ and the isomorphism class $Q$ of $\mathcal{Q}^\bullet$ in $D(\mathcal{O}_W)$ only depends on the isomorphism class of $Lp^*E$ in $D(\mathcal{O}_{\mathbf{P}^1_X})$. \end{proof} \begin{proof}[Second proof] The only choice we made above was the choice of the bounded complex $\mathcal{E}^\bullet$ of finite locally free $\mathcal{O}_X$-modules representing $E$. Let us temporarily denote @@ -7563,18 +7588,10 @@ \section{Localized chern classes} } $$where all morphisms are blowing ups which are isomorphisms over \mathbf{A}^1_X\footnote{In fact, this step isn't necessary as the reader can show that although the ideals defining the blowups b and b' are not the same, the blowing ups W and W' are isomorphic over \mathbf{P}^1_X. Namely, a simple local calculation shows the ideals defining b and b' will be off locally on \mathbf{P}^1_X by a power of the ideal defining the effective Cartier divisor (\mathbf{P}^1_X)_\infty. We won't use this, if only to demonstrate that there is a lot of flexibility in our construction.}. In fact, the lemma shows that W'' is the strict transform of W with respect to the blowing up b' and W'' is the strict transform of W' with respect to the blowing up b. \mathbf{A}^1_X. Note that W'' is the strict transform of W with respect to the blowing up b' and W'' is the strict transform of W' with respect to the blowing up b (see Divisors, Lemma \ref{divisors-lemma-strict-transform}). \medskip\noindent By Lemma \ref{lemma-localized-chern-pre-independent} @@ -7604,6 +7621,58 @@ \section{Localized chern classes} some bounded complex of finite locally free modules, we conclude. \end{proof} \noindent Here is another sanity check. \begin{lemma} \label{lemma-base-change-loc-chern} In the situation above let f : X' \to X be a morphism of schemes which is locally of finite type. Denote E' = Lf^*E and Z' = f^{-1}(Z). Then the bivariant class$$ P_p(Z' \to X', E') \in A^p(Z' \to X'), \quad\text{resp.}\quad c_p(Z' \to X', E') \in A^p(Z' \to X') $$constructed above using X', Z', E' is the restriction (Remark \ref{remark-restriction-bivariant}) of the bivariant class P_p(Z \to X, E) \in A^p(Z \to X), resp.\ c_p(Z \to X, E) \in A^p(Z \to X). \end{lemma} \begin{proof} Choose a bounded complex \mathcal{E}^\bullet of finite locally free \mathcal{O}_X-modules representing E. Denote (\mathcal{E}')^\bullet = f^*\mathcal{E}^\bullet. Observe that \mathbf{P}^1_{X'} \to \mathbf{P}^1_X is a morphism of schemes such that the pullback of the effective Cartier divisor (\mathbf{P}^1_X)_\infty is the effective Cartier divisor (\mathbf{P}^1_{X'})_\infty. By More on Flatness, Lemma \ref{flat-lemma-complex-and-divisor-blowup-base-change} we obtain a commutative diagram$$ \xymatrix{ W' \ar[rd]_{b'} \ar[r]_-g & \mathbf{P}^1_{X'} \times_{\mathbf{P}^1_X} W \ar[d]_r \ar[r]_-q & W \ar[d]^b \\ & \mathbf{P}^1_{X'} \ar[r] & \mathbf{P}^1_X } $$such that W' is the strict transform of \mathbf{P}^1_{X'} with respect to b and such that (\mathcal{Q}')^\bullet = g^*q^*\mathcal{Q}^\bullet. The restriction of the bivariant class P_p(Z \to X, E), resp.\ c_p(Z \to X, E) corresponds to the class constructed in Lemma \ref{lemma-localized-chern-pre} using the proper morphism r and the complex q^*\mathcal{Q}^\bullet. On the other hand, the bivariant class P_p(Z' \to X', E'), resp.\ c_p(Z' \to X', E') corresponds to the proper morphism b' and the complex (\mathcal{Q}')^\bullet. Thus we conclude by Lemma \ref{lemma-localized-chern-pre-independent}. \end{proof} \begin{definition} \label{definition-localized-chern} Let (S, \delta) be as in Situation \ref{situation-setup}. Let X be a scheme 116 flat.tex  @@ -13532,7 +13532,7 @@ \section{Blowing up complexes, II} \begin{lemma} \label{lemma-complex-and-divisor-blowup} In Situation \ref{situation-complex-and-divisor} let b : X' \longrightarrow X In Situation \ref{situation-complex-and-divisor} let b : X' \to X be the blowing up of the product of the ideals \mathcal{J}_i from Remark \ref{remark-complex-and-divisor-ideal}. Denote D' = b^{-1}D with ideal sheaf @@ -13622,9 +13622,8 @@ \section{Blowing up complexes, II} D' \subset X', and \mathcal{Q}^\bullet be as in Lemma \ref{lemma-complex-and-divisor-blowup}. Let U \subset X be as in Lemma \ref{lemma-complex-and-divisor-blowup-good}. Then there exists a closed subscheme T \subset D' cut out by a finite type quasi-coherent sheaf of ideals which scheme theoretically contains D' \cap b^{-1}(U) Then there exists a closed immersion T \to D' of finite presentation with D' \cap b^{-1}(U) \subset T scheme theoretically such that \mathcal{Q}^\bullet|_T has finite locally free cohomology sheaves. \end{lemma} @@ -13657,7 +13656,7 @@ \section{Blowing up complexes, II} \medskip\noindent To glue this affine local construction, we remark that in the proof of More on Algebra, Lemma \ref{more-algebra-lemma-eta-vanishing-beta} More on Algebra, Lemma \ref{more-algebra-lemma-eta-vanishing-beta-plus} the ideal cutting out T is constructed with a certain universal property. Namely, the result of More on Algebra, Lemma \ref{more-algebra-lemma-eta-locally-free} @@ -13693,6 +13692,108 @@ \section{Blowing up complexes, II} This is obvious. \end{proof} \begin{lemma} \label{lemma-complex-and-divisor-derived} Let X be a scheme and let D \subset X be an effective Cartier divisor. Let E \in D(\mathcal{O}_X) be a perfect object. Let U \subset X be the maximal open over which the cohomology sheaves H^i(E) are locally free. There exists a proper morphism b : X' \longrightarrow X and an object Q \in D(\mathcal{O}_{X'}) with the following properties \begin{enumerate} \item D' = b^{-1}D is an effective Cartier divisor, \item Q = L\eta_{\mathcal{I}'}Lb^*E where \mathcal{I}' is the ideal sheaf of D', \item Q is a perfect object of D(\mathcal{O}_{X'}), \item there exists a closed immersion T \to D' of finite presentation with D' \cap b^{-1}(U) \subset T scheme theoretically such that Q|_T has finite locally free cohomology sheaves, \item for any open subscheme V \subset X such that E|_V can be represented by a bounded complex \mathcal{E}^\bullet of finite locally free \mathcal{O}_V-modules, the base changes of X' \to X, Q, D', and T to V are given by the constructions of Lemmas \ref{lemma-complex-and-divisor-blowup} and \ref{lemma-complex-and-divisor-blowup-T}. \end{enumerate} \end{lemma} \begin{proof} We first construct the morphism b : X' \to X by glueing the blowings up constructed over opens V \subset X as in (5). By Constructions, Lemma \ref{constructions-lemma-relative-glueing} to do this it suffices to show that given V \subset X open and two bounded complexes \mathcal{E}^\bullet and (\mathcal{E}')^\bullet of finite locally free \mathcal{O}_V-modules representing E|_V the resulting blowing ups are canonically isomorphic. To do this, it suffices, by the universal property of blowing up of Divisors, Lemma \ref{divisors-lemma-universal-property-blowing-up}, to show that the ideals \mathcal{J}_i and \mathcal{J}'_i from Remark \ref{remark-complex-and-divisor-ideal} constructed using \mathcal{E}^\bullet and (\mathcal{E}')^\bullet locally differ by multiplication by an invertible ideal. We will in fact show that they differ locally by a power of the ideal sheaf \mathcal{I} of D. By More on Algebra, Lemma \ref{more-algebra-lemma-compare-representatives-perfect} working locally it suffices to prove the relationship when$$ (\mathcal{E}')^\bullet = \mathcal{E}^\bullet \oplus ( \ldots \to 0 \to \mathcal{O}_V \xrightarrow{1} \mathcal{O}_V \to 0 \to \ldots) $$with the two summands \mathcal{O}_V placed in degrees i and i + 1 say. Computing minors explicitly one finds that \mathcal{J}'_{i + 1} = \mathcal{I}\mathcal{J}_{i + 1} and all other ideals stay the same. \medskip\noindent Thus we have the morphism b : X' \to X agreeing locally with the blowing ups in (5). Of course this immediately gives us the effective Cartier divisor D' = b^{-1}D, its invertible ideal sheaf \mathcal{I}' and the object Q = L\eta_{\mathcal{I}'}Lb^*E. See Remark \ref{remark-Leta} for the construction of L\eta_{\mathcal{I}'}. Since the construction commutes with restricting to opens we find that Q|_{V'} is represented by the complex \mathcal{Q}^\bullet over the open V' = b^{-1}(V) constructed using \mathcal{E}^\bullet over V. \medskip\noindent To finish the proof it suffices to show that the closed subschemes T_V \subset V' constructed in Lemma \ref{lemma-complex-and-divisor-blowup-T} glue. Again by relative glueing, it suffices to show that the construction of T does not depend on the choice of the complex \mathcal{E}^\bullet representing E|_V. Again we reduce to the case where$$ (\mathcal{E}')^\bullet = \mathcal{E}^\bullet \oplus ( \ldots \to 0 \to \mathcal{O}_V \xrightarrow{1} \mathcal{O}_V \to 0 \to \ldots) $$with the two summands \mathcal{O}_V placed in degrees i and i + 1 say. Note that in this case (\mathcal{Q}')^\bullet and \mathcal{Q}^\bullet differ as follows$$ (\mathcal{Q}')^\bullet = \mathcal{Q}^\bullet \oplus ( \ldots \to 0 \to (\mathcal{I}')^{i + 1}|_{V'} \xrightarrow{1} (\mathcal{I}')^{i + 1}|_{V'} \to 0 \to \ldots) $$In the proof of Lemma \ref{lemma-complex-and-divisor-blowup-T} we defined T \subset D' as the largest closed subscheme of D' such that \mathcal{Q}^i|_T is a direct sum of two parts compatible with the restriction to T of the canonical split injective maps$$ c^i : \mathcal{Q}^i \longrightarrow (\mathcal{I}')^ib^*\mathcal{E}^i \oplus (\mathcal{I}')^{i + 1}b^*\mathcal{E}^{i + 1} $$for all i. The direct sum decomposition for (\mathcal{Q}')^\bullet in terms of \mathcal{Q}^\bullet and the explicit complex (\mathcal{I}')^{i + 1}|_{V'} \to (\mathcal{I}')^{i + 1}|_{V'} implies in a straightforward manner that T plays the same role for (\mathcal{Q}')^\bullet and the proof is complete. \end{proof} @@ -13731,11 +13832,10 @@ \section{Blowing up complexes, III} (\mathbf{P}^1_X)_\infty \subset \mathbf{P}^1_X and the bounded complex p^*\mathcal{E}^\bullet of finite locally free modules. We also denote$$ \mathcal{Q}^\bullet = \eta_\infty b^*p^*\mathcal{E}^\bullet \mathcal{Q}^\bullet = \eta_\mathcal{I} b^*p^*\mathcal{E}^\bullet $$the complex considered in Lemma \ref{lemma-complex-and-divisor-blowup} where \eta_\infty = \eta_\mathcal{I} is the operator of Section \ref{section-eta} where \eta_\mathcal{I} is the operator of Section \ref{section-eta} associated to the ideal sheaf \mathcal{I} of the effective Cartier divisor W_\infty = b^{-1}(\mathbf{P}^1_X)_\infty on W.  Some details omitted. \end{proof} \begin{lemma} \label{lemma-compare-representatives-perfect} Let R be a ring. Let \mathfrak p \subset R be a prime. Let M^\bullet and N^\bullet be bounded complexes of finite projective R-modules representing the same object of D(R). Then there exists an f \in R, f \not \in \mathfrak p such that there is an isomorphism (!) of complexes$$ M^\bullet_f \oplus P^\bullet \cong N^\bullet_f \oplus Q^\bullet  where $P^\bullet$ and $Q^\bullet$ are finite direct sums of trivial complexes, i.e., complexes of the form the form $\ldots \to 0 \to R_f \xrightarrow{1} R_f \to 0 \to \ldots$ (placed in arbitrary degrees). \end{lemma} \begin{proof} If we have an isomorphism of the type described over the localization $R_\mathfrak p$, then using that $R_\mathfrak p = \colim R_f$ (Algebra, Lemma \ref{algebra-lemma-localization-colimit}) we can descend the isomorphism to an isomorphism over $R_f$ for some $f$. Thus we may assume $R$ is local and $\mathfrak p$ is the maximal ideal. In this case the result follows from the uniqueness of a minimal'' complex representing a perfect object, see Lemma \ref{lemma-lift-pseudo-coherent-from-residue-field}, and the fact that any complex is a direct sum of a trivial complex and a minimal one (Algebra, Lemma \ref{algebra-lemma-add-trivial-complex}). \end{proof} \begin{lemma} \label{lemma-lift-complex-finite-projectives} Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $E^\bullet$