# stacks/stacks-project

A calculation with loc chern classes

 @@ -6666,9 +6666,9 @@ \section{Additivity of chern classes} $c_i(\mathcal{E}) = c_i(\mathcal{F}) + c_1(\mathcal{L})c_{i - 1}(\mathcal{F})$. By Lemma \ref{lemma-get-rid-of-trivial-subbundle} we have $c_j(\mathcal{E} \otimes \mathcal{L}^{\otimes -1}) = c_j(\mathcal{E} \otimes \mathcal{L}^{\otimes -1})$ for $j = 0, \ldots, r$ (were we set $c_r(\mathcal{F}) = 0$ by convention). = c_j(\mathcal{F} \otimes \mathcal{L}^{\otimes -1})$for$j = 0, \ldots, r$were we set$c_r(\mathcal{F} \otimes \mathcal{L}^{-1}) = 0$by convention. Applying Lemma \ref{lemma-chern-classes-E-tensor-L} we deduce $$\sum_{j = 0}^i @@ -9759,7 +9759,14 @@ \section{Blowing up at infinity} \item \label{item-cone-is-open} E \setminus X' = E \setminus (X' \cap E) = \underline{\text{Spec}}_Z(\mathcal{C}_{Z/X, *}) = C_ZX. \underline{\text{Spec}}_Z(\mathcal{C}_{Z/X, *}) = C_ZX, \item \label{item-find-Z-in-blowup} there is a closed immersion \mathbf{P}^1_Z \to W whose composition with b is the inclusion morphism \mathbf{P}^1_Z \to \mathbf{P}^1_X and whose base change by \infty is the composition Z \to C_ZX \to E \to W_\infty where the first arrow is the vertex of the cone. \end{enumerate} We recall that \mathcal{C}_{Z/X, *} is the conormal algebra of Z in X, see Divisors, Definition \ref{divisors-definition-conormal-sheaf} and @@ -9791,6 +9798,16 @@ \section{Blowing up at infinity} (for example because both are cut out by the pullback of the ideal sheaf of Z to X'). This proves (3). \medskip\noindent The intersection of \infty(Z) with \mathbf{P}^1_Z is the effective Cartier divisor (\mathbf{P}^1_Z)_\infty hence the strict transform of \mathbf{P}^1_Z by the blowing up b maps isomorphically to \mathbf{P}^1_Z (see Divisors, Lemmas \ref{divisors-lemma-strict-transform} and \ref{divisors-lemma-blow-up-effective-Cartier-divisor}). This gives us the morphism \mathbf{P}^1_Z \to W mentioned in (8). It is a closed immersion as b is separated, see Schemes, Lemma \ref{schemes-lemma-section-immersion}. \medskip\noindent Suppose that \Spec(A) \subset X is an affine open and that Z \cap \Spec(A) corresponds to the finitely generated ideal I \subset A. @@ -9801,7 +9818,7 @@ \section{Blowing up at infinity} of (A[s], J). Observe that$$ J^n = I^n \oplus sI^{n - 1} \oplus sI^{n - 2} \ldots \oplus s^nA I^n \oplus sI^{n - 1} \oplus s^2I^{n - 2} \ldots \oplus s^nA \oplus s^{n + 1}A \oplus \ldots $$as an A-submodule of A[s] for all n \geq 0. Consider the open subscheme @@ -9876,13 +9893,22 @@ \section{Blowing up at infinity} This proves D = X' as desired. \medskip\noindent It remains to prove (6). Observe that the zero scheme of \frac{s}{f} Let us prove (6). Observe that the zero scheme of \frac{s}{f} in the previous paragraph is the restriction of the zero scheme of S on the affine open D_+(f^{(1)}). Hence we see that S = 0 defines X' \cap E on E. Thus (5) follows from (6). X' \cap E on E. Thus (6) follows from (5). \medskip\noindent Finally, we have to prove the last part of (8). This is clear because the map \mathbf{P}^1_Z \to W is affine locally given by the surjection$$ B \to B \otimes_{A[s]} A/I = (A/I \oplus I/I^2 \oplus I^2/I^3 \oplus \ldots)[S] \to A/I[S] $$and the identification \text{Proj}(A/I[S]) = \Spec(A/I). Some details omitted. @@ -10253,6 +10279,200 @@ \section{Higher codimension gysin homomorphisms} \section{Calculating some classes} \label{section-calculate} \noindent To get further we need to compute the values of some of the classes we've constructed above. \begin{lemma} \label{lemma-compute-koszul} Let (S, \delta) be as in Situation \ref{situation-setup}. Let X be a scheme locally of finite type over S. Let \mathcal{E} be a locally free \mathcal{O}_X-module of rank r. Then$$ \prod\nolimits_{n = 0, \ldots, r} c(\wedge^n \mathcal{E})^{(-1)^n} = 1 - (r - 1)! c_r(\mathcal{E}) + \ldots $$\end{lemma} \begin{proof} By the splitting principle we can turn this into a calculation in the polynomial ring on the chern roots x_1, \ldots, x_r of \mathcal{E}. See Section \ref{section-splitting-principle}. Observe that$$ c(\wedge^n \mathcal{E}) = \prod\nolimits_{1 \leq i_1 < \ldots < i_n \leq r} (1 + x_{i_1} + \ldots + x_{i_n}) $$Thus the logarithm of the left hand side of the equation in the lemma is$$ - \sum\nolimits_{p \geq 1} \sum\nolimits_{n = 0}^r \sum\nolimits_{1 \leq i_1 < \ldots < i_n \leq r} \frac{(-1)^{p + n}}{p}(x_{i_1} + \ldots + x_{i_n})^p $$Please notice the minus sign in front. However, we have$$ \sum\nolimits_{p \geq 0} \sum\nolimits_{n = 0}^r \sum\nolimits_{1 \leq i_1 < \ldots < i_n \leq r} \frac{(-1)^{p + n}}{p!}(x_{i_1} + \ldots + x_{i_n})^p = \prod (1 - e^{-x_i}) $$Hence we see that the first nonzero term in our chern class is in degree r and equal to the predicted value. \end{proof} \begin{lemma} \label{lemma-compute-section} Let (S, \delta) be as in Situation \ref{situation-setup}. Let X be a scheme locally of finite type over S. Let \mathcal{C} be a locally free \mathcal{O}_X-module of rank r. Consider the morphisms$$ X = \underline{\text{Proj}}_X(\mathcal{O}_X[T]) \xrightarrow{i} E = \underline{\text{Proj}}_X(\text{Sym}^*(\mathcal{C})[T]) \xrightarrow{\pi} X $$Then p^* \circ \pi_* \circ c_r(i_*\mathcal{O}_X) = (-1)^{r - 1}(r - 1)! j^* where j : C \to E and p : C \to X are the inclusion and structure morphism of the vector bundle C = \underline{\Spec}(\text{Sym}^*(\mathcal{C})). \end{lemma} \begin{proof} To prove the equality it suffices to assume that X is integral and prove that both sides give the same result when capping with [E], see Lemma \ref{lemma-bivariant-zero}. The canonical map \pi^*\mathcal{C} \to \mathcal{O}_E(1) vanishes exactly along i(X). Hence the Koszul complex on the map$$ \pi^*\mathcal{C} \otimes \mathcal{O}_E(-1) \to \mathcal{O}_E $$is a resolution of i_*\mathcal{O}_X. In particular we see that i_*\mathcal{O}_X is a perfect object of D(\mathcal{O}_E) whose chern classes are defined. By Lemma \ref{lemma-compute-koszul} we conclude$$ c_r(i_*\mathcal{O}_X) = - (r - 1)! c_r(\pi^*\mathcal{C} \otimes \mathcal{O}_E(-1)) $$On the other hand, by Lemma \ref{lemma-chern-classes-dual} we have$$ c_r(\pi^*\mathcal{C} \otimes \mathcal{O}_E(-1)) = (-1)^r c_r(\pi^*\mathcal{C}^\vee \otimes \mathcal{O}_E(1)) $$and \pi^*\mathcal{C}^\vee \otimes \mathcal{O}_E(1) has a section vanishing exactly along i(X). Thus c_r(\pi^*\mathcal{C}^\vee \otimes \mathcal{O}_E(1)) \cap [E] is [i(X)] by Lemma \ref{lemma-top-chern-class}. The result follows easily from this; details omitted. \end{proof} \begin{lemma} \label{lemma-agreement-with-loc-chern} Let (S, \delta) be as in Situation \ref{situation-setup}. Let i : Z \to X be a regular closed immersion of codimension r between schemes locally of finite type over S. Let \mathcal{N} = \mathcal{C}_{Z/X}^\vee be the normal sheaf. If X is quasi-compact and has the resolution property, then$$ c_r(Z \to X, i_*\mathcal{O}_Z) = (-1)^{r - 1} (r - 1)! c(Z \to X, \mathcal{N}) $$in A^r(Z \to X). The left hand side is the localized chern class of Definition \ref{definition-localized-chern}. \end{lemma} \begin{proof} For any x \in Z we can choose an affine open neighbourhood \Spec(A) \subset X such that Z \cap \Spec(A) = V(f_1, \ldots, f_r) where f_1, \ldots, f_r \in A is a regular sequence. See Divisors, Definition \ref{divisors-definition-regular-immersion} and Lemma \ref{divisors-lemma-Noetherian-scheme-regular-ideal}. Then we see that the Koszul complex on f_1, \ldots, f_r is a resolution of A/(f_1, \ldots, f_r) for example by More on Algebra, Lemma \ref{more-algebra-lemma-regular-koszul-regular}. Hence A/(f_1, \ldots, f_r) is perfect as an A-module. It follows that F = i_*\mathcal{O}_Z is a perfect object of D(\mathcal{O}_X) whose restriction to X \setminus Z is zero. Since X is quasi-compact and quasi-separated (Properties, Lemma \ref{properties-lemma-locally-Noetherian-quasi-separated}) and has the resolution property we see that F = i_*\mathcal{O}_Z can be represented by a bounded complex of finite locally free modules (Derived Categories of Schemes, Lemma \ref{perfect-lemma-construct-strictly-perfect}) and hence the chern classes of F = i_*\mathcal{O}_Z are defined (Definition \ref{definition-defined-on-perfect}). All in all we conclude that the statement makes sense. \medskip\noindent Denote b : W \to \mathbf{P}^1_X the blowing up in \infty(Z) as in Section \ref{section-blowup-Z-first}. By (\ref{item-find-Z-in-blowup}) we have a closed immersion$$ i' : \mathbf{P}^1_Z \longrightarrow W $$We claim that Q = i'_*\mathcal{O}_{\mathbf{P}^1_Z} is a perfect object of D(\mathcal{O}_W) and that F and Q satisfy the assumptions of Lemma \ref{lemma-independent-loc-chern-bQ}. \medskip\noindent Assume the claim. The output of Lemma \ref{lemma-independent-loc-chern-bQ} is that we have$$ c_p(Z \to X, F) = c'_p(Q) = (E \to Z)_* \circ c'_p(Q|_E) \circ C $$On the other hand, by construction of c(Z \to X, \mathcal{N}) and because \mathcal{C}_{Z/X} = \mathcal{N}^\vee is locally free we have$$ j^* \circ C = p^* \circ c(Z \to X, \mathcal{N}) $$where j : C_ZX \to W_\infty and p : C_ZX \to Z are the given maps. Thus the relationship follows from Lemma \ref{lemma-compute-section} applied to E \to Z. \medskip\noindent Proof of the claim. Let A and f_1, \ldots, f_r be as above. Consider the affine open \Spec(A[s]) \subset \mathbf{P}^1_X as in Section \ref{section-blowup-Z-first}. Recall that s = 0 defines (\mathbf{P}^1_X)_\infty over this open. Hence over \Spec(A[s]) we are blowing up in the ideal generated by the regular sequence s, f_1, \ldots, f_r. By More on Algebra, Lemma \ref{more-algebra-lemma-blowup-regular-sequence} the r + 1 affine charts are global complete intersections over A[s]. The chart corresponding to the affine blowup algebra$$ A[s][f_1/s, \ldots, f_r/s] = A[s, y_1, \ldots, y_r]/(sy_i - f_i) $$contains i'(Z \cap \Spec(A)) as the closed subscheme cut out by y_1, \ldots, y_r. Since y_1, \ldots, y_r, sy_1 - f_1, \ldots, sy_r - f_r is a regular sequence in the polynomial ring A[s, y_1, \ldots, y_r] we find that i' is a regular immersion. Some details omitted. As above we conclude that Q = i'_*\mathcal{O}_{\mathbf{P}^1_Z} is a perfect object of D(\mathcal{O}_W). Since W also has the resolution property (Derived Categories of Schemes, Lemma \ref{perfect-lemma-resolution-property-ample-relative}) we find that the chern classes of Q are defined. All the other assumptions on F and Q in Lemma \ref{lemma-independent-loc-chern-bQ} (and Lemma \ref{lemma-localized-chern-pre}) are immediately verified. \end{proof} \section{Gysin maps for diagonals} \label{section-gysin-for-diagonal}  @@ -7197,6 +7197,50 @@ \section{The resolution property} \ref{properties-proposition-characterize-ample}. \end{proof} \begin{lemma} \label{lemma-resolution-property-ample-relative} Let f : X \to Y be a morphism of schemes. Assume \begin{enumerate} \item Y is quasi-compact and quasi-separated and has the resolution property, \item there exists an f-ample invertible module on X. \end{enumerate} Then X has the resolution property. \end{lemma} \begin{proof} Let \mathcal{F} be a finite type quasi-coherent \mathcal{O}_X-module. Let \mathcal{L} be an f-ample invertible module. Choose an affine open covering Y = V_1 \cup \ldots \cup V_m. Set U_j = f^{-1}(V_j). By Properties, Proposition \ref{properties-proposition-characterize-ample} for each j we know there exists finitely many maps s_{j, i} : \mathcal{L}^{\otimes n_{j, i}}|_{U_j} \to \mathcal{F}|_{U_j} which are jointly surjective. Consider the quasi-coherent \mathcal{O}_Y-modules$$ \mathcal{H}_n = f_*(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}) $$We may think of s_{j, i} as a section over V_j of the sheaf \mathcal{H}_{-n_{j, i}}. Suppose we can find finite locally free \mathcal{O}_Y-modules \mathcal{E}_{i, j} and maps \mathcal{E}_{i, j} \to \mathcal{H}_{-n_{j, i}} such that s_{j, i} is in the image. Then the corresponding maps$$ f^*\mathcal{E}_{i, j} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n_{i, j}} \longrightarrow \mathcal{F}$$are going to be jointly surjective and the lemma is proved. By Properties, Lemma \ref{properties-lemma-quasi-coherent-colimit-finite-type} for each$i, j$we can find a finite type quasi-coherent submodule$\mathcal{H}'_{i, j} \subset \mathcal{H}_{-n_{j, i}}$which contains the section$s_{i, j}$over$V_j$. Thus using the resolution property of$Y$to get surjections$\mathcal{E}_{i, j} \to \mathcal{H}'_{j, i}$and we conclude. \end{proof} \begin{lemma} \label{lemma-resolution-property-goes-up-affine} Let$f : X \to Y$be an affine morphism of schemes with$Y\$ quasi-compact