The Stacks project

Proof. Let $\{ W_ j\} _{j \in J}$ be the collection of irreducible components of $\text{Supp}(\mathcal{F})$ which have $\delta$-dimension $k + 1$. Note that $\{ W_ j\}$ is a locally finite collection of closed subsets of $X$ by Lemma 42.10.1. For every $j$, let $\xi _ j \in W_ j$ be the generic point. Set

See Definition 42.68.13 for notation. We claim that

If we prove this then the lemma follows.

Let $Z \subset X$ be an integral closed subscheme of $\delta$-dimension $k$. To prove the equality above it suffices to show that the coefficient $n$ of $[Z]$ in $[H^0(\mathcal{F}, \varphi , \psi )]_ k - [H^1(\mathcal{F}, \varphi , \psi )]_ k$ is the same as the coefficient $m$ of $[Z]$ in $\sum (W_ j \to X)_*\text{div}(f_ j)$. Let $\xi \in Z$ be the generic point. Consider the local ring $A = \mathcal{O}_{X, \xi }$. Let $M = \mathcal{F}_\xi$ as an $A$-module. Denote $\varphi , \psi : M \to M$ the action of $\varphi , \psi$ on the stalk. By our choice of $\xi \in Z$ we have $\delta (\xi ) = k$ and hence $\dim (\text{Supp}(M)) = 1$. Finally, the integral closed subschemes $W_ j$ passing through $\xi$ correspond to the minimal primes $\mathfrak q_ i$ of $\text{Supp}(M)$. In each case the element $f_ j \in R(W_ j)^*$ corresponds to the element $\det _{\kappa (\mathfrak q_ i)}(M_{\mathfrak q_ i}, \varphi , \psi )$ in $\kappa (\mathfrak q_ i)^*$. Hence we see that

and

Thus the result follows from Proposition 42.68.43. $\square$

Proof. By Lemma 42.23.2 we have $Z_ k(X) = K_0(\textit{Coh}_{\leq k}(X)/\textit{Coh}_{\leq k - 1}(X))$ compatible with the map of Lemma 42.23.4. Thus, suppose we have an element $[A] - [B]$ of $K_0(\textit{Coh}_{\leq k}(X)/\textit{Coh}_{\leq k - 1}(X))$ which maps to zero in $B_ k(X)$, i.e., maps to zero in $K_0(\textit{Coh}_{\leq k + 1}(X)/\textit{Coh}_{\leq k - 1}(X))$. We have to show that $[A] - [B]$ corresponds to a cycle rationally equivalent to zero on $X$. Suppose $[A] = [\mathcal{A}]$ and $[B] = [\mathcal{B}]$ for some coherent sheaves $\mathcal{A}, \mathcal{B}$ on $X$ supported in $\delta$-dimension $\leq k$. The assumption that $[A] - [B]$ maps to zero in the group $K_0(\textit{Coh}_{\leq k + 1}(X)/\textit{Coh}_{\leq k - 1}(X))$ means that there exists coherent sheaves $\mathcal{A}', \mathcal{B}'$ on $X$ supported in $\delta$-dimension $\leq k - 1$ such that $[\mathcal{A} \oplus \mathcal{A}'] - [\mathcal{B} \oplus \mathcal{B}']$ is zero in $K_0(\textit{Coh}_{k + 1}(X))$ (use part (1) of Homology, Lemma 12.11.3). By part (2) of Homology, Lemma 12.11.3 this means there exists a $(2, 1)$-periodic complex $(\mathcal{F}, \varphi , \psi )$ in the category $\textit{Coh}_{\leq k + 1}(X)$ such that $\mathcal{A} \oplus \mathcal{A}' = H^0(\mathcal{F}, \varphi , \psi )$ and $\mathcal{B} \oplus \mathcal{B}' = H^1(\mathcal{F}, \varphi , \psi )$. By Lemma 42.69.2 this implies that

This proves that $[A] - [B]$ maps to a cycle rationally equivalent to zero by the map

of Lemma 42.23.2. This is what we had to prove and the proof is complete. $\square$

Proof. Since $M$ has no embedded associated primes, and since the support of $M$ is $\mathop{\mathrm{Spec}}(A)$ we see that $\text{Ass}(M) = \{ \mathfrak q_1, \ldots , \mathfrak q_ t\}$. Hence $a$, $b$ are nonzerodivisors on $M$. Note that

as the injective map $b : M \to bM$ maps $(a/b)IM$ to $aIM$ and the injective map $a : M \to aM$ maps $IM$ to $aIM$. Hence the left hand side of the equation of the lemma is equal to

Applying the second formula of Lemma 42.3.2 with $x = a, b$ respectively and using Algebra, Definition 10.121.2 of the $\text{ord}$-functions we get the result. $\square$

Proof. Recall from Divisors, Lemma 31.24.5 the existence of injective maps $1 : \mathcal{I}\mathcal{F} \to \mathcal{F}$ and $s : \mathcal{I}\mathcal{F} \to \mathcal{F} \otimes _{\mathcal{O}_ X}\mathcal{L}$ whose cokernels are supported on a closed nowhere dense subsets $T$. Denote $\mathcal{Q}_ i$ there cokernels as in the lemma. We conclude that $\dim _\delta (\text{Supp}(\mathcal{Q}_ i)) \leq k$. By Divisors, Lemmas 31.23.5 and 31.23.8 the pullbacks $s_ i$ are defined and are regular meromorphic sections for $\mathcal{L}|_{X_ i}$. The equality of cycles in (4) implies the equality of cycle classes in (4). Hence the only remaining thing to show is that

holds in $Z_ k(X)$. To see this, let $Z \subset X$ be an integral closed subscheme of $\delta$-dimension $k$. Let $\xi \in Z$ be the generic point. Let $A = \mathcal{O}_{X, \xi }$ and $M = \mathcal{F}_\xi$. Moreover, choose a generator $s_\xi \in \mathcal{L}_\xi$. Then we can write $s = (a/b) s_\xi$ where $a, b \in A$ are nonzerodivisors. In this case $I = \mathcal{I}_\xi = \{ x \in A \mid x(a/b) \in A\}$. In this case the coefficient of $[Z]$ in the left hand side is

and the coefficient of $[Z]$ in the right hand side is

where $\mathfrak q_1, \ldots , \mathfrak q_ t$ are the minimal primes of the $1$-dimensional local ring $A$. Hence the result follows from Lemma 42.69.5. $\square$

Proof. Let

be the short exact sequence constructed in Divisors, Lemma 31.4.6. This in particular means that $\mathcal{F}'$ has no embedded associated points. Since the support of $\mathcal{K}$ is nowhere dense in the support of $\mathcal{F}$ we see that $\dim _\delta (\text{Supp}(\mathcal{K})) \leq k$. We may re-apply Divisors, Lemma 31.4.6 starting with $\mathcal{K}$ to get a short exact sequence

where now $\dim _\delta (\text{Supp}(\mathcal{K}'')) < k$ and $\mathcal{K}'$ has no embedded associated points. Suppose we can prove the lemma for the coherent sheaves $\mathcal{F}'$ and $\mathcal{K}'$. Then we see from the equations

(use Lemma 42.10.4),

(use the $\otimes \mathcal{L}$ is exact) and the trivial vanishing of $[\mathcal{K}'']_{k + 1}$ and $[\mathcal{K}'' \otimes _{\mathcal{O}_ X} \mathcal{L}] - [\mathcal{K}'']$ in $K_0(\textit{Coh}_{\leq k + 1}(X)/\textit{Coh}_{\leq k - 1}(X))$ that the result holds for $\mathcal{F}$. What this means is that we may assume that the sheaf $\mathcal{F}$ has no embedded associated points.

Assume $X$, $\mathcal{F}$ as in the lemma, and assume in addition that $\mathcal{F}$ has no embedded associated points. Consider the sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ X$, the corresponding closed subscheme $i : Z \to X$ and the coherent $\mathcal{O}_ Z$-module $\mathcal{G}$ constructed in Divisors, Lemma 31.4.7. Recall that $Z$ is a locally Noetherian scheme without embedded points, $\mathcal{G}$ is a coherent sheaf without embedded associated points, with $\text{Supp}(\mathcal{G}) = Z$ and such that $i_*\mathcal{G} = \mathcal{F}$. Moreover, set $\mathcal{N} = \mathcal{L}|_ Z$.

By Divisors, Lemma 31.25.4 the invertible sheaf $\mathcal{N}$ has a regular meromorphic section $s$ over $Z$. Let us denote $\mathcal{J} \subset \mathcal{O}_ Z$ the sheaf of denominators of $s$. By Lemma 42.69.6 there exist short exact sequences

such that $\dim _\delta (\text{Supp}(\mathcal{Q}_ i)) \leq k$ and such that the cycle $[\mathcal{Q}_2]_ k - [\mathcal{Q}_1]_ k$ is a representative of $c_1(\mathcal{N}) \cap [\mathcal{G}]_{k + 1}$. We see (using the fact that $i_*(\mathcal{G} \otimes \mathcal{N}) = \mathcal{F} \otimes \mathcal{L}$ by the projection formula, see Cohomology, Lemma 20.54.2) that

in $K_0(\textit{Coh}_{\leq k + 1}(X)/\textit{Coh}_{\leq k - 1}(X))$. This already shows that $[\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}] - [\mathcal{F}]$ is an element of $B_ k(X)$. Moreover we have

by the above and Lemmas 42.26.4 and 42.12.4. And this agree with the image of the element under $B_ k(X) \to \mathop{\mathrm{CH}}\nolimits _ k(X)$ by definition. Hence the lemma is proved. $\square$