42.69 Appendix B: Alternative approaches

In this appendix we first briefly try to connect the material in the main text with $K$-theory of coherent sheaves. In particular we describe how cupping with $c_1$ of an invertible module is related to tensoring by this invertible module, see Lemma 42.69.7. This material is obviously very interesting and deserves a much more detailed and expansive exposition.

42.69.1 Rational equivalence and K-groups

This section is a continuation of Section 42.23. The motivation for the following lemma is Homology, Lemma 12.11.3.

Lemma 42.69.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a scheme locally of finite type over $S$. Let $\mathcal{F}$ be a coherent sheaf on $X$. Let

$\xymatrix{ \ldots \ar[r] & \mathcal{F} \ar[r]^\varphi & \mathcal{F} \ar[r]^\psi & \mathcal{F} \ar[r]^\varphi & \mathcal{F} \ar[r] & \ldots }$

be a complex as in Homology, Equation (12.11.2.1). Assume that

1. $\dim _\delta (\text{Supp}(\mathcal{F})) \leq k + 1$.

2. $\dim _\delta (\text{Supp}(H^ i(\mathcal{F}, \varphi , \psi ))) \leq k$ for $i = 0, 1$.

Then we have

$[H^0(\mathcal{F}, \varphi , \psi )]_ k \sim _{rat} [H^1(\mathcal{F}, \varphi , \psi )]_ k$

as $k$-cycles on $X$.

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

$f_ j = \det \nolimits _{\kappa (\xi _ j)} (\mathcal{F}_{\xi _ j}, \varphi _{\xi _ j}, \psi _{\xi _ j}) \in R(W_ j)^*.$

See Definition 42.68.13 for notation. We claim that

$- [H^0(\mathcal{F}, \varphi , \psi )]_ k + [H^1(\mathcal{F}, \varphi , \psi )]_ k = \sum (W_ j \to X)_*\text{div}(f_ j)$

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

$n = - e_ A(M, \varphi , \psi )$

and

$m = \sum \text{ord}_{A/\mathfrak q_ i} (\det \nolimits _{\kappa (\mathfrak q_ i)}(M_{\mathfrak q_ i}, \varphi , \psi ))$

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

Lemma 42.69.3. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a scheme locally of finite type over $S$. The map

$\mathop{\mathrm{CH}}\nolimits _ k(X) \longrightarrow K_0(\textit{Coh}_{\leq k + 1}(X)/\textit{Coh}_{\leq k - 1}(X))$

from Lemma 42.23.4 induces a bijection from $\mathop{\mathrm{CH}}\nolimits _ k(X)$ onto the image $B_ k(X)$ of the map

$K_0(\textit{Coh}_{\leq k}(X)/\textit{Coh}_{\leq k - 1}(X)) \longrightarrow K_0(\textit{Coh}_{\leq k + 1}(X)/\textit{Coh}_{\leq k - 1}(X)).$

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

$[\mathcal{A} \oplus \mathcal{A}']_ k \sim _{rat} [\mathcal{B} \oplus \mathcal{B}']_ k$

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

$K_0(\textit{Coh}_{\leq k}(X)/\textit{Coh}_{\leq k - 1}(X)) \longrightarrow Z_ k(X)$

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

42.69.4 Cartier divisors and K-groups

In this section we describe how the intersection with the first Chern class of an invertible sheaf $\mathcal{L}$ corresponds to tensoring with $\mathcal{L} - \mathcal{O}$ in $K$-groups.

Lemma 42.69.5. Let $A$ be a Noetherian local ring. Let $M$ be a finite $A$-module. Let $a, b \in A$. Assume

1. $\dim (A) = 1$,

2. both $a$ and $b$ are nonzerodivisors in $A$,

3. $A$ has no embedded primes,

4. $M$ has no embedded associated primes,

5. $\text{Supp}(M) = \mathop{\mathrm{Spec}}(A)$.

Let $I = \{ x \in A \mid x(a/b) \in A\}$. Let $\mathfrak q_1, \ldots , \mathfrak q_ t$ be the minimal primes of $A$. Then $(a/b)IM \subset M$ and

$\text{length}_ A(M/(a/b)IM) - \text{length}_ A(M/IM) = \sum \nolimits _ i \text{length}_{A_{\mathfrak q_ i}}(M_{\mathfrak q_ i}) \text{ord}_{A/\mathfrak q_ i}(a/b)$

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

\begin{align*} & \text{length}_ A(M/(a/b)IM) \\ & = \text{length}_ A(bM/aIM) \\ & = \text{length}_ A(M/aIM) - \text{length}_ A(M/bM) \\ & = \text{length}_ A(M/aM) + \text{length}_ A(aM/aIM) - \text{length}_ A(M/bM) \\ & = \text{length}_ A(M/aM) + \text{length}_ A(M/IM) - \text{length}_ A(M/bM) \end{align*}

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

$\text{length}_ A(M/aM) - \text{length}_ A(M/bM).$

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. $\square$

Lemma 42.69.6. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Let $s \in \Gamma (X, \mathcal{K}_ X(\mathcal{L}))$ be a meromorphic section of $\mathcal{L}$. Assume

1. $\dim _\delta (X) \leq k + 1$,

2. $X$ has no embedded points,

3. $\mathcal{F}$ has no embedded associated points,

4. the support of $\mathcal{F}$ is $X$, and

5. the section $s$ is regular meromorphic.

In this situation let $\mathcal{I} \subset \mathcal{O}_ X$ be the ideal of denominators of $s$, see Divisors, Definition 31.23.10. Then we have the following:

1. there are short exact sequences

$\begin{matrix} 0 & \to & \mathcal{I}\mathcal{F} & \xrightarrow {1} & \mathcal{F} & \to & \mathcal{Q}_1 & \to & 0 \\ 0 & \to & \mathcal{I}\mathcal{F} & \xrightarrow {s} & \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L} & \to & \mathcal{Q}_2 & \to & 0 \end{matrix}$
2. the coherent sheaves $\mathcal{Q}_1$, $\mathcal{Q}_2$ are supported in $\delta$-dimension $\leq k$,

3. the section $s$ restricts to a regular meromorphic section $s_ i$ on every irreducible component $X_ i$ of $X$ of $\delta$-dimension $k + 1$, and

4. writing $[\mathcal{F}]_{k + 1} = \sum m_ i[X_ i]$ we have

$[\mathcal{Q}_2]_ k - [\mathcal{Q}_1]_ k = \sum m_ i(X_ i \to X)_*\text{div}_{\mathcal{L}|_{X_ i}}(s_ i)$

in $Z_ k(X)$, in particular

$[\mathcal{Q}_2]_ k - [\mathcal{Q}_1]_ k = c_1(\mathcal{L}) \cap [\mathcal{F}]_{k + 1}$

in $\mathop{\mathrm{CH}}\nolimits _ k(X)$.

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

$[\mathcal{Q}_2]_ k - [\mathcal{Q}_1]_ k = \sum m_ i(X_ i \to X)_*\text{div}_{\mathcal{L}|_{X_ i}}(s_ i)$

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

$\text{length}_ A(M/(a/b)IM) - \text{length}_ A(M/IM)$

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

$\sum \text{length}_{A_{\mathfrak q_ i}}(M_{\mathfrak q_ i}) \text{ord}_{A/\mathfrak q_ i}(a/b)$

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. $\square$

Lemma 42.69.7. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Assume $\dim _\delta (\text{Supp}(\mathcal{F})) \leq k + 1$. Then the element

$[\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}] - [\mathcal{F}] \in K_0(\textit{Coh}_{\leq k + 1}(X)/\textit{Coh}_{\leq k - 1}(X))$

lies in the subgroup $B_ k(X)$ of Lemma 42.69.3 and maps to the element $c_1(\mathcal{L}) \cap [\mathcal{F}]_{k + 1}$ via the map $B_ k(X) \to \mathop{\mathrm{CH}}\nolimits _ k(X)$.

Proof. Let

$0 \to \mathcal{K} \to \mathcal{F} \to \mathcal{F}' \to 0$

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

$0 \to \mathcal{K}'' \to \mathcal{K} \to \mathcal{K}' \to 0$

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

$[\mathcal{F}]_{k + 1} = [\mathcal{F}']_{k + 1} + [\mathcal{K}']_{k + 1} + [\mathcal{K}'']_{k + 1}$

(use Lemma 42.10.4),

$[\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}] - [\mathcal{F}] = [\mathcal{F}' \otimes _{\mathcal{O}_ X} \mathcal{L}] - [\mathcal{F}'] + [\mathcal{K}' \otimes _{\mathcal{O}_ X} \mathcal{L}] - [\mathcal{K}'] + [\mathcal{K}'' \otimes _{\mathcal{O}_ X} \mathcal{L}] - [\mathcal{K}'']$

(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

$\begin{matrix} 0 & \to & \mathcal{J}\mathcal{G} & \xrightarrow {1} & \mathcal{G} & \to & \mathcal{Q}_1 & \to & 0 \\ 0 & \to & \mathcal{J}\mathcal{G} & \xrightarrow {s} & \mathcal{G} \otimes _{\mathcal{O}_ Z} \mathcal{N} & \to & \mathcal{Q}_2 & \to & 0 \end{matrix}$

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

$[\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}] - [\mathcal{F}] = [i_*\mathcal{Q}_2] - [i_*\mathcal{Q}_1]$

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

\begin{eqnarray*} [i_*\mathcal{Q}_2]_ k - [i_*\mathcal{Q}_1]_ k & = & i_*\left( [\mathcal{Q}_2]_ k - [\mathcal{Q}_1]_ k \right) \\ & = & i_*\left(c_1(\mathcal{N}) \cap [\mathcal{G}]_{k + 1} \right) \\ & = & c_1(\mathcal{L}) \cap i_*[\mathcal{G}]_{k + 1} \\ & = & c_1(\mathcal{L}) \cap [\mathcal{F}]_{k + 1} \end{eqnarray*}

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. $\square$

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