### 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

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

$\dim _\delta (\text{Supp}(H^ i(\mathcal{F}, \varphi , \psi ))) \leq k$ 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.
$\square$

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