The Stacks project

[Chapter V, Serre_algebre_locale]

Lemma 43.19.4. Let $X$ be a nonsingular variety. Let $\mathcal{F}$ and $\mathcal{G}$ be coherent sheaves on $X$ with $\dim (\text{Supp}(\mathcal{F})) \leq r$, $\dim (\text{Supp}(\mathcal{G})) \leq s$, and $\dim (\text{Supp}(\mathcal{F}) \cap \text{Supp}(\mathcal{G}) ) \leq r + s - \dim X$. In this case $[\mathcal{F}]_ r$ and $[\mathcal{G}]_ s$ intersect properly and

\[ [\mathcal{F}]_ r \cdot [\mathcal{G}]_ s = \sum (-1)^ p [\text{Tor}_ p^{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})]_{r + s - \dim (X)}. \]

Proof. The statement that $[\mathcal{F}]_ r$ and $[\mathcal{G}]_ s$ intersect properly is immediate. Since we are proving an equality of cycles we may work locally on $X$. (Observe that the formation of the intersection product of cycles, the formation of $\text{Tor}$-sheaves, and forming the cycle associated to a coherent sheaf, each commute with restriction to open subschemes.) Thus we may and do assume that $X$ is affine.


\[ RHS(\mathcal{F}, \mathcal{G}) = [\mathcal{F}]_ r \cdot [\mathcal{G}]_ s \quad \text{and}\quad LHS(\mathcal{F}, \mathcal{G}) = \sum (-1)^ p [\text{Tor}_ p^{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})]_{r + s - \dim (X)} \]

Consider a short exact sequence

\[ 0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0 \]

of coherent sheaves on $X$ with $\text{Supp}(\mathcal{F}_ i) \subset \text{Supp}(\mathcal{F})$, then both $LHS(\mathcal{F}_ i, \mathcal{G})$ and $RHS(\mathcal{F}_ i, \mathcal{G})$ are defined for $i = 1, 2, 3$ and we have

\[ RHS(\mathcal{F}_2, \mathcal{G}) = RHS(\mathcal{F}_1, \mathcal{G}) + RHS(\mathcal{F}_3, \mathcal{G}) \]

and similarly for LHS. Namely, the support condition guarantees that everything is defined, the short exact sequence and additivity of lengths gives

\[ [\mathcal{F}_2]_ r = [\mathcal{F}_1]_ r + [\mathcal{F}_3]_ r \]

(Chow Homology, Lemma 42.10.4) which implies additivity for RHS. The long exact sequence of $\text{Tor}$s

\[ \ldots \to \text{Tor}_1(\mathcal{F}_3, \mathcal{G}) \to \text{Tor}_0(\mathcal{F}_1, \mathcal{G}) \to \text{Tor}_0(\mathcal{F}_2, \mathcal{G}) \to \text{Tor}_0(\mathcal{F}_3, \mathcal{G}) \to 0 \]

and additivity of lengths as before implies additivity for LHS.

By Algebra, Lemma 10.62.1 and the fact that $X$ is affine, we can find a filtration of $\mathcal{F}$ whose graded pieces are structure sheaves of closed subvarieties of $\text{Supp}(\mathcal{F})$. The additivity shown in the previous paragraph, implies that it suffices to prove $LHS = RHS$ with $\mathcal{F}$ replaced by $\mathcal{O}_ V$ where $V \subset \text{Supp}(\mathcal{F})$. By symmetry we can do the same for $\mathcal{G}$. This reduces us to proving that

\[ LHS(\mathcal{O}_ V, \mathcal{O}_ W) = RHS(\mathcal{O}_ V, \mathcal{O}_ W) \]

where $W \subset \text{Supp}(\mathcal{G})$ is a closed subvariety. If $\dim (V) = r$ and $\dim (W) = s$, then this equality is the definition of $V \cdot W$. On the other hand, if $\dim (V) < r$ or $\dim (W) < s$, i.e., $[V]_ r = 0$ or $[W]_ s = 0$, then we have to prove that $RHS(\mathcal{O}_ V, \mathcal{O}_ W) = 0$ 1.

Let $Z \subset V \cap W$ be an irreducible component of dimension $r + s - \dim (X)$. This is the maximal dimension of a component and it suffices to show that the coefficient of $Z$ in $RHS$ is zero. Let $\xi \in Z$ be the generic point. Write $A = \mathcal{O}_{X, \xi }$, $B = \mathcal{O}_{X \times X, \Delta (\xi )}$, and $C = \mathcal{O}_{V \times W, \Delta (\xi )}$. By Lemma 43.19.1 we have

\[ \text{coeff of }Z\text{ in } RHS(\mathcal{O}_ V, \mathcal{O}_ W) = \sum (-1)^ i \text{length}_ B \text{Tor}_ i^ B(A, C) \]

Since $\dim (V) < r$ or $\dim (W) < s$ we have $\dim (V \times W) < r + s$ which implies $\dim (C) < \dim (X)$ (small detail omitted). Moreover, the kernel $I$ of $B \to A$ is generated by a regular sequence of length $\dim (X)$ (Lemma 43.13.3). Hence vanishing by Lemma 43.16.2 because the Hilbert function of $C$ with respect to $I$ has degree $\dim (C) < n$ by Algebra, Proposition 10.60.9. $\square$

[1] The reader can see that this is not a triviality by taking $r = s = 1$ and $X$ a nonsingular surface and $V = W$ a closed point $x$ of $X$. In this case there are $3$ nonzero $\text{Tor}$s of lengths $1, 2, 1$ at $x$.

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0B0W. Beware of the difference between the letter 'O' and the digit '0'.