The Stacks project

Lemma 43.14.2. Let $X$ be a nonsingular variety. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_ X$-modules. The $\mathcal{O}_ X$-module $\text{Tor}_ p^{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})$ is coherent, has stalk at $x$ equal to $\text{Tor}_ p^{\mathcal{O}_{X, x}}(\mathcal{F}_ x, \mathcal{G}_ x)$, is supported on $\text{Supp}(\mathcal{F}) \cap \text{Supp}(\mathcal{G})$, and is nonzero only for $p \in \{ 0, \ldots , \dim (X)\} $.

Proof. The result on stalks was discussed above and it implies the support condition. The $\text{Tor}$'s are coherent by Lemma 43.14.1. The vanishing of negative $\text{Tor}$'s is immediate from the construction. The vanishing of $\text{Tor}_ p$ for $p > \dim (X)$ can be seen as follows: the local rings $\mathcal{O}_{X, x}$ are regular (as $X$ is nonsingular) of dimension $\leq \dim (X)$ (Algebra, Lemma 10.116.1), hence $\mathcal{O}_{X, x}$ has finite global dimension $\leq \dim (X)$ (Algebra, Lemma 10.110.8) which implies that $\text{Tor}$-groups of modules vanish beyond the dimension (More on Algebra, Lemma 15.66.19). $\square$

Comments (2)

Comment #2779 by Ko Aoki on

Typo in the proof: "he local rings" should be replaced by "the local rings".

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