Tor-amplitude [0, 1] in the setting of intersection theory on a regular surface?

Question

The question is coming from Definition 1.5 in Deligne's Expose X in SGA 7 on intersection theory.

Let $X$ be a connected regular scheme of dimension $2$ and $Y \subset X$ a reduced divisor that admits a proper morphism $Y \rightarrow \mathrm{Spec}(k)$ for some field $k$. Suppose also that $D$ and $E$ are two effective divisors on $X$ with the reduced closed scheme associated to $D$ being a subscheme of $Y$. The definition mentioned above defines the intersection number of $D$ and $E$ (with respect to $k$), and the definition involves $\mathcal{O}_D \otimes^{\mathbb{L}} \mathcal{O}_E$ (in the derived category of $\mathcal{O}_X$-modules). The claim is that this complex has vanishing cohomology in degrees other than $0$ and $1$, i.e., that $\mathrm{Tor}^n_{\mathcal{O}_X}(\mathcal{O}_D, \mathcal{O}_E) = 0$ for $n \ge 2$. Why is this so?

Do we somehow know that $\mathcal{O}_D$ is of tor-amplitude $[0, 1]$ because it is supported on a $1$-dimensional closed subscheme? Is there some relation of this sort between tor-amplitude and the dimension of the support?

Answer 1

You did not explicitly ask the following, but it is related to your second question, and it comes up in practice quite often. So I would like to state this at any rate. Let $X$ be a locally Noetherian scheme. Let $D\subset X$ be a closed subscheme such that there exists a complex $E^\bullet$ of (finite rank) locally free $\mathcal{O}_X$-modules concentrated in degrees $[-c,0]$, and there exists a chain homomorphism, $$\phi:E^\bullet \to \mathcal{O}_D[0],$$ such that the induced map $$h^0(\phi):h^0(E^\bullet) \to \mathcal{O}_D$$ is an isomorphism. Moreover, assume that the restriction of $E^\bullet$ to $X\setminus D$ is acyclic. Then every irreducible component of $D$ has "codimension $\leq c$ in $X$".

Precisely, let $\eta$ be any generic point of $D$. Form the Noetherian local ring $R=\mathcal{O}_{X,\eta}$. Then the stalk $E^\bullet_\eta$ satisfies the hypotheses of the New Intersection Theorem. Therefore, the New Intersection Theorem says that the Krull dimension of $R$ is $\leq c$.

How does this come up? Here is a typical application. Let $Y$ be a regular locally Noetherian scheme, and let $C\subset Y$ be a closed subscheme that is Cohen-Macaulay and everywhere has codimension $c$. Then (at least locally), there exists a locally free resolution $$\psi:F^\bullet \xrightarrow{\text{qism}} \mathcal{O}_C[0],$$ with $F^\bullet$ concentrated in degrees $[-c,0]$. Now let $f:X\to Y$ be a morphism, let $D$ be $X\times_Y C$, let $E^\bullet$ be $f^*F^\bullet$, and let $\phi$ be $f^*\psi$. Then $E^\bullet$ and $\phi$ satisfy the hypotheses from above. Therefore every irreducible component of $D$ has codimension $\leq c$ in $X$. Note that this setup is more general than local complete intersection morphisms, the class of morphisms that arises most often in intersection theory.