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?