The Stacks project

Proof. We are going to use Lemma 43.19.4 without further mention. This implies that

where $\dim (U) = a$, $\dim (V) = b$, $\dim (W) = c$, $\dim (X) = n$. The assumptions in the lemma guarantee that the coherent sheaves in the formulae above satisfy the required bounds on dimensions of supports in order to make sense of these. Now consider the object

of the derived category $D_{\textit{Coh}}(\mathcal{O}_ X)$. We claim that the expressions obtained above for $U \cdot (V \cdot W)$ and $(U \cdot V) \cdot W$ are equal to

This will prove the lemma. By symmetry it suffices to prove one of these equalities. To do this we represent $\mathcal{O}_ U$ and $\mathcal{O}_ V \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{O}_ W$ by K-flat complexes $M^\bullet$ and $L^\bullet$ and use the spectral sequence associated to the double complex $M^\bullet \otimes _{\mathcal{O}_ X} L^\bullet$ in Homology, Section 12.25. This is a spectral sequence with $E_2$ page

converging to $H^{p + q}(K)$ (details omitted; compare with More on Algebra, Example 15.62.4). Since lengths are additive in short exact sequences we see that the result is true. $\square$