Lemma 43.20.1. Let $X$ be a nonsingular variety. Let $U, V, W$ be closed subvarieties. Assume that $U, V, W$ intersect properly pairwise and that $\dim (U \cap V \cap W) \leq \dim (U) + \dim (V) + \dim (W) - 2\dim (X)$. Then

as cycles on $X$.

It is clear that proper intersections as defined above are commutative. Using the key Lemma 43.19.4 we can prove that (proper) intersection products are associative.

Lemma 43.20.1. Let $X$ be a nonsingular variety. Let $U, V, W$ be closed subvarieties. Assume that $U, V, W$ intersect properly pairwise and that $\dim (U \cap V \cap W) \leq \dim (U) + \dim (V) + \dim (W) - 2\dim (X)$. Then

\[ U \cdot (V \cdot W) = (U \cdot V) \cdot W \]

as cycles on $X$.

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

\begin{align*} V \cdot W & = \sum (-1)^ i [\text{Tor}_ i(\mathcal{O}_ V, \mathcal{O}_ W)]_{b + c - n} \\ U \cdot (V \cdot W) & = \sum (-1)^{i + j} [ \text{Tor}_ j(\mathcal{O}_ U, \text{Tor}_ i(\mathcal{O}_ V, \mathcal{O}_ W)) ]_{a + b + c - 2n} \\ U \cdot V & = \sum (-1)^ i [\text{Tor}_ i(\mathcal{O}_ U, \mathcal{O}_ V)]_{a + b - n} \\ (U \cdot V) \cdot W & = \sum (-1)^{i + j} [ \text{Tor}_ j(\text{Tor}_ i(\mathcal{O}_ U, \mathcal{O}_ V), \mathcal{O}_ W)) ]_{a + b + c - 2n} \end{align*}

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

\[ K = \mathcal{O}_ U \otimes ^\mathbf {L}_{\mathcal{O}_ X} \mathcal{O}_ V \otimes ^\mathbf {L}_{\mathcal{O}_ X} \mathcal{O}_ W \]

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

\[ \sum (-1)^ k [H^ k(K)]_{a + b + c - 2n} \]

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

\[ E_2^{p, q} = \text{Tor}_{-p}(\mathcal{O}_ U, \text{Tor}_{-q}(\mathcal{O}_ V, \mathcal{O}_ W)) \]

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. $\square$

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