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The Stacks project

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


Comments (2)

Comment #7548 by Hao Peng on

[Typo]In the last paragraph of the proof, should be replaced by .


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