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.
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.
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
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