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 as cycles on $X$.
\[ U \cdot (V \cdot W) = (U \cdot V) \cdot W \]
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. $\square$
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)