nLab Serre intersection formula

Idea

Serre’s intersection formula is a refinement of the intersection theory statement of Bézout's theorem. By replacing the plain tensor product of structure sheaves of intersecting subvarieties with the derived tensor product – the Tor – it holds also when the subvarieties do not intersect transversally.

This may be understood as a first step towards formulating intersection theory properly in derived algebraic geometry.

Statement

Given a regular scheme $X$ and subschemes $Y,Z$ with defining ideal sheaves $\mathcal{I},\mathcal{J}$, respectively, the intersection multiplicity $m(x,Y,Z)$ at a generic point $x$ of (an irreducible component of) the intersection $Y\cap Z$ is given by the Serre intersection (or multiplicity) formula in terms of torsion groups:

$$m(x,Y,Z) = \sum_{i\geq 0} (-1)^i length_{\mathcal{O}_{X,x}} Tor_i^{\mathcal{O}_{X,x}} (\mathcal{O}_{X,x}/\mathcal{I}_x,\mathcal{O}_{X,x}/\mathcal{J}_x)$$

The formula can be interpreted by the Hochschild homology, or within the derived category of coherent sheaves on $X$. It has been one of the motivating results for the development of derived algebraic geometry.

References

• MathOverflow: serre-intersection-formula-and-derived-algebraic-geometry