nLab
Serre intersection formula

Contents

Contents

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

m(x,Y,Z)= i0(1) ilength 𝒪 X,xTor i 𝒪 X,x(𝒪 X,x/ x,𝒪 X,x/𝒥 x) 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 XX. It has been one of the motivating results for the development of derived algebraic geometry.

References

Last revised on December 18, 2018 at 16:48:30. See the history of this page for a list of all contributions to it.