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.
Given a regular scheme and subschemes with defining ideal sheaves , respectively, the intersection multiplicity at a generic point of (an irreducible component of) the intersection is given by the Serre intersection (or multiplicity) formula in terms of torsion groups:
The formula can be interpreted by the Hochschild homology, or within the derived category of coherent sheaves on . It has been one of the motivating results for the development of derived algebraic geometry.
Last revised on December 18, 2018 at 21:48:30. See the history of this page for a list of all contributions to it.