Quadratic complete intersections

Abstract

We study Betti numbers of graded finitely generated modules over a quadratic complete intersection. In the case of codimension 1, we give a natural class of quadratic forms Q whose Clifford algebras are division rings, and deduce the possible Betti numbers of modules over the hypersurfaces . Our approach leads to a new version of the Betti degree Conjecture. In higher codimensions, we obtain formulas for the Betti numbers in terms of the ranks of certain free modules in a higher matrix factorization.