Peder Thompson: Serre's dimension inequality over a complete intersection

Abstract: For a pair of modules M and N over a complete intersection ring R, there is a natural inequality whenever the tensor product of M and N has finite length: the quantity dim(M)+dim(N) is at most dim(R)+codim(R). This is an extension of Serre’s fundamental dimension inequality for regular local rings, motivated by the geometric intersection of varieties. We will discuss an extension of Hochster’s theta invariant to complete intersections and consider when the non-vanishing of this invariant detects equality. We will also explore versions of this inequality involving depth and complexity, and their relation to some open questions. This is based on joint work with Petter Andreas Bergh and David Jorgensen.