Multiplicities and Betti numbers in local algebra via lim Ulrich points

Abstract

This work concerns finite free complexes with finite-length homology over a commutative noetherian local ring $R$. The focus is on complexes that have length $\text{dim} R$, which is the smallest possible value, and, in particular, on free resolutions of modules of finite length and finite projective dimension. Lower bounds are obtained on the Euler characteristic of such short complexes when $R$ is a strict complete intersection, and also on the Dutta multiplicity, when $R$ is the localization at its maximal ideal of a standard graded algebra over a field of positive prime characteristic. The key idea in the proof is the construction of a suitable Ulrich module, or, in the latter case, a sequence of modules that have the Ulrich property asymptotically, and with good convergence properties in the rational Grothendieck group of $R$. Such a sequence is obtained by constructing an appropriate sequence of sheaves on the associated projective variety.