Vol. 16, No. 5, 2022

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Multiplicities and Betti numbers in local algebra via lim Ulrich points

Srikanth B. Iyengar, Linquan Ma and Mark E. Walker

Vol. 16 (2022), No. 5, 1213–1257
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 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.

Keywords
complete intersection ring, Dutta multiplicity, Euler characteristic, finite free complex, finite projective dimension, lim Ulrich sequence
Mathematical Subject Classification
Primary: 13D40
Secondary: 13A35, 13C14, 13D15, 14F06
Milestones
Received: 1 May 2021
Revised: 18 August 2021
Accepted: 17 September 2021
Published: 16 August 2022
Authors
Srikanth B. Iyengar
Department of Mathematics
University of Utah
Salt Lake City, UT
United States
Linquan Ma
Department of Mathematics
Purdue University
West Lafayette, IN
United States
Mark E. Walker
Department of Mathematics
University of Nebraska
Lincoln, NE
United States