Multiplicities and Betti numbers in local algebra via lim Ulrich points

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

doi:10.2140/ant.2022.16.1213

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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

DOI: 10.2140/ant.2022.16.1213

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

Vol. 16, No. 5, 2022