Vol. 16, No. 5, 2022

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print) Author Index To Appear Other MSP Journals
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 $\mathrm{dim}R$, 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