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Multiplicities and
Betti numbers in local algebra via lim Ulrich points
Srikanth B. Iyengar, Linquan Ma and Mark E. Walker |
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Vol. 16 (2022), No. 5, 1213–1257
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Abstract |
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This work concerns finite free complexes with finite-length homology over a commutative noetherian local ring . The focus is on complexes that have length , 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 is a strict complete intersection, and also on the Dutta multiplicity, when 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 . Such a sequence is obtained by constructing an appropriate sequence of sheaves on the associated projective variety. |
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Keywords
complete intersection ring, Dutta multiplicity, Euler
characteristic, finite free complex, finite projective
dimension, lim Ulrich sequence
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Mathematical Subject Classification
Primary: 13D40
Secondary: 13A35, 13C14, 13D15, 14F06
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Milestones
Received: 1 May 2021
Revised: 18 August 2021
Accepted: 17 September 2021
Published: 16 August 2022
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Authors |
| Srikanth B. Iyengar | |
| Department of Mathematics University of Utah Salt Lake City, UT United States |
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| Linquan Ma | |
| Department of Mathematics Purdue University West Lafayette, IN United States |
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| Mark E. Walker | |
| Department of Mathematics University of Nebraska Lincoln, NE United States |
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