Koszul homomorphisms and universal resolutions in local algebra
File(s)
Author(s)
Briggs, Benjamin
Cameron, James C
Letz, Janina C
Pollitz, Josh
Type
Journal Article
Abstract
We define a local homomorphism (Q,k)→(R,ℓ)
to be Koszul if its derived fiber R⊗LQk
is formal, and if TorQ(R,k)
is Koszul in the classical sense. This recovers the classical definition when Q is a field, and more generally includes all flat deformations of Koszul algebras. The non-flat case is significantly more interesting, and there is no need for examples to be quadratic: all complete intersection and all Golod quotients are Koszul homomorphisms. We show that the class of Koszul homomorphisms enjoys excellent homological properties, and we give many more examples, especially various monomial and Gorenstein examples. We then study Koszul homomorphisms from the perspective of A∞
-structures on resolutions. We use this machinery to construct universal free resolutions of R-modules by generalizing a classical construction of Priddy. The resulting (infinite) free resolution of an R-module M is often minimal and can be described by a finite amount of data whenever M and R have finite projective dimension over Q. Our construction simultaneously recovers the resolutions of Shamash and Eisenbud over a complete intersection ring, and the bar resolutions of Iyengar and Burke over a Golod ring, and produces analogous resolutions for various other classes of local rings.
to be Koszul if its derived fiber R⊗LQk
is formal, and if TorQ(R,k)
is Koszul in the classical sense. This recovers the classical definition when Q is a field, and more generally includes all flat deformations of Koszul algebras. The non-flat case is significantly more interesting, and there is no need for examples to be quadratic: all complete intersection and all Golod quotients are Koszul homomorphisms. We show that the class of Koszul homomorphisms enjoys excellent homological properties, and we give many more examples, especially various monomial and Gorenstein examples. We then study Koszul homomorphisms from the perspective of A∞
-structures on resolutions. We use this machinery to construct universal free resolutions of R-modules by generalizing a classical construction of Priddy. The resulting (infinite) free resolution of an R-module M is often minimal and can be described by a finite amount of data whenever M and R have finite projective dimension over Q. Our construction simultaneously recovers the resolutions of Shamash and Eisenbud over a complete intersection ring, and the bar resolutions of Iyengar and Burke over a Golod ring, and produces analogous resolutions for various other classes of local rings.
Date Issued
2025-03-28
Date Acceptance
2025-02-16
Citation
Forum of Mathematics, Sigma, 2025, 13
ISSN
2050-5094
Publisher
Cambridge University Press
Journal / Book Title
Forum of Mathematics, Sigma
Volume
13
Copyright Statement
© The Author(s), 2025. Published by Cambridge University Press. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
License URL
Subjects
COMPLETE INTERSECTION
COMPLEXES
HOMOLOGICAL ALGEBRA
IDEALS
Mathematics
Mathematics, Applied
MODULES
Physical Sciences
POINCARE-SERIES
PROPERTY
QUILLEN
Science & Technology
SOCLE
Publication Status
Published
Article Number
e63
Date Publish Online
2025-03-28