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Abstract We define a local homomorphism$$(Q,k)\to (R,\ell )$$to be Koszul if its derived fiber$$R\otimes ^{\mathsf {L}}_Q k$$is formal, and if$$\operatorname {Tor}^{Q}(R,k)$$is Koszul in the classical sense. This recovers the classical definition whenQis 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$$\mathrm {A}_{\infty }$$-structures on resolutions. We use this machinery to construct universal free resolutions ofR-modules by generalizing a classical construction of Priddy. The resulting (infinite) free resolution of anR-moduleMis often minimal and can be described by a finite amount of data wheneverMandRhave finite projective dimension overQ. 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.
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Quadratic Gorenstein Rings and the Koszul Property II
Abstract Conca–Rossi–Valla [6] ask if every quadratic Gorenstein ring $$R$$ of regularity three is Koszul. In [15], we use idealization to answer their question, proving that in nine or more variables there exist quadratic Gorenstein rings of regularity three, which are not Koszul. In this paper, we study the analog of the Conca–Rossi–Valla question when the regularity of $$R$$ is four or more. Let $$R$$ be a quadratic Gorenstein ring having $${\operatorname {codim}} \ R = c$$ and $${\operatorname {reg}} \ R = r \ge 4$$. We prove that if $c = r+1$ then $$R$$ is always Koszul, and for every $$c \geq r+2$$, we construct quadratic Gorenstein rings that are not Koszul, answering questions of Matsuda [16] and Migliore–Nagel [19].
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- Award ID(s):
- 2048906
- PAR ID:
- 10318135
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imrn/rnab297
