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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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A category $\mathcal{O}$ for oriented matroids
We associate to a sufficiently generic oriented matroid program and choice of linear system of parameters a finite-dimensional algebra, whose representation theory is analogous to blocks of Bernstein—Gelfand—Gelfand category\mathcal{O}. When the data above comes from a generic linear program for a hyperplane arrangement, we recover the algebra defined by Braden—Licata—Proudfoot—Webster. Applying our construction to non-linear oriented matroid programs provides a large new class of algebras. For Euclidean oriented matroid programs, the resulting algebras are quasi-hereditary and Koszul, as in the linear setting. In the non-Euclidean case, we obtain algebras that are not quasi-hereditary and not known to be Koszul, but still have a natural class of standard modules and satisfy numerical analogues of quasi-heredity and Koszulity on the level of graded Grothendieck groups.
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- Award ID(s):
- 1802299
- PAR ID:
- 10550365
- Publisher / Repository:
- EMS Press
- Date Published:
- Journal Name:
- Journal of Combinatorial Algebra
- Volume:
- 7
- Issue:
- 1
- ISSN:
- 2415-6302
- Page Range / eLocation ID:
- 159 to 226
- 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/10.4171/JCA/71
