Koszul homomorphisms and universal resolutions in local algebra

TL;DR

The paper develops a comprehensive relative Koszul duality framework for local rings by defining Koszul local homomorphisms via the formality of the derived fiber $R\otimes^{\mathsf{L}}_Q k$ and the Koszulity of $\operatorname{Tor}^Q(R,k)$. It shows broad ubiquity, encompassing flat maps with Koszul fibers, complete intersections, Golod maps, and Cohen presentations, with Cohen Koszul rings exhibiting rational Poincaré series and strong homological control. A central advance is encoding resolutions through $\mathrm{A}_\infty$-structures, transferring resolutions from $Q$ to $R$, and constructing universal (Priddy) resolutions via twisted tensor products using a strictly Koszul or strictly Koszul-presented data $(A,V,W)$. The second half develops a robust framework for $A_\infty$-algebras, cyclic and curved coalgebras, and the Priddy resolution, unifying Shamash–Eisenbud and Iyengar–Burke resolutions as special cases, and providing explicit, computable resolutions in large classes of local rings, including almost Golod Gorenstein rings. Overall, the work integrates homological algebra, $A_\infty$-techniques, and Koszul duality to produce a flexible, widely applicable method for constructing and understanding infinite free resolutions in local algebra. It highlights deep connections to rational homotopy and toric topology, offering a broad toolkit for future exploration of resolutions and their invariants.

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 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 $\mathrm{A}_\infty$-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.

Figures (1)

• Figure 1: Illustration of the weight and homological gradings of the $\mathrm{A}_{\infty}$-resolution $A$ for various examples.

Theorems & Definitions (96)

• Theorem A
• Theorem B
• Definition 2.1
• Remark 2.2
• Definition 2.4
• Remark 2.5
• Definition 2.6
• Remark 2.7
• Definition 2.9
• Remark 2.10