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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. 

In this article we study base change of Poincaré series along a quasi-complete intersection homomorphism $\mathrm{\phi <\#comment/>}:<\#comment/>Q\to <\#comment/>R$, where $Q$is a local ring with maximal ideal $\mathfrak{m}$. In particular, we give a precise relationship between the Poincaré series ${\mathrm{P}}_M^Q\left(t\right)$of a finitely generated $R$-module $M$to ${\mathrm{P}}_M^R\left(t\right)$when the kernel of $\mathrm{\phi <\#comment/>}$is contained in $\mathfrak{m}{\mathrm{a}\mathrm{n}\mathrm{n}}_Q\left(M\right)$. This generalizes a classical result of Shamash for complete intersection homomorphisms. Our proof goes through base change formulas for Poincaré series under the map of dg algebras $Q\to <\#comment/>E$, with $E$the Koszul complex on a minimal set of generators for the kernel of $\mathrm{\phi <\#comment/>}$. 

Over a local ring $R$, the theory of cohomological support varieties attaches to any bounded complex $M$of finitely generated $R$-modules an algebraic variety ${\mathrm{V}}_R\left(M\right)$that encodes homological properties of $M$. We give lower bounds for the dimension of ${\mathrm{V}}_R\left(M\right)$in terms of classical invariants of $R$. In particular, when $R$is Cohen–Macaulay and not complete intersection we find that there are always varieties that cannot be realized as the cohomological support of any complex. When $M$has finite projective dimension, we also give an upper bound for $\dim <\#comment/>{\mathrm{V}}_R\left(M\right)$in terms of the dimension of the radical of the homotopy Lie algebra of $R$. This leads to an improvement of a bound due to Avramov, Buchweitz, Iyengar, and Miller on the Loewy lengths of finite free complexes, and it recovers a result of Avramov and Halperin on the homotopy Lie algebra of $R$. Finally, we completely classify the varieties that can occur as the cohomological support of a complex over a Golod ring.