Matlis category equivalences for a ring epimorphism

Abstract

Under mild assumptions, we construct the two Matlis additive category equivalences for an associative ring epimorphism u : R −→ U . Assuming that the ring epimorphism is homological of flat/projective dimension 1, we discuss the abelian categories of u-comodules and u-contramodules and construct the recollement of unbounded derived categories of R-modules, U -modules, and complexes of R-modules with u-co/contramodule cohomology. Further assumptions allow to describe the third category in the recollement as the unbounded derived category of the abelian categories of u-comodules and u-contramodules. For commutative rings, we also prove that any homological epimorphism of projective dimension 1 is flat. Injectivity of the map u is not required.The following result can be also found in [5, Corollary 4.4].Corollary 7.3. Let u : R −→ U be a homological ring epimorphism. Assume that fd U R ≤ 1 and pd R U ≤ 1. Suppose further that (U/R) ⊗ R J = 0 for all injective left R-modules J and Hom R (U/R, F ) = 0 for all projective left R-modules F . Then for every conventional derived category symbol ⋆ = b, +, −, or ∅, there is a triangulated equivalence between the derived categories of the abelian categories R-mod u-co and R-mod u-ctra of left u-comodules and left u-contramodules,Proof. According to Corollary 6.2 and Theorem 7.1(a-b), we have a chain of triangulated equivalences D ⋆ (R-mod u-co ) ∼ = D ⋆ u-co (R-mod) ∼ = D ⋆ u-ctra (R-mod) ∼ = D ⋆ (R-mod u-ctra ).

Covering classes and $1$-tilting cotorsion pairs over commutative rings

“…The material covered in this section is a combination of ideas from [Pos18], [BP20], and [BP19a], and covers mostly methods using contramodules and topological rings.…”

“…In [BP20] it is considered the case of a (not necessarily injective nor flat nor commutative) ring epimorphism u : R → U such that Tor R 1 (U, U ) = 0. In [BP20, Theorem 1.2] it is shown that the adjunction (− ⊗ R K), Hom R (K, −) (where K = U/u(R)) defines an equivalence between the class of u-divisible right u-comodules and the class of u-torsion-free left u-contramodules.…”

“…In our situation, that is when u is a flat injective epimorphism with associated Gabriel topology G, the class of u-comodules coincides with the class of G-torsion modules and the class of u-torsion-free modules coincides with the class of G-torsion-free modules. Thus, in our setting, Theorem 1.2 in [BP20] becomes: Theorem 5.13. [BP20, Theorem 1.2] Let u : R → U be a flat injective ring epimorphism of commutative rings.…”

Categories of modules, comodules and contramodules over representations

“…Accordingly, a contramodule M over a coalgebra C consists of a space M as well as a morphism Hom(C, M ) −→ M satisfying certain coassociativity and counit conditions (see Section 5). While contramodules were introduced much earlier by Eilenberg and Moore [14, § IV.5], the subject has seen a lot of interest in recent years (see, for instance, [7], [8], [10], [26], [27], [28], [29], [30], [31], [32], [37], [44]). One important aspect of our paper is that for comodules over a coalgebra representation C : X −→ Coalg or modules over an algebra representation A : X −→ Alg, it becomes necessary to work with objects of two different orientations, which we refer to as "cis-objects" and "trans-objects."…”

On representation categories of $A_\infty$-algebras and $A_\infty$-coalgebras

“…In a categorical sense, it may be said therefore that both comodules and contramodules dualize the notion of module over an algebra. Even though the study of contramodules in the literature is not as developed as that of comodules, the topic has seen a lot of renewed interest in recent years (see, for instance, [2], [3], [4], [6], [17], [18], [19], [20], [25]). Accordingly, our notion of an A ∞ -contramodule over an A ∞ -coalgebra C consists of a graded vector space M ∈ V ect Z along with a collection of structure maps…”