Injective modules over the Jacobson algebra

For a field K, let $\mathcal {R}$ denote the Jacobson algebra $K\langle X, Y \ | \ XY=1\rangle $ . We give an explicit construction of the injective envelope of each of the (infinitely many) simple left $\mathcal {R}$ -modules. Consequently, we obtain an explicit description of a minimal injective cogenerator for $\mathcal {R}$ . Our approach involves realizing $\mathcal {R}$ up to isomorphism as the Leavitt path K-algebra of an appropriate graph $\mathcal {T}$ , which thereby allows us to utilize important machinery developed for that class of algebras.

The Tilting–Cotilting Correspondence

To a big $n$-tilting object in a complete, cocomplete abelian category ${\textsf{A}}$ with an injective cogenerator we assign a big $n$-cotilting object in a complete, cocomplete abelian category ${\textsf{B}}$ with a projective generator and vice versa. Then we construct an equivalence between the (conventional or absolute) derived categories of ${\textsf{A}}$ and ${\textsf{B}}$. Under various assumptions on ${\textsf{A}}$, which cover a wide range of examples (for instance, if ${\textsf{A}}$ is a module category or, more generally, a locally finitely presentable Grothendieck abelian category), we show that ${\textsf{B}}$ is the abelian category of contramodules over a topological ring and that the derived equivalences are realized by a contramodule-valued variant of the usual derived Hom functor.

Balance for relative cohomology of complexes

Let [Formula: see text] be an arbitrary ring. We use a strict [Formula: see text]-resolution [Formula: see text] of a complex [Formula: see text] with finite [Formula: see text]-projective dimension, where [Formula: see text] denotes a subcategory of right [Formula: see text]-modules closed under extensions and direct summands and admits an injective cogenerator [Formula: see text], to define the [Formula: see text]th relative cohomology functor [Formula: see text] as [Formula: see text]. If a complex [Formula: see text] has finite [Formula: see text]-injective dimension, then one can use a dual argument to define a notion of a relative cohomology functor [Formula: see text], where [Formula: see text] is a subcategory of right [Formula: see text]-modules closed under extensions and direct summands and admits a projective generator. Under several orthogonal conditions, we show that there exists an isomorphism [Formula: see text] of relative cohomology groups for each [Formula: see text]. This result simultaneously encompasses a balance result of Holm on Gorenstein projective and injective modules, a balance result of Sather-Wagstaff, Sharif and White on Gorenstein projective and injective modules with respect to semidualizing modules, and a balance result of Liu on Gorenstein projective and injective complexes. In particular, as an application of this result, we extend the above balance result of Sather-Wagstaff, Sharif and White to the setting of complexes.

Purity and flatness in symmetric monoidal closed exact categories

Let [Formula: see text] be a symmetric monoidal closed exact category. This category is a natural framework to define the notions of purity and flatness. When [Formula: see text] is endowed with an injective cogenerator with respect to the exact structure, we show that an object [Formula: see text] in [Formula: see text] is flat if and only if any conflation ending in [Formula: see text] is pure. Furthermore, we prove a generalization of the Lambek Theorem (J. Lambek, A module is flat if and only if its character module is injective, Canad. Math. Bull. 7 (1964) 237–243) in [Formula: see text]. In the case [Formula: see text] is a quasi-abelian category, we prove that [Formula: see text] has enough pure injective objects.