This paper is a sequel to arXiv:1503.05523 and arXiv:1605.03934. We extend the classical Harrison-Matlis module category equivalences to a triangulated equivalence between the derived categories of the abelian categories of torsion modules and contramodules over a Matlis domain. This generalizes to the case of any commutative ring $R$ with a fixed multiplicative system $S$ such that the $R$-module $S^{-1}R$ has projective dimension $1$. The latter equivalence connects complexes of $R$-modules with $S$-torsion and $S$-contramodule cohomology modules. It takes a nicer form of an equivalence between the derived categories of abelian categories when $S$ consists of nonzero-divisors or the $S$-torsion in $R$ is bounded.

Infinite Abelian groups

Triangulated Matlis equivalence

https://arrow.tudublin.ie/cgi/viewcontent.cgi?article=1273&context=scschmatart

The Kaplansky Test Problems - An approach via radicals

Modern developments in representation theory rely heavily on homological methods. This book for advanced graduate students and researchers introduces these methods from their foundations up and discusses several landmark results that illustrate their power and beauty. Categorical foundations include abelian and derived categories, with an emphasis on localisation, spectra, and purity. The representation theoretic focus is on module categories of Artin algebras, with discussions of the representation theory of finite groups and finite quivers. Also covered are Gorenstein and quasi-hereditary algebras, including Schur algebras, which model polynomial representations of general linear groups, and the Morita theory of derived categories via tilting objects. The final part is devoted to a systematic introduction to the theory of purity for locally finitely presented categories, covering pure-injectives, definable subcategories, and Ziegler spectra. With its clear, detailed exposition of important topics in modern representation theory, many of which were unavailable in one volume until now, it deserves a place in every representation theorist's library.

Homological Theory of Representations

background

/pdf/modules-over-bounded-dedekind-prime-rings-ii-54yzbioxdj.pdf

Modules over bounded Dedekind prime rings. II

In this paper, we shall discuss some recent contributions to the project [15, 14, 2, 18, 22, 23] of explaining why no satisfactory system of complete invariants has yet been found for the torsion-free abelian groups of finite rank n ≥ 2. Recall that, up to isomorphism, the torsion-free abelian groups of rank n are exactly the additive subgroups of the n-dimensional vector space ℚ n which contain n linearly independent elements. Thus the collection of torsion-free abelian groups of rank at most n can be naturally identified with the set S (ℚ n ) of all nontrivial additive subgroups of ℚ n . In 1937, Baer [4] solved the classification problem for the class S(ℚ)of rank 1 groups as follows. Let ℙ be the set of primes. If G is a torsion-free abelian group and 0 ≠ x ϵ G, then the p-height of x is defined to be hx (p) = sup{n ϵ ℕ ∣ There exists y ϵ G such that pny = x} ϵ ℕ ∪{∞}; and the characteristic χ (x) of x is defined to be the function

Infinite Abelian groups

Citations

Triangulated Matlis equivalence

The Kaplansky Test Problems - An approach via radicals

Homological Theory of Representations

Modules over bounded Dedekind prime rings. II

On the Complexity of the Classification Problem for Torsion-Free Abelian Groups of Finite Rank

Related Papers (5)

Finite Rank Torsion Free Abelian Groups and Rings

Abstract Harmonic Analysis

Abelian groups

Abelian groups without elements of finite order

Rings and Categories of Modules