Jan Šaroch

The countable Telescope Conjecture for module categories

Abstract By the Telescope Conjecture for Module Categories, we mean the following claim: “Let R be any ring and ( A , B ) be a hereditary cotorsion pair in Mod-R with A and B closed under direct limits. Then ( A , B ) is of finite type.” We prove a modification of this conjecture with the word ‘finite’ replaced by ‘countable.’ We show that a hereditary cotorsion pair ( A , B ) of modules over an arbitrary ring R is generated by a set of strongly countably presented modules provided that B is closed under unions of well-ordered chains. We also characterize the modules in B and the countably presented modules in A in terms of morphisms between finitely presented modules, and show that ( A , B ) is cogenerated by a single pure-injective module provided that A is closed under direct limits. Then we move our attention to strong analogies between cotorsion pairs in module categories and localizing pairs in compactly generated triangulated categories.

Completeness of cotorsion pairs

Abstract Complete cotorsion pairs are among the main sources of module approximations. Given a ring R and a cotorsion pair ℭ = (𝒜, ℬ), we consider closure properties of the classes 𝒜 and ℬ that imply completeness of ℭ. Assuming Gödels Axiom of Constructibility (V = L) we prove that ℭ is complete provided ℭ is generated by a set, and either (i) 𝒜 is closed under pure submodules, or (ii) ℭ is hereditary and ℬ consists of modules of finite injective dimension. These two results are independent of ZFC + GCH. However, (i) or (ii) implies completeness of ℭ in ZFC provided ℬ is closed under arbitrary direct sums. In ZFC, we also show that ℭ is complete whenever ℭ is hereditary, 𝒜 closed under arbitrary direct products, and ℬ consists of modules of finite injective dimension. This yields a characterization of n-cotilting cotorsion pairs as the hereditary cotorsion pairs (𝒞, 𝒟) such that 𝒞 is closed under arbitrary direct products and 𝒟 consists of modules of injective dimension ≤ n.

Approximations and Mittag-Leffler conditions the tools

Mittag-Leffler modules occur naturally in algebra, algebraic geometry, and model theory, [20], [14], [19]. If R is a non-right perfect ring, then it is known that in contrast with the classes of all projective and flat modules, the class of all flat Mittag-Leffler modules is not deconstructible [16], and it does not provide for approximations when R has cardinality ≤ ℵ0, [8]. We remove the cardinality restriction on R in the latter result. We also prove an extension of the Countable Telescope Conjecture [23]: a cotorsion pair (A, B) is of countable type whenever the class B is closed under direct limits.In order to prove these results, we develop new general tools combining relative Mittag-Leffler conditions with set-theoretic homological algebra. They make it possible to trace the above facts to their ultimate, countable, origins in the properties of Bass modules. These tools have already found a number of applications: e.g., they yield a positive answer to Enochs’ problem on module approximations for classes of modules associated with tilting [4], and enable investigation of new classes of flat modules occurring in algebraic geometry [26]. Finally, the ideas from Section 3 have led to the solution of a long-standing problem due to Auslander on the existence of right almost split maps [22].

Quasi-Euclidean subrings of ℚ[x]

Using a nonstandard model of Peano arithmetic, we show that there are quasi-Euclidean subrings of ℚ[x] which are not k-stage Euclidean for any norm and positive integer k. These subrings can be either PID or non-UFD, depending on the choice of parameters in our construction. In both cases, there are 2ω such domains up to ring isomorphism.