Baer’s Criterion for Injectivity is a useful tool of the theory of modules.
Its dual version (DBC) is known to hold for all right perfect rings, but its validity for the non-right perfect ones is a complex problem (first formulated by C. Faith
[Algebra. II. Ring Theory,
Springer, Berlin, 1976]).
Recently, it has turned out that there are two classes of non-right perfect rings:
(1) those for which DBC fails in ZFC, and
(2) those for which DBC is independent of ZFC.
First examples of rings in the latter class were constructed in
Faith’s problem on R-projectivity is undecidable,
Proc. Amer. Math. Soc. 147 2019, 2, 497–504];
here, we show that this class contains all small semiartinian von Neumann regular rings with primitive factors artinian.
Kaplansky classes emerged in the context of Enochs' solution of the Flat Cover Conjecture. Their connection to abstract model theory goes back to Baldwin et al.: a class of roots of Ext is a Kaplansky class closed under direct limits if and only if the pair is an abstract elementary class (AEC) in the sense of Shelah. We prove that this AEC has finite character in case for a class of pure-injective modules. In particular, all AECs of roots of Ext over any right noetherian right hereditary ring R have finite character (but the case of general rings remains open).
If is an AEC of roots of Ext, then is known to be a covering class. However, Kaplansky classes need not even be precovering in general: We prove that the class of all ℵ1-projective modules (which is equal to the class of all flat Mittag-Leffler modules) is a Kaplansky class for any ring R, but it fails to be precovering in case R is not right perfect, the class equals the class of all flat modules and consists of modules of projective dimension . Assuming the Singular Cardinal Hypothesis, we prove that is not precovering for each countable non-right perfect ring R.
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 Gödel's 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.
We apply tilting theory to study modules of finite projective dimension. We introduce
the notion of finite and cofinite type for tilting and cotilting classes of modules, respectively,
showing that, for each dimension, there is a bijection between these classes and resolving
classes of modules.
We then focus on Iwanaga-Gorenstein rings. Using tilting theory, we prove the first finitistic
dimension conjecture for these rings. Moreover, we characterize them among noetherian rings
by the property that Gorenstein injective modules form a tilting class. Finally, we give an
explicit construction of families of (co)tilting modules of (co)finite type for one-dimensional
commutative Gorenstein rings.