Our main tool in the proof of results is Gabriel–Roiter (co)measure, an invariant assigned to modules of finite length, and defined by Gabriel and Ringel.
![locally presentable category locally presentable category](https://usercontent.village.co/cs420/s340x340/16278ad6-b8da-4c67-8963-3052839af86f-image.jpg)
In particular, a Cohen–Macaulay algebra Λ is of finite CM-type if and only if every ω-Gorenstein projective module is of finite CM-type, which generalizes a result of Chen for Gorenstein algebras. It will turn out that, ω-Gorenstein projective modules with bounded CM-support are fully decomposable. Finally, we examine the mentioned results in the context of Cohen–Macaulay artin algebras admitting a dualizing bimodule ω, as defined by Auslander and Reiten. Namely, it is shown that R is of finite CM-type if and only if R is an isolated singularity and the category of all fully decomposable balanced big Cohen–Macaulay modules is closed under kernels of epimorphisms. In addition, the pure-semisimplicity of a subcategory of balanced big Cohen–Macaulay modules is settled. While the first Brauer–Thrall conjecture fails in general by a counterexample of Dieterich dealing with multiplicities to measure the size of maximal Cohen–Macaulay modules, our formalism establishes the validity of the conjecture for complete Cohen–Macaulay local rings. it is a direct sum of finitely generated modules. Among other results, it is proved that, for a given balanced big Cohen–Macaulay R-module M with an m-primary cohomological annihilator, if there is a bound on the h̲-length of all modules appearing in CM-support of M, then it is fully. In this paper, we assign a numerical invariant, for any balanced big Cohen–Macaulay module, called h̲-length.
![locally presentable category locally presentable category](http://www.mat.uc.pt/~cmuc/CMUCnews/012019/images/stanislav.png)
Let (R, m, k) be a complete Cohen–Macaulay local ring. For example, étale fundamental groups are not “true” groups but only profinite groups, and one cannot hope to recover more: the “Tannakian” functor represented by the étale fundamental group of a scheme preserves finite products but not all products. More generally, we introduce a new notion of “commutative 2-ring” that includes both Grothendieck topoi and symmetric monoidal categories of modules, and define a notion of πġ for the corresponding “affine 2-schemes.” These results help to simplify and clarify some of the peculiarities of the étale fundamental group. This gives a new definition for étale πġ(spec(R)) in terms of the category of R-modules rather than the category of étale covers. the separable absolute Galois group of R when it is a field.
#Locally presentable category pro#
The second construction gives an indication that one can possibly develop a noncommutative proper homotopy theory in the context of topological algebras, e.g., pro C ∗-algebras.A natural question in the theory of Tannakian categories is: What if you don’t remember Forget? Working over an arbitrary commutative ring R, we prove that an answer to this question is given by the functor represented by the étale fundamental groupoid πġ(spec(R)), i.e. The first result can be used to deduce derived Morita equivalence between DG categories of topological bundles associated to separable C ∗-algebras up to a K-theoretic identification from the knowledge of KK-equivalence between the C ∗-algebras. This construction respects homotopy between proper maps after enforcing matrix stability on the category of pro C ∗-algebras. Motivated by a construction of Cuntz we associate a pro C ∗-algebra to any simplicial set, which is functorial with respect to proper maps of simplicial sets and those of pro C ∗-algebras. We construct an additive functor from the category of separable C ∗-algebras with morphisms enriched over Kasparov’s KK0-groups to the noncommutative correspondence category NCC K dg, whose objects are small DG categories and morphisms are given by the equivalence classes of some DG bimodules up to a certain K-theoretic identification.