Finally, a theorem of Ohm-Rush is applied to prove that any $R$ with only finitely many minimal primes has the property that $R\lbrack X \rbrack/I$ is $R$-flat implies $I$ is finitely generated. An example is given to show that this conjecture needs some tightening. a ring such that finitely generated flat modules are projective and they have asked whether conversely any $A(0)$ ring has this property. Ohm-Rush have also observed that a ring $R$ with the property "$R\lbrack X \rbrack/I$ is $R$-flat implies $I$ is finitely generated" is necessarily an $A(0)$ ring, i.e. $I$ is even locally principal at primes of $R\lbrack X \rbrack$. It is shown that this is indeed the case, and it then follows easily that. Here $R$ is a commutative ring with identity, $X$ is an indeterminate, and $I$ is an ideal of $R\lbrack X \rbrack$. A question which was raised by them is whether $R\lbrack X \rbrack/I$ is a flat $R$-module implies $I$ is locally finitely generated at primes of $R\lbrack X \rbrack$. This, in fact, provides an application of the Gabriel–Roiter (co)measure in the category of maximal Cohen–Macaulay modules. 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. 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. 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. 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.
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