Complete Local Noetherian Ring Research Articles

Auslander-Reiten Conjecture in a Dual Vein

We consider a dual notion of the famous Auslander-Reiten Conjecture in case of Noetherian algebras over commutative Noetherian rings. Firstly, in the introduction, we will examine its relevance by showing that in an standard situation, the validity of this dual implies the validity of the Auslander-Reiten Conjecture itself. Moreover, in two important cases these two notions coincide: Artin algebras, and Noetherian algebras over complete local Noetherian rings. In this regard we will prove the following theorem: Let $(R, \mathfrak {m})$ be d-Gorenstein, d ≥ 2, and let Λ be a Noetherian R-algebra which is Gorenstein and (maximal) Cohen-Macaulay as R-module. If M is an Artinian self-orthogonal Gorenstein injective Λ-module such that ${\text {Hom}}_{\Lambda }({\Lambda }_{\mathfrak {p}}, M)$ is an injective ${\Lambda }_{\mathfrak {p}}$ -module for every nonmaximal prime ideal $\mathfrak {p}$ of R, then M is injective. Some applications are discussed afterwards.

Based quasi-hereditary algebras

A notion of a split quasi-hereditary algebra has been defined by Cline, Parshall and Scott. Du and Rui describe a based approach to split quasi-hereditary algebras. We develop this approach further to show that over a complete local Noetherian ring, one can achieve even stronger basis properties. This is important for ‘schurifying’ quasi-hereditary algebras as developed in our subsequent work. The schurification procedure associates to an algebra A a new algebra, which is the classical Schur algebra if A is a field. Schurification produces interesting new quasi-hereditary and cellular algebras. It is important to work over an integral domain of characteristic zero, taking into account a super-structure on the input algebra A. So we pay attention to super-structures on quasi-hereditary algebras and investigate a subtle conforming property of heredity data which is crucial to guarantee that the schurification of A is quasi-hereditary if so is A. We establish a Morita equivalence result which allows us to pass to basic quasi-hereditary algebras preserving conformity.

A Block Refinement of the Green-Puig Parameterization of the Isomorphism Types of Indecomposable Modules

Let $p$ be a prime integer, let $\mathcal{O}$ be a commutative complete local Noetherian ring with an algebraically closed residue field $k$ of charateristic $p$ and let$G$ be a finite group. Let $P$ be a $p$-subgroup of $G$ and let $X$ be an indecomposable $\mathcal{O} P$-module with vertex $P$. Let $\Lambda (G,P,X)$ denote a set of representatives for the isomorphism classes of indecomposable $\mathcal{O} G$-modules with vertex-source pair $(P,X)$ (so that $\Lambda(G,P,X)$ is a finite set by the Green correspondence). As mentioned in [5, Notes on Section~26], L. Puig asserted that a defect multiplicity module determined by $(P,X)$ can be used to obtain an extended parameterization of $\Lambda(G,P,X)$. In [5, Proposition 26.3], J. Thevenaz completed this program under the hypotheses that $X$ is $\mathcal{O}$-free. Here we use the methods of proof of [5, Theorem 26.3] to show that the $\mathcal{O}$-free hypothesis on $X$ is superfluous. (M. Linckelmann has also proved this, cf. [3]). Let $B$ be a block of $\mathcal{O} G$. Then we obtain a corresponding paramaterization of the $(\mathcal{O} G)B$-modules in $\Lambda(G,P,X)$.

The Density of Ramified Primes

Let $F$ be a number field, $\mathcal{O}$ be a domain with fraction field $\mathcal{K}$ of characteristic zero and $\rho: \mathrm{Gal}(\overline F/F) \to \mathrm{GL}_n(\mathcal{O})$ be a representation such that $\rho\otimes\overline{\mathcal{K}}$ is semisimple. If $\mathcal{O}$ admits a finite monomorphism from a power series ring with coefficients in a $p$-adic integer ring (resp. $\mathcal{O}$ is an affinoid algebra over a $p$-adic number field) and $\rho$ is continuous with respect to the maximal ideal adic topology (resp. the Banach algebra topology), then we prove that the set of ramified primes of $\rho$ is of density zero. If $\mathcal{O}$ is a complete local Noetherian ring over $\mathbb{Z}_p$ with finite residue field of characteristic $p$, $\rho$ is continuous with respect to the maximal ideal adic topology and the kernels of pure specializations of $\rho$ form a Zariski-dense subset of $\mathrm{Spec} \mathcal{O}$, then we show that the set of ramified primes of $\rho$ is of density zero. These results are analogues, in the context of big Galois representations, of a result of Khare and Rajan, and are proved relying on their result.

On faithfulness of the lifting for Hopf algebras and fusion categories

We use a version of Haboush's theorem over complete local Noetherian rings to prove faithfulness of the lifting for semisimple cosemisimple Hopf algebras and separable (braided, symmetric) fusion categories from characteristic $p$ to characteristic zero (arXiv/math:0203060, Section 9), showing that, moreover, any isomorphism between such structures can be reduced modulo $p$. This fills a gap in arXiv/math:0203060, Subsection 9.3. We also show that lifting of semisimple cosemisimple Hopf algebras is a fully faithful functor, and prove that lifting induces an isomorphism on Picard and Brauer-Picard groups. Finally, we show that a subcategory or quotient category of a separable multifusion category is separable (resolving an open question from arXiv/math:0203060, Subsection 9.4), and use this to show that certain classes of tensor functors between lifts of separable categories to characteristic zero can be reduced modulo $p$.

Universal valued fields and lifting points in local tropical varieties

ABSTRACTLet k be a field with a real valuation ν and R a k-algebra. We show that there exist a k-algebra K and a valuation μ on K extending ν such that any real valuation of R is induced by μ through some homomorphism from R to K. Let ν be trivial and R a complete local Noetherian ring with the residue field k. Let K be the ring of Hahn series with its natural valuation μ and coefficients in . We prove the following weak universality property: For any local valuation v and a finite set of elements of R, there exists a homomorphism f:R→K such that , i = 1,…,n. This implies that if for an ideal I, then every point of the local tropicalization of I lifts to a K-point of R.

The inverse deformation problem

Given a commutative complete local noetherian ring $A$ with finite residue field $\boldsymbol{k}$, we show that there is a topologically finitely generated profinite group $\unicode[STIX]{x1D6E4}$ and an absolutely irreducible continuous representation $\overline{\unicode[STIX]{x1D70C}}:\unicode[STIX]{x1D6E4}\rightarrow \text{GL}_{n}(\boldsymbol{k})$ such that $A$ is a universal deformation ring for $\unicode[STIX]{x1D6E4},\overline{\unicode[STIX]{x1D70C}}$.

A structure theorem for subgroups of $GL_n$ over complete local Noetherian rings with large residual image

Given a complete local Noetherian ring (A, mA) with finite residue field and a subfield k of A/mA, we show that every closed subgroup G of GLn(A) such that G mod mA ⊇ SLn(k) contains a conjugate of SLn(W(k)A) under some small restrictions on k. Here W(k)A is the closed subring of A generated by the Teichm¨uller lifts of elements of the subfield k.

Obstructions for deformations of complexes

We develop two approaches to obstruction theory for deformations of derived isomorphism classes of complexes of modules for a profinite group G over a complete local Noetherian ring A of positive residue characteristic.

Isomorphism of complete local noetherian rings and strong approximation

About a year ago Angus Macintyre raised the following question. Let A and B be complete local noetherian rings with maximal ideals m and n such that A/m is isomorphic to B/n n for every n. Does it follow that A and B are isomorphic? We show that the answer is yes if the residue field is algebraic over its prime field. The proof uses a strong approximation theorem of Pfister and Popescu, or rather a variant of it, which we obtain by a method due to Denef and Lipshitz. Examples by Gabber show that the answer is no in general.

Krull dimension of Iwasawa algebras

for example, because of their connections with number theory and arithmetic geometry; see the paper by Coates, Schneider and Sujatha [4] for more details. When G is a uniform pro-p group, ΛG is a concrete example of a complete local Noetherian ring (noncommutative, in general) with good homological properties: it is known that ΛG has finite global dimension and is an Auslander regular ring. Thus, ΛG falls into the class of rings studied by Brown, Hajarnavis and MacEacharn in [1]. There they consider various properties of Noetherian rings R of finite global dimension, including the Krull(–Gabriel–Rentschler) dimension K(R)—a module-theoretic dimension which measures how far R is from being Artinian. They also posed the following question:

The structure of positively graded locala artinian rings and extended graded complete local noetherian rings

The structure of positively graded locala artinian rings and extended graded complete local noetherian rings