Finiteness Theorems for Deformations of Complexes

Frauke M. Bleher; Ted Chinburg

Displaying similar documents to “Finiteness Theorems for Deformations of Complexes”

Obstructions for deformations of complexes

Frauke M. Bleher, Ted Chinburg (2013)

Annales de l’institut Fourier

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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.

Lifting $D$ -modules from positive to zero characteristic

João Pedro P. dos Santos (2011)

Bulletin de la Société Mathématique de France

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We study liftings or deformations of $D$ -modules ( $D$ is the ring of differential operators from EGA IV) from positive characteristic to characteristic zero using ideas of Matzat and Berthelot’s theory of arithmetic $D$ -modules. We pay special attention to the growth of the differential Galois group of the liftings. We also apply formal deformation theory (following Schlessinger and Mazur) to analyze the space of all liftings of a given $D$ -module in positive characteristic. At the end we compare...

The correspondence between Barsotti-Tate groups and Kisin modules when $p = 2$

Tong Liu (2013)

Journal de Théorie des Nombres de Bordeaux

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Let $K$ be a finite extension over $ℚ_{2}$ and $𝒪_{K}$ the ring of integers. We prove the equivalence of categories between the category of Kisin modules of height 1 and the category of Barsotti-Tate groups over $𝒪_{K}$ .

Some Results About Holonomic $ε$ -Modules

Jan-Erik Björk (1983)

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Category $𝒪$ for quantum groups

Henning Haahr Andersen, Volodymyr Mazorchuk (2015)

Journal of the European Mathematical Society

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In this paper we study the BGG-categories $𝒪_{q}$ associated to quantum groups. We prove that many properties of the ordinary BGG-category $𝒪$ for a semisimple complex Lie algebra carry over to the quantum case. Of particular interest is the case when $q$ is a complex root of unity. Here we prove a tensor decomposition for both simple modules, projective modules, and indecomposable tilting modules. Using the known Kazhdan-Lusztig conjectures for $𝒪$ and for finite dimensional $U_{q}$ -modules we are able...

Melkersson condition on Serre subcategories

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Let R be a commutative noetherian ring, let be an ideal of R, and let be a subcategory of the category of R-modules. The condition $C_{}$ , defined for R-modules, was introduced by Aghapournahr and Melkersson (2008) in order to study when the local cohomology modules relative to belong to . In this paper, we define and study the class $_{}$ consisting of all modules satisfying $C_{}$ . If and are ideals of R, we get a necessary and sufficient condition for to satisfy $C_{}$ and $C_{}$ simultaneously. We also...

A note on Frobenius divided modules in mixed characteristics

Pierre Berthelot (2012)

Bulletin de la Société Mathématique de France

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If $X$ is a smooth scheme over a perfect field of characteristic $p$ , and if $𝒟_{X}^{(\infty)}$ is the sheaf of differential operators on $X$ [7], it is well known that giving an action of $𝒟_{X}^{(\infty)}$ on an $𝒪_{X}$ -module $ℰ$ is equivalent to giving an infinite sequence of $𝒪_{X}$ -modules descending $ℰ$ via the iterates of the Frobenius endomorphism of $X$ [5]. We show that this result can be generalized to any infinitesimal deformation $f : X \to S$ of a smooth morphism in characteristic $p$ , endowed with Frobenius liftings. We also show that it...

On the composition structure of the twisted Verma modules for $𝔰𝔩 (3, ℂ)$

Libor Křižka, Petr Somberg (2015)

Archivum Mathematicum

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We discuss some aspects of the composition structure of twisted Verma modules for the Lie algebra $𝔰𝔩 (3, ℂ)$ , including the explicit structure of singular vectors for both $𝔰𝔩 (3, ℂ)$ and one of its Lie subalgebras $𝔰𝔩 (2, ℂ)$ , and also of their generators. Our analysis is based on the use of partial Fourier tranform applied to the realization of twisted Verma modules as $D$ -modules on the Schubert cells in the full flag manifold for $SL (3, ℂ)$ .

Separable $ℵ_{k}$ -free modules with almost trivial dual

Daniel Herden, Héctor Gabriel Salazar Pedroza (2016)

Commentationes Mathematicae Universitatis Carolinae

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An $R$ -module $M$ has an almost trivial dual if there are no epimorphisms from $M$ to the free $R$ -module of countable infinite rank $R^{(ω)}$ . For every natural number $k > 1$ , we construct arbitrarily large separable $ℵ_{k}$ -free $R$ -modules with almost trivial dual by means of Shelah’s Easy Black Box, which is a combinatorial principle provable in ZFC.

Base change for Picard-Vessiot closures

Andy R. Magid (2011)

Banach Center Publications

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The differential automorphism group, over F, Π₁(F₁) of the Picard-Vessiot closure F₁ of a differential field F is a proalgebraic group over the field $C_{F}$ of constants of F, which is assumed to be algebraically closed of characteristic zero, and its category of $C_{F}$ modules is equivalent to the category of differential modules over F. We show how this group and the category equivalence behave under a differential extension E ⊃ F, where $C_{E}$ is also algebraically closed.

$n$ - $gr$ -coherent rings and Gorenstein graded modules

Mostafa Amini, Driss Bennis, Soumia Mamdouhi (2022)

Czechoslovak Mathematical Journal

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Let $R$ be a graded ring and $n \geq 1$ be an integer. We introduce and study the notions of Gorenstein $n$ -FP-gr-injective and Gorenstein $n$ -gr-flat modules by using the notion of special finitely presented graded modules. On $n$ -gr-coherent rings, we investigate the relationships between Gorenstein $n$ -FP-gr-injective and Gorenstein $n$ -gr-flat modules. Among other results, we prove that any graded module in $R$ -gr (or gr- $R$ ) admits a Gorenstein $n$ -FP-gr-injective (or Gorenstein $n$ -gr-flat) cover and preenvelope,...

Derived endo-discrete artin algebras

Raymundo Bautista (2006)

Colloquium Mathematicae

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Let Λ be an artin algebra. We prove that for each sequence ${(h_{i})}_{i \in ℤ}$ of non-negative integers there are only a finite number of isomorphism classes of indecomposables $X \in^{b} (Λ)$ , the bounded derived category of Λ, with $l e n g t h_{E (X)} H^{i} (X) = h_{i}$ for all i ∈ ℤ and E(X) the endomorphism ring of X in $^{b} (Λ)$ if and only if $^{b} (M o d Λ)$ , the bounded derived category of the category $M o d Λ$ of all left Λ-modules, has no generic objects in the sense of [4].

Endotrivial modules over groups with quaternion or semi-dihedral Sylow 2-subgroup

Jon F. Carlson, Nadia Mazza, Jacques Thévenaz (2013)

Journal of the European Mathematical Society

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Let $G$ be a finite group with a Sylow 2-subgroup $P$ which is either quaternion or semi-dihedral. Let $k$ be an algebraically closed field of characteristic 2. We prove the existence of exotic endotrivial $k G$ -modules, whose restrictions to $P$ are isomorphic to the direct sum of the known exotic endotrivial $k P$ -modules and some projective modules. This provides a description of the group $T (G)$ of endotrivial $k G$ -modules.

Generalised functions of bounded deformation

Gianni Dal Maso (2013)

Journal of the European Mathematical Society

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We introduce the space $G B D$ of generalized functions of bounded deformation and study the structure properties of these functions: the rectiability and the slicing properties of their jump sets, and the existence of their approximate symmetric gradients. We conclude by proving a compactness results for $G B D$ , which leads to a compactness result for the space $G S B D$ of generalized special functions of bounded deformation. The latter is connected to the existence of solutions to a weak formulation...

Real deformations and invariants of map-germs

J. H. Rieger, M. A. S. Ruas, R. Wik Atique (2008)

Banach Center Publications

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A stable deformation $f^{t}$ of a real map-germ $f : ℝ ⁿ, 0 \to ℝ^{p}, 0$ is said to be an M-deformation if all isolated stable (local and multi-local) singularities of its complexification $f_{ℂ}^{t}$ are real. A related notion is that of a good real perturbation $f^{t}$ of f (studied e.g. by Mond and his coworkers) for which the homology of the image (for n < p) or discriminant (for n ≥ p) of $f^{t}$ coincides with that of $f_{C}^{t}$ . The class of map germs having an M-deformation is, in some sense, much larger than the one having a good...

Models of group schemes of roots of unity

A. Mézard, M. Romagny, D. Tossici (2013)

Annales de l’institut Fourier

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Let $𝒪_{K}$ be a discrete valuation ring of mixed characteristics $(0, p)$ , with residue field $k$ . Using work of Sekiguchi and Suwa, we construct some finite flat $𝒪_{K}$ -models of the group scheme $μ_{p^{n}, K}$ of $p^{n}$ -th roots of unity, which we call . We carefully set out the general framework and algebraic properties of this construction. When $k$ is perfect and $𝒪_{K}$ is a complete totally ramified extension of the ring of Witt vectors $W (k)$ , we provide a parallel study of the Breuil-Kisin modules of finite flat models of $μ_{p^{n}, K}$ ,...

Irregularity of an analogue of the Gauss-Manin systems

Céline Roucairol (2006)

Bulletin de la Société Mathématique de France

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In $𝒟$ -modules theory, Gauss-Manin systems are defined by the direct image of the structure sheaf $𝒪$ by a morphism. A major theorem says that these systems have only regular singularities. This paper examines the irregularity of an analogue of the Gauss-Manin systems. It consists in the direct image complex $f_{+} (𝒪 e^{g})$ of a $𝒟$ -module twisted by the exponential of a polynomial $g$ by another polynomial $f$ , where $f$ and $g$ are two polynomials in two variables. The analogue of the Gauss-Manin systems can...

Prescribing endomorphism algebras of $ℵ_{n}$ -free modules

Rüdiger Göbel, Daniel Herden, Saharon Shelah (2014)

Journal of the European Mathematical Society

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It is a well-known fact that modules over a commutative ring in general cannot be classified, and it is also well-known that we have to impose severe restrictions on either the ring or on the class of modules to solve this problem. One of the restrictions on the modules comes from freeness assumptions which have been intensively studied in recent decades. Two interesting, distinct but typical examples are the papers by Blass [1] and Eklof [8], both jointly with Shelah. In the first case...

On the structure theory of the Iwasawa algebra of a p-adic Lie group

Otmar Venjakob (2002)

Journal of the European Mathematical Society

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This paper is motivated by the question whether there is a nice structure theory of finitely generated modules over the Iwasawa algebra, i.e. the completed group algebra, $Λ$ of a $p$ -adic analytic group $G$ . For $G$ without any $p$ -torsion element we prove that $Λ$ is an Auslander regular ring. This result enables us to give a good definition of the notion of a pseudo-null $Λ$ -module. This is classical when $G = ℤ_{p}^{k}$ for some integer $k \geq 1$ , but was previously unknown in the non-commutative case. Then the category...

The multiplicity problem for indecomposable decompositions of modules over domestic canonical algebras

Piotr Dowbor, Andrzej Mróz (2008)

Colloquium Mathematicae

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Given a module M over a domestic canonical algebra Λ and a classifying set X for the indecomposable Λ-modules, the problem of determining the vector $m (M) = {(m_{x})}_{x \in X} \in ℕ^{X}$ such that $M ≅ ⨁_{x \in X} X_{x}^{m_{x}}$ is studied. A precise formula for $d i m_{k} H o m_{Λ} (M, X)$ , for any postprojective indecomposable module X, is computed in Theorem 2.3, and interrelations between various structures on the set of all postprojective roots are described in Theorem 2.4. It is proved in Theorem 2.2 that a general method of finding vectors m(M) presented by the authors...

On $τ$ -extending modules

Y. Talebi, R. Mohammadi (2016)

Commentationes Mathematicae Universitatis Carolinae

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In this paper we introduce the concept of $τ$ -extending modules by $τ$ -rational submodules and study some properties of such modules. It is shown that the set of all $τ$ -rational left ideals of $_{R} R$ is a Gabriel filter. An $R$ -module $M$ is called $τ$ -extending if every submodule of $M$ is $τ$ -rational in a direct summand of $M$ . It is proved that $M$ is $τ$ -extending if and only if $M = R e j_{M} E (R / τ (R)) \oplus N$ , such that $N$ is a $τ$ -extending submodule of $M$ . An example is given to show that the direct sum of $τ$ -extending modules need not...