Injective Resolutions Research Articles

Comodule theories in Grothendieck categories and relative Hopf objects

We develop the categorical algebra of the noncommutative base change of a comodule category by means of a Grothendieck category S. We describe when the resulting category of comodules is locally finitely generated, locally noetherian or may be recovered as a coreflective subcategory of the noncommutative base change of a module category. We also introduce the category SHA of relative (A,H)-Hopf modules in S, where H is a Hopf algebra and A is a right H-comodule algebra. We study the cohomological theory in SHA by means of spectral sequences. Using coinduction functors and functors of coinvariants, we study torsion theories and how they relate to injective resolutions in SHA. Finally, we use the theory of associated primes and support in noncommutative base change of module categories to give direct sum decompositions of minimal injective resolutions in SHA.

Strongly Gorenstein-injective modules over Morita rings

In this paper, we consider Morita rings with zero bimodule homomorphisms. We establish necessary and sufficient conditions for all strongly complete injective resolutions over a Morita ring [Formula: see text]. We also obtain necessary and sufficient conditions for all strongly Gorenstein-injective modules over [Formula: see text].

The Representation Theory of the Increasing Monoid

We study the representation theory of the increasing monoid. Our results provide a fairly comprehensive picture of the representation category: for example, we describe the Grothendieck group (including the effective cone), classify injective objects, establish properties of injective and projective resolutions, construct a derived auto-duality, and so on. Our work is motivated by numerous connections of this theory to other areas, such as representation stability, commutative algebra, simplicial theory, and shuffle algebras.

Injective and Tilting Resolutions and a Kazhdan-Lusztig Theory for the General Linear and Symplectic Group

Let k be an algebraically closed field of characteristic p > 0 and let G be a symplectic or general linear group over k. We consider induced modules for G under the assumption that p is bigger than the greatest hook length in the partitions involved. We give explicit constructions of left resolutions of induced modules by tilting modules. Furthermore, we give injective resolutions for induced modules in certain truncated categories. We show that the multiplicities of the indecomposable tilting and injective modules in these resolutions are the coefficients of certain Kazhdan-Lusztig polynomials. We also show that our truncated categories have a Kazhdan-Lusztig theory in the sense of Cline, Parshall and Scott. This builds further on work of Cox-De Visscher and Brundan-Stroppel.

Superized Troesch complexes and cohomology for strict polynomial superfunctors

We adapt a construction due to Troesch to the category of strict polynomial superfunctors in order to construct complexes of injective objects whose cohomology is isomorphic to Frobenius twists of the (super)symmetric power functors . We apply these complexes to construct injective resolutions of the even and odd Frobenius twist functors, to investigate the structure of the Yoneda algebra of the Frobenius twist functor, and to compute other extension groups between strict polynomial superfunctors. By an equivalence of categories, this also provides cohomology calculations in the category of left modules over Schur superalgebras.

Resolutions and homological dimensions of DG-modules

Recently, Yekutieli introduced projective dimension, injective dimension and flat dimension of DG-modules by generalizing the characterization of projective dimension, injective dimension and flat dimension of ordinary modules by vanishing of Ext or Tor-groups. In this paper, we introduce a DG-version of projective, injective and flat resolution for DG-modules over a connective DG-algebra which are different from the known DG-version of projective, injective and flat resolutions. An important feature of these resolutions is that, roughly speaking, the “length” of these resolutions gives projective, injective or flat dimensions. We show that these resolutions allow us to investigate basic properties of projective, injective and flat dimensions of DG-modules. As an application we introduce the global dimension of a connective DG-algebra and show that finiteness of the global dimension is derived invariant.

Littlewood complexes for symmetric groups

We construct a complex L ∙ λ \mathcal {L}_\bullet ^\lambda resolving the irreducible representations S λ [ n ] \mathcal {S}^{\lambda [n]} of the symmetric groups S n S_n by representations restricted from G L n ( k ) GL_n(k) . This construction lifts to R e p ( S ∞ ) \mathrm {Rep}(S_\infty ) , where it yields injective resolutions of simple objects. It categorifies stable Specht polynomials, and allows us to understand evaluations of these polynomials for all n n .

Homological identities and dualizing complexes of commutative differential graded algebras

In this paper we study a connective commutative differential graded algebra (CDGA) which is piecewise Noetherian. The principal aim is to analyze a dualizing complex of CDGA’s and investigate a Gorenstein CDGA. We also establish many other results which can be expected to play basic roles in the study of a CDGA, such as the Auslaner-Buchsbaum formula and Bass formula without any unnecessary assumptions. The key notion is the sup-projective (sppj) and inf-injective (ifij) resolutions introduced by the author, which are DG-versions of the projective and injective resolutions for ordinary modules. In the paper, we show that sppj and ifij resolutions are powerful tools to study DG-modules.

A version of purity on local abelian groups

In [6], generalizations of the standard notion of purity on p -local abelian groups were defined using functorial methods to create injective resolutions. For example, if \lambda is a limit ordinal, then for a group G the completion functor L_\lambda G determines the notion of L_\lambda -purity. Another way of constructing a type of purity, called p^{ < \lambda}_{\mathrm w} - purity , is defined using the functor \prod_{\alpha < \lambda} (G/p^\alpha G) . Properties of this second type of purity are studied; for example, it is shown to be hereditary if and only if \lambda has countable cofinality. In addition, L_\lambda and p^{< \lambda}_{\mathrm w} -purity are compared in a variety of contexts, for example, in the category of Warfield groups.

New Characterizations of Restricted Injective Dimensions for Complexes

In this paper, we study the restricted injective dimensions of complexes. Some new characterizations of the restricted injective dimensions are obtained. In particular, it is shown that the restricted injective dimensions can be computed in terms of the restricted injective resolutions. As applications, we get some results on the behavior of the restricted injective dimensions under change of rings. In addition, a characterization of almost Cohen-Macaulay ring is obtained.

Minimal resolutions of graded modules over an exterior algebra

Let K be a field, E the exterior algebra of a n --dimensional K -vector space V . We study projective and injective resolutions over E . More precisely, given a category M of finitely generated Z-graded left and right E -modules, we give upper bounds for the graded Betti numbers and the graded Bass numbers of classes of modules in M .

$\boldsymbol{n}$-coherent rings and semidualizing bimodules

In this paper, we characterize right $n$-coherent rings by $C$-$FP_n$-injective modules and $C$-$FP_n$-flat modules, where $_SC_R$ is a semidualizing bimodule and $n$ is an integer with $n>1$. We investigate right derived functors of $-\otimes-$ defined via proper right $C$-$FP_n$-flat resolutions and proper right $C$-$FP_n$-injective resolutions, and study left derived functorsof Hom$(-,-)$ defined via proper right (left) $C$-$FP_n$-injective resolutions. As applications, we give some characterizations of relative homological dimensions defined by $C$-$FP_n$-injective modules and $C$-$FP_n$-flat modules.

A general construction of n-angulated categories using periodic injective resolutions

Let C be an additive category equipped with an automorphism Σ. We show how to obtain n-angulations of (C,Σ) using some particular periodic injective resolutions. We give necessary and sufficient conditions on (C,Σ) admitting an n-angulation. Then we apply these characterizations to explain the standard construction of n-angulated categories and the n-angulated categories arising from some local rings. Moreover, we obtain a class of new examples of n-angulated categories from quasi-periodic selfinjective algebras.

Derived categories of N-complexes

We study the homotopy category $\mathsf{K}_{N}(\mathcal{B})$ of $N$-complexes of an additive category $\mathcal{B}$ and the derived category $\mathsf{D}_{N}(\mathcal{A})$ of an abelian category $\mathcal{A}$. First we show that both $\mathsf{K}_N(\mathcal{B})$ and $\mathsf{D}_N(\mathcal{A})$ have natural structures of triangulated categories. Then we establish a theory of projective (resp., injective) resolutions and derived functors. Finally, under some conditions of an abelian category $\mathcal{A}$, we show that $\mathsf{D}_{N}(\mathcal{A})$ is triangle equivalent to the ordinary derived category $\mathsf{D}(\mathsf{Morph}_{N-2}(\mathcal{A}))$ where $\mathsf{Morph}_{N-2}(\mathcal{A})$ is the category of sequential $N-2$ morphisms of $\mathcal{A}$.

English

Let $$\Lambda = \left( {\begin{array}{*{20}{c}} A&M 0&B \end{array}} \right)$$ be an Artin algebra. In view of the characterization of finitely generated Gorenstein injective Λ-modules under the condition that M is a cocompatible (A,B)-bimodule, we establish a recollement of the stable category $$\overline {Ginj\left( \Lambda \right)} $$ . We also determine all strongly complete injective resolutions and all strongly Gorenstein injective modules over Λ.

STABILITY PATTERNS IN REPRESENTATION THEORY

We develop a comprehensive theory of the stable representation categories of several sequences of groups, including the classical and symmetric groups, and their relation to the unstable categories. An important component of this theory is an array of equivalences between the stable representation category and various other categories, each of which has its own flavor (representation theoretic, combinatorial, commutative algebraic, or categorical) and offers a distinct perspective on the stable category. We use this theory to produce a host of specific results: for example, the construction of injective resolutions of simple objects, duality between the orthogonal and symplectic theories, and a canonical derived auto-equivalence of the general linear theory.

Residues and duality for singularity categories of isolated Gorenstein singularities

AbstractWe study Serre duality in the singularity category of an isolated Gorenstein singularity and find an explicit formula for the duality pairing in terms of generalised fractions and residues. For hypersurfaces we recover the residue formula of the string theorists Kapustin and Li. These results are obtained from an explicit construction of complete injective resolutions of maximal Cohen–Macaulay modules.

Ore localization and minimal injective resolutions

In this paper, we describe the structure of the localization of ExtRi(R/P,M), where P is a prime ideal and M is a module, at certain Ore sets X. We first study the situation for FBN rings, and then consider matters for a large class of Auslander–Gorenstein rings. We need to impose suitable homological regularity conditions to get results in the more general situation. The results obtained are then used to study the shape of minimal injective resolutions of modules over noetherian rings.

RESOLUTIONS AND DIMENSIONS OF RELATIVE INJECTIVE MODULES AND RELATIVE FLAT MODULES

Abstract. Let m and n be fixed positive integers and M a right R-module. Recall that M is said to be (m,n)-injective if Ext 1 (P,M) = 0for any (m,n)-presented right R-module P; M is said to be (m,n)-flatif Tor 1 (N,P) = 0 for any (m,n)-presented left R-module P. In termsof some derived functors, relative injective or relative flat resolutions anddimensions are investigated. As applications, some new characterizationsof von Neumann regular rings and p.p. rings are given. 1. IntroductionLet C be a class of left R-modules and M a left R-module. Following ([7]),we say that a homomorphism ϕ : M → C is a C-preenvelope of M if C ∈ C andthe abelian group homomorphism Hom(ϕ,C ′ ) : Hom (C,C ′ ) → Hom (M,C ′ ) issurjective for each C ′ ∈ C. A C-preenvelope ϕ : M → C is called a C-envelopeif every endomorphism f : C → C such that fϕ = ϕ is an isomorphism. AC-envelope ϕ : M → C is said to have the unique mapping property (see[6]) if for any homomorphism f : M → C ′ with C ′ ∈ C, there is a uniquehomomorphism g : C → C

Finiteness aspects of Gorenstein homological dimensions

We present an alternative way of measuring the Gorenstein projective (resp., injective) dimension of modules via a new type of complete projective (resp., injective) resolutions. As an application, we easily recover well known theorems such as the Ausland