Self-equivalences of stable module categories

Let P be an abelian p-group, E a cyclic -group acting freely on P and k an algebraically closed field of characteristic . In this work, we prove that every self-equivalence of the stable module category of comes from a self-equivalence of the derived category of . Work of Puig and Rickard allows us to deduce that if a block B with defect group P and inertial quotient E is Rickard equivalent to , then they are splendidly Rickard equivalent. That is, Broué's original conjecture implies Rickard's refinement of the conjecture in this case. All of this follows from a general result concerning the self-equivalences of the thick subcategory generated by the trivial module. Received January 22, 1998; in final form June 23, 1998     

Comparisons between singularity categories and relative stable categories of finite groups

We consider the relationship between the relative stable category of and the usual singularity category for group algebras with coefficients in a commutative noetherian ring. When the coefficient ring is self-injective we show that these categories share a common, relatively large, Verdier quotient. At the other extreme, when the coefficient ring has finite global dimension, there is a semi-orthogonal decomposition, due to Poulton, relating the two categories. We prove that this decomposition is partially compatible with the monoidal structure and study the morphism it induces on spectra.     

Splitting monoidal stable model categories

If C is a stable model category with a monoidal product then the set of homotopy classes of self-maps of the unit forms a commutative ring, [S,S]^C. An idempotent e of this ring will split the homotopy category: [X,Y]^C≅e[X,Y]^C⊕(1−e)[X,Y]^C. We prove that provided the localised model structures exist, this splitting of the homotopy category comes from a splitting of the model category, that is, C is Quillen equivalent to L_eSC×L_(1−e)SC and [X,Y]^L_eSC≅e[X,Y]^C. This Quillen equivalence is strong monoidal and is symmetric when the monoidal product of C is.     

Semi-dualizing complexes and their Auslander categories

Let be a commutative Noetherian ring. We study -modules, and complexes of such, with excellent duality properties. While their common properties are strong enough to admit a rich theory, we count among them such, potentially, diverse objects as dualizing complexes for on one side, and on the other, the ring itself. In several ways, these two examples constitute the extremes, and their well-understood properties serve as guidelines for our study; however, also the employment, in recent studies of ring homomorphisms, of complexes ``lying between' these extremes is incentive.

     

Compact objects in stable module categories

Let R be a perfect ring, the stable module category of right R-modules. We show that any compact object in is isomorphic to some finitely generated R-module. Moreover, we apply the above to stable equivalences between module categories. Received: 10 April 2006     

Frobenius morphisms and stable module categories of repetitive algebras

Let k be the algebraic closure of a finite field F_q and A be a finite dimensional k-algebra with a Frobenius morphism F.In the present paper we establish a relation between the stable module category of the repetitive algebra  of A and that of the repetitive algebra of the fixed-point algebra A~F.As an application,it is shown that the derived category of A~F is equivalent to the subcategory of F-stable objects in the derived category of A when A has a finite global dimension.     

Morphism spaces in stable categories of Frobenius algebras

We present an explicit formula computing morphism spaces in the stable category of a Frobenius algebra.     

Resolution of unbounded complexes in Grothendieck categories

As Spaltenstein showed, the category of unbounded complexes of sheaves on a topological space has enough K-injective complexes. We extend this result to the category of unbounded complexes of an arbitrary Grothendieck category. This is important for a construction, by the author, of a triangulated category of equivariant motives.     

Tchebyshev triangulations of stable simplicial complexes

We generalize the notion of the Tchebyshev transform of a graded poset to a triangulation of an arbitrary simplicial complex in such a way that, at the level of the associated F-polynomials j_∑fj−1(j(x−1)/2), the triangulation induces taking the Tchebyshev transform of the first kind. We also present a related multiset of simplicial complexes whose association induces taking the Tchebyshev transform of the second kind. Using the reverse implication of a theorem by Schelin we observe that the Tchebyshev transforms of Schur stable polynomials with real coefficients have interlaced real roots in the interval (−1,1), and present ways to construct simplicial complexes with Schur stable F-polynomials. We show that the order complex of a Boolean algebra is Schur stable. Using and expanding the recently discovered relation between the derivative polynomials for tangent and secant and the Tchebyshev polynomials we prove that the roots of the corresponding pairs of derivative polynomials are all pure imaginary, of modulus at most one, and interlaced.     

Exact categories and vector space categories

In a series of papers additive subbifunctors of the bifunctor are studied in order to establish a relative homology theory for an artin algebra . On the other hand, one may consider the elements of as short exact sequences. We observe that these exact sequences make into an exact category if and only if is closed in the sense of Butler and Horrocks.

Concerning the axioms for an exact category we refer to Gabriel and Roiter's book. In fact, for our general results we work with subbifunctors of the extension functor for arbitrary exact categories.

In order to study projective and injective objects for exact categories it turns out to be convenient to consider categories with almost split exact pairs, because many earlier results can easily be adapted to this situation.

Exact categories arise in representation theory for example if one studies categories of representations of bimodules. Representations of bimodules gained their importance in studying questions about representation types. They appear as domains of certain reduction functors defined on categories of modules. These reduction functors are often closely related to the functor and in general do not preserve at all the usual exact structure of .

By showing the closedness of suitable subbifunctors of we can equip with an exact structure such that some reduction functors actually become `exact'. This allows us to derive information about the projective and injective objects in the respective categories of representations of bimodules appearing as domains, and even show that almost split sequences for them exist.

Examples of such domains appearing in practice are the subspace categories of a vector space category with bonds. We provide an example showing that existence of almost split sequences for them is not a general fact but may even fail if the vector space category is finite.

     

Mutation pairs and quotient categories of Abelian categories

The notion of 𝒟-mutation pairs of subcategories in an abelian category is defined in this article. When (𝒵,𝒵) is a 𝒟-mutation pair in an abelian category 𝒜, the quotient category 𝒵∕𝒟 carries naturally a triangulated structure. Moreover, our result generalize the construction of the quotient triangulated category by Happel [10 Happel, D. (1988). Triangulated Categories in the Representation of Finite Dimensional Algebras. London Mathematical Society, LMN, Vol. 119. Cambridge: Cambridge University Press.[Crossref] , [Google Scholar], Theorem 2.6]. Finally, we find a one-to-one correspondence between cotorsion pairs in 𝒜 and cotorsion pairs in the quotient category 𝒵∕𝒟, and study homological finiteness of subcategories in a mutation pair.     

Finitely semisimple spherical categories and modular categories are self-dual

We show that every essentially small finitely semisimple k-linear additive spherical category for which k=End(1) is a field, is equivalent to its dual over the long canonical forgetful functor. This includes the special case of modular categories. In order to prove this result, we show that the universal coend of the spherical category, with respect to the long forgetful functor, is self-dual as a Weak Hopf Algebra.     

Special precovered categories of Gorenstein categories

Let A be an abelian category and P(A)be the subcategory of A consisting of projective objects.Let C be a full,additive and self-orthogonal subcategory of A with P(A)a generator,and let G(C)be the Gorenstein subcategory of A.Then the right 1-orthogonal category G(C)~⊥1 of G(C)is both projectively resolving and injectively coresolving in A.We also get that the subcategory SPC(G(C))of A consisting of objects admitting special G(C)-precovers is closed under extensions and C-stable direct summands(*).Furthermore,if C is a generator for G(C)~⊥1,then we have that SPC(G(C))is the minimal subcategory of A containing G(C)~⊥1∪G(C)with respect to the property(*),and that SPC(G(C))is C-resolving in A with a C-proper generator C.     

Relation categories and coproduct congruence categories in universal algebra

It is known that a categoryV-Rel ofadmissible relations can be formed for any variety of algebrasV, such that morphismsAB correspond to subalgebras ofA x B. We adapt the relation category construction of Hilton and Wu to categoriesC with finite limits and colimits and an image factorization system. The existence ofC-Rel and a dualcograph constructionC-Cogr are proved equivalent to certain stability properties of pullbacks or pushouts forC. For algebraic varietiesV,V -Cogr exists iffV satisfies the amalgamation property (AP) and the congruence extension property (CEP). MorphismsAB inV-Cogr correspond to congruences on the coproductA + B. It is showed that congruence permutability (CP), the intersection property for amalgamations (IPA), the Hamiltonian property, and the property that congruences 6 are determined by the equivalence class [0] can be given characterizations in terms of interlocked pullbacks and pushouts in such a categoryC. A new property IDA (intersections determine amalgamations) is defined, which is dual to CP in this context. Familiar results, such as CP implies congruence modularity, can be proved in such categories. Dually, ifV satisfies AP, CEP, IPA and IDA, it has modular lattices of subalgebras. These results are related to order duality for Su and Con. (For certain varietiesV, the subalgebras ofA are in one-one correspondence with the morphisms below 1_A inV -Rel orV-Cogr, and the congruences correspond to the morphisms above 1_A.) IfV is pointed (eachA in V has a smallest trivial subalgebra), then a category formulation is obtained for: CP implies the Jónsson-Tarski decomposition properties. The dual shows that pointed varieties satisfying IDA have a restricted form, with pointed unary varieties and varieties ofR-modules as special cases.Dedicated to Bjarni Jónsson on his 70th birthday ntprbPresented by G. McNulty.     

Equivalences between categories of modules and categories of comodules

We show the close connection between apparently different Galois theories for comodules introduced recently in [J. Gomez-Torrecillas and J. Vercruysse, Comatrix corings and Galois Comodules over firm rings, Algebr. Represent. Theory, 10 （2007）, 271 306] and [Wisbauer, On Galois comodules, Comm. Algebra 34 （2006）, 2683-2711]. Furthermore we study equivalences between categories of comodules over a coring and modules over a firm ring. We show that these equivalences are related to Galois theory for comodules.