Derived Equivalences for Triangular Matrix Rings

We generalize derived equivalences for triangular matrix rings induced by a certain type of classical tilting module introduced by Auslander, Platzeck and Reiten to generalize reflection functors in the representation theory of quivers due to Bernstein, Gelfand and Ponomarev.     

On Derived Equivalences for Selfinjective Algebras

We show that if A is a representation-finite selfinjective Artin algebra, then every P ^? ? K ^b(𝒫 _A ) with Hom _K(Mod?A)(P ^?,P ^?[i]) = 0 for i ≠ 0 and add(P ^?) = add(νP ^?) is a direct summand of a tilting complex, and that if A, B are derived equivalent representation-finite selfinjective Artin algebras, then there exists a sequence of selfinjective Artin algebras A = B ₀, B ₁,…, B _m  = B such that, for any 0 ≤ i < m, B _i+1 is the endomorphism algebra of a tilting complex for B _i of length ≤ 1.     

Derived Equivalences Between Subrings

In this paper, we construct derived equivalences between two subrings of relevant Φ-Yoneda rings from an arbitrary short exact sequence in an abelian category. As a consequence, any short exact sequence in an abelian category gives rise to a derived equivalence between two subrings of endomorphism rings.     

Repetitive代数的导出等价

若代数A和B导出等价,则它们的Repetitive代数A和B也是导出等价,从而是稳定等价.这样利用更直接的代数方法完全回答了H.Asashiba提出的问题,并推广了Rickard,杜先能,Tachikawa-Wakamatsu等人的相应结果.进而,把以上结果推广到上有界复形的导出范畴的对称recollement的情形.     

Morita Equivalent Blocks in Non-Normal Subgroups and p-Radical Blocks in Finite Groups

Let O be a complete discrete valuation ring with unique maximalideal J(O), let K be its quotient field of characteristic 0,and let k be its residue field O/J(O) of prime characteristicp. We fix a finite group G, and we assume that K is big enoughfor G, that is, K contains all the |G|-th roots of unity, where|G| is the order of G. In particular, K and k are both splittingfields for all subgroups of G. Suppose that H is an arbitrarysubgroup of G. Consider blocks (block ideals) A and B of thegroup algebras RG and RH, respectively, where R{O, k}. We considerthe following question: when are A and B Morita equivalent?Actually, we deal with ‘naturally Morita equivalent blocksA and B’, which means that A is isomorphic to a full matrixalgebra of B, as studied by B. Külshammer. However, Külshammerassumes that H is normal in G, and we do not make this assumption,so we get generalisations of the results of Külshammer.Moreover, in the case H is normal in G, we get the same resultsas Külshammer; however, he uses the results of E. C. Dade,and we do not.     

Derived Equivalences in n-Angulated Categories

In this paper, we consider n-perforated Yoneda algebras for n-angulated categories, and show that the construction of Hu, König and Xi is still available for n-angulated categories. Namely, under some conditions, n-angles induce derived equivalences between the quotient algebras of n-perforated Yoneda algebras. This result generalizes the corresponding results for triangulated categories of Hu, König and Xi.     

Restriction Between Cohomology Algebras of Blocks of Finite Groups

Let k be an algebraically closed field of characteristic p and G be a finite group. Let N be a normal subgroup of G and c be a G-stable block of kN. We shall discuss the cohomology algebra of the block c, defined by M. Linckelmann and, in this case a generalized block cohomology which can be defined using some generalized Brauer pairs, denoted (c, G)-Brauer pairs, which are introduced by R. Kessar and R. Stancu. We also analyze the restriction map between these two cohomology algebras associated to the block c through transfer maps between the Hochschild cohomology algebras of kGc and of the block c.     

Transfer in Hochschild Cohomology of Blocks of Finite Groups

We develop the notion of a cohomology ring of blocks of finite groups and study its basic properties by means of transfer maps between the Hochschild cohomology rings of symmetric algebras associated with bounded complexes of finitely generated bimodules which are projective on either side.     

Generalized Auslander-Reiten Conjecture and Derived Equivalences

In this note, we prove that the generalized Auslander-Reiten conjecture is preserved under derived equivalences between Artin algebras.     

Cohomology Algebras of Blocks of Finite Groups and Brauer Correspondence

M. Linckelmann defined the cohomology algebras of blocks of finite groups. This note is an attempt to analyze an inclusion of cohomology algebras of blocks that corresponds under Brauer correspondence through transfer maps between the Hochschild cohomology algebras of the blocks.Presented by Jon Carlson.     

Equivalent Blocks of Finite General Linear Groups in Non-describing Characteristic

J. Chuang, R. Kessar, and J. Rickard have proved Broué's Abelian defect group conjecture for many symmetric groups. We adapt the ideas of Kessar and Chuang towards finite general linear groups (represented over non-describing characteristic). We then describe Morita equivalences between certain p-blocks of GL_n(q) with defect group C_p^α × C_p^α, as q varies (see Theorem 2). Here p and q are coprime. This generalizes work of S. Koshitani and M. Hyoue, who proved the same result for principal blocks of GL_n(q) when p = 3, α = 1, in a different way.     

Derived Equivalences of Actions of a Category

Let $\mathbb{k}$ be a commutative ring and I a category. As a generalization of a $\mathbb{k}$ -category with a (pseudo) action of a group we consider a family of $\mathbb{k}$ -categories with a (pseudo, lax, or oplax) action of I, namely an oplax functor from I to the 2-category of small $\mathbb{k}$ -categories. We investigate derived equivalences of those oplax functors, and establish a Morita type theorem for them. This gives a base of investigations of derived equivalences of Grothendieck constructions of those oplax functors.     

On Standard Derived Equivalences of Orbit Categories

Let k be a commutative ring, \(\mathcal {A}\) and \(\mathcal {B}\) – two k-linear categories with an action of a group G. We introduce the notion of a standard G-equivalence from \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {B}\) to \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}\), where \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}\) is the homotopy category of finitely generated projective \(\mathcal {A}\)-complexes. We construct a map from the set of standard G-equivalences to the set of standard equivalences from \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {B}\) to \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}\) and a map from the set of standard G-equivalences from \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {B}\) to \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}\) to the set of standard equivalences from \(\mathcal {K}_{p}^{\mathrm {b}}(\mathcal {B}/G)\) to \(\mathcal {K}_{p}^{\mathrm {b}}(\mathcal {A}/G)\), where \(\mathcal {A}/G\) denotes the orbit category. We investigate the properties of these maps and apply our results to the case where \(\mathcal {A}=\mathcal {B}=R\) is a Frobenius k-algebra and G is the cyclic group generated by its Nakayama automorphism ν. We apply this technique to obtain the generating set of the derived Picard group of a Frobenius Nakayama algebra over an algebraically closed field.     

Cohomology Algebras of Blocks of Finite Groups and Brauer Correspondence II

Let k be an algebraically closed field of characteristic p. We shall discuss the cohomology algebras of a block ideal B of the group algebra kG of a finite group G and a block ideal C of the block ideal of kH of a subgroup H of G which are in Brauer correspondence and have a common defect group, continuing (Kawai and Sasaki, Algebr Represent Theory 9(5):497–511, 2006). We shall define a (B,C)-bimodule L. The k-dual L ^* induces the transfer map between the Hochschild cohomology algebras of B and C, which restricts to the inclusion map of the cohomology algebras of B into that of C under some condition. Moreover the module L induces a kind of refinement of Green correspondence between indecomposable modules lying in the blocks B and C; the block varieties of modules lying in B and C which are in Green correspondence will also be discussed.     

Derived Equivalences of Upper Triangular Differential Graded Algebras

This paper generalises a result for upper triangular matrix rings to the situation of upper triangular matrix differential graded algebras. An upper triangular matrix DGA has the form (R, S, M) where R and S are differential graded algebras and M is a DG-left-R-right-S-bimodule. We show that under certain conditions on the DG-module M and with the existance of a DG-R-module X, from which we can build the derived category D(R), that there exists a derived equivalence between the upper triangular matrix DGAs (R, S, M) and (S, M′, R′), where the DG-bimodule M′ is obtained from M and X and R′ is the endomorphism differential graded algebra of a K-projective resolution of X.     

Formal Groups of Building Blocks Completely Defined Over Finite Abelian Extensions of Q

Let C be an elliptic curve defined over Q. We can associatetwo formal groups with C: the formal group (X, Y) determinedby the formal completion of the Néron model of C overZ along the zero section, and the formal group F_L(X, Y) of theL-series attached to l-adic representations on C of the absoluteGalois group of Q. Honda shows that F_L(X, Y) is defined overZ, and it is strongly isomorphic over Z to (X, Y). In this paperwe give a generalization of the result of Honda to buildingblocks over finite abelian extensions of Q. The difficulty isto define new matrix L-series of building blocks. Our generalizationcontains the generalization of Deninger and Nart to abelianvarieties of GL₂-type. It also contains the generalization ofour previous paper to Q-curves over quadratic fields. 2000 MathematicsSubject Classification 11G10 (primary), 11F11 (secondary).     

拓扑偶的自同伦等价群

研究拓扑偶范畴中的自同伦等价群.对于同伦可结合与同伦可逆的Co-H空间X与Y,我们将在一定的条件下得到一条可裂的短正合序列:1→(?)(X)→(?)(X(?)Y)→(?)(Y)→1.     

The Grothendieck Groups and Stable Equivalences of Mesh Algebras

We deal with the finite-dimensional mesh algebras given by stable translation quivers. These algebras are self-injective, and thus the stable module categories have a structure of triangulated categories. Our main result determines the Grothendieck groups of these stable module categories. As an application, we give a complete classification of the mesh algebras up to stable equivalences.