Injective Presentations of Induced Modules over Cluster-Tilted Algebras

Every cluster-tilted algebra B is the relation extension \(C\ltimes \textup {Ext}^{2}_{C}(DC,C)\) of a tilted algebra C. A B-module is called induced if it is of the form M? _C B for some C-module M. We study the relation between the injective presentations of a C-module and the injective presentations of the induced B-module. Our main result is an explicit construction of the modules and morphisms in an injective presentation of any induced B-module. In the case where the C-module, and hence the B-module, is projective, our construction yields an injective resolution. In particular, it gives a module theoretic proof of the well-known 1-Gorenstein property of cluster-tilted algebras.     

Stratified modules over an extension algebra

Let A be a standard Koszul standardly stratified algebra and X an A-module. The paper investigates conditions which imply that the module Ext* _A (X) over the Yoneda extension algebra A* is filtered by standard modules. In particular, we prove that the Yoneda extension algebra of A is also standardly stratified. This is a generalization of similar results on quasi-hereditary and on graded standardly stratified algebras.     

Invariants and rings of quotients of <Emphasis Type="Italic">H</Emphasis>-semiprime <Emphasis Type="Italic">H</Emphasis>-module algebras satisfying a polynomial identity

We consider an action of a finite-dimensional Hopf algebra H on a PI-algebra. We prove that an H-semiprime H-module algebra A has a Frobenius artinian classical ring of quotients Q, provided that A has a finite set of H-prime ideals with zero intersection. The ring of quotients Q is an H-semisimple H-module algebra and a finitely generated module over the subalgebra of central invariants. Moreover, if algebra A is a projective module of constant rank over its center, then A is integral over its subalgebra of central invariants.     

<Emphasis Type="Italic">V</Emphasis>-Gorenstein injective modules preenvelopes and related dimension

Let R and S be associative rings and _S V _R a semidualizing (S-R)-bimodule. An R-module N is said to be V-Gorenstein injective if there exists a Hom _R (I _V (R),?) and Hom _R (?,I _V (R)) exact exact complex \( \cdots \to {I_1}\xrightarrow{{{d_0}}}{I_0} \to {I^0}\xrightarrow{{{d_0}}}{I^1} \to \cdots \) of V-injective modules I _i and I ⁱ , i ∈ N₀, such that N ? Im(I ₀ → I ⁰). We will call N to be strongly V-Gorenstein injective in case that all modules and homomorphisms in the above exact complex are equal, respectively. It is proved that the class of V-Gorenstein injective modules are closed under extension, direct summand and is a subset of the Auslander class A _V (R) which leads to the fact that V-Gorenstein injective modules admit exact right I _V (R)-resolution. By using these facts, and thinking of the fact that the class of strongly V-Gorenstein injective modules is not closed under direct summand, it is proved that an R-module N is strongly V-Gorenstein injective if and only if N ⊕ E is strongly V-Gorenstein injective for some V-injective module E. Finally, it is proved that an R-module N of finite V-Gorenstein injective injective dimension admits V-Gorenstein injective preenvelope which leads to the fact that, for a natural integer n, Gorenstein V-injective injective dimension of N is bounded to n if and only if \(Ext_{{I_V}\left( R \right)}^{ \geqslant n + 1}\left( {I,N} \right) = 0\) for all modules I with finite I _V (R)-injective dimension.     

The Nodal Cubic is a Quantum Homogeneous Space

The cusp was recently shown to admit the structure of a quantum homogeneous space, that is, its coordinate ring B can be embedded as a right coideal subalgebra into a Hopf algebra A such that A is faithfully flat as a B-module. In the present article such a Hopf algebra A is constructed for the coordinate ring B of the nodal cubic, thus further motivating the question which affine varieties are quantum homogeneous spaces.     

<Emphasis Type="Italic">τ</Emphasis>-Tilting Modules Over One-Point Extensions by a Projective Module

Let A be the one point extension of an algebra B by a projective B-module. We prove that the extension of a given support τ-tilting B-module is a support τ-tilting A-module; and, conversely, the restriction of a given support τ-tilting A-module is a support τ-tilting B-module. Moreover, we prove that there exists a full embedding of quivers between the corresponding poset of support τ-tilting modules.     

Spectral sequence and finitely presented dimension for weak Gopf–Galois extensions

Let H be a weak Hopf algebra, A a right weak H-comodule algebra, and B the subalgebra of the H-coinvariant elements of A. Let A/B be a right weak H-Galois extension. In this paper, a spectral sequence for Ext which yields an estimate for the global dimension of A in terms of the corresponding data for H and B is constructed. Next, the relationship between the finitely presented dimensions of A and its subalgebra B are given. Further, the case in which A is an n-Gorenstein algebra is studied.     

Invariants of the action of a semisimple Hopf algebra on PI-algebra

We extend several classical results in the theory of invariants of finite groups to the case of action of a finite-dimensional Hopf algebra H on an algebra satisfying a polynomial identity. In particular, we prove that an H-module algebra A over an algebraically closed field k is integral over the subalgebra of invariants, if H is a semisimple and cosemisimple Hopf algebra. We show that for char k > 0, the algebra Z\({\left( A \right)^{{H_0}}}\) is integral over the subalgebra of central invariants Z(A)^H, where Z(A) is the center of algebra A, H⁰ is the coradical of H. This result allowed us to prove that the algebra A is integral over the subalgebra Z(A)^H in some special case. We also construct a counterexample to the integrality of the algebra \({A^{{H_0}}}\) over the subalgebra of invariants A^H for a pointed Hopf algebra over a field of non-zero characteristic.     

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.     

The number of simple modules of a cellular algebra

Let n be a natural number, and let A be an indecomposable cellular algebra such that the spectrum of its Cartan matrix C is of the form {n, 1, …., 1}. In general, not every natural number could be the number of non-isomorphic simple modules over such a cellular algebra. Thus, two natural questions arise: (1) which numbers could be the number of non-isomorphic simple modules over such a cellular algebra A ? (2) Given such a number, is there a cellular algebra such that its Cartan matrix has the desired property ? In this paper, we shall completely answer the first question, and give a partial answer to the second question by constructing cellular algebras with the pre-described Cartan matrix.     

Weak module amenability of triangular Banach algebras

Let A and B be unital Banach algebras and let M be a unital Banach A,B-module. Forrest and Marcoux [6] have studied the weak amenability of triangular Banach algebra \(\mathcal{T} = \left[ {_B^{AM} } \right]\) and showed that T is weakly amenable if and only if the corner algebras A and B are weakly amenable. When \(\mathfrak{A}\) is a Banach algebra and A and B are Banach \(\mathfrak{A}\)-module with compatible actions, and M is a commutative left Banach \(\mathfrak{A}\)-A-module and right Banach \(\mathfrak{A}\)-B-module, we show that A and B are weakly \(\mathfrak{A}\)-module amenable if and only if triangular Banach algebra T is weakly \(\mathfrak{T}\)-module amenable, where \(\mathfrak{T}: = \{ [^\alpha _\alpha ]:\alpha \in \mathfrak{A}\} \).     

On a Variety Related to the Commuting Variety of a Reductive Lie Algebra

For a reductive Lie algbera over an algbraically closed field of charasteristic zero, we consider a Borel subgroup B of its adjoint group, a Cartan subalgebra contained in the Lie algebra of B and the closure X of its orbit under B in the Grassmannian. The variety X plays an important role in the study of the commuting variety. In this note, we prove that X is Gorenstein with rational singularities.     

Twisted Frobenius Extensions of Graded Superrings

We define twisted Frobenius extensions of graded superrings. We develop equivalent definitions in terms of bimodule isomorphisms, trace maps, bilinear forms, and dual sets of generators. The motivation for our study comes from categorification, where one is often interested in the adjointness properties of induction and restriction functors. We show that A is a twisted Frobenius extension of B if and only if induction of B-modules to A-modules is twisted shifted right adjoint to restriction of A-modules to B-modules. A large (non-exhaustive) class of examples is given by the fact that any time A is a Frobenius graded superalgebra, B is a graded subalgebra that is also a Frobenius graded superalgebra, and A is projective as a left B-module, then A is a twisted Frobenius extension of B.     

Koszulity and Koszul modules of dual extension algebras

Let A and B be algebras, and let T be the dual extension algebra of A and B. We provide a different method to prove that T is Koszul if and only if both A and B are Koszul. Furthermore, we prove that an algebra is Koszul if and only if one of its iterated dual extension algebras is Koszul, if and only if all its iterated dual extension algebras are Koszul. Finally, we give a necessary and sufficient condition for a dual extension algebra to have the property that all linearly presented modules are Koszul modules, which provides an effective way to construct algebras with such a property.     

(Strongly) Gorenstein injective modules over upper triangular matrix Artin algebras

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.     

Characteristic Modules of Dual Extensions and Groebner Bases

Let C be a finite dimensional directed algebra over an algebraically closed field k and A=A(C) the dual extension of C. The characteristic modules of A are constructed explicitly for a class of directed algebras, which generalizes the results of Xi. Furthermore, it is shown that the characteristic modules of dual extensions of a certain class of directed algebras admit the left Groebner basis theory in the sense of E. L. Green.     

Finitistic dimension and restricted injective dimension

We study the relations between finitistic dimensions and restricted injective dimensions. Let R be a ring and T a left R-module with A = End _R T. If _R T is selforthogonal, then we show that rid(T _A ) ? findim(A _A ) ? findim( _R T) + rid(T _A ). Moreover, if R is a left noetherian ring and T is a finitely generated left R-module with finite injective dimension, then rid(T _A ) ? findim(A _A ) ? fin.inj.dim( _R R)+rid(T _A ). Also we show by an example that the restricted injective dimensions of a module may be strictly smaller than the Gorenstein injective dimension.     

RESTRICTED LIE ALGEBRAS WITH MAXIMAL 0-PIM

In this paper it is shown that the projective cover of the trivial irreducible module of a finite-dimensional solvable restricted Lie algebra is induced from the one dimensional trivial module of a maximal torus. As a consequence, the number of the isomorphism classes of irreducible modules with a fixed p-character for a finite-dimensional solvable restricted Lie algebra L is bounded above by p ^MT(L), where MT(L) denotes the maximal dimension of a torus in L. Finally, it is proved that in characteristic p > 3 the projective cover of the trivial irreducible L-module is induced from the one-dimensional trivial module of a torus of maximal dimension, only if L is solvable.     

Tensor products of tilting modules

We consider whether the tilting properties of a tilting A-module T and a tilting B-module T′ can convey to their tensor product T ? T′: The main result is that T ? T′ turns out to be an (n + m)-tilting A ? B-module, where T is an m-tilting A-module and T′ is an n-tilting B-module.     

Realization of Poisson enveloping algebra

For a Poisson algebra, the category of Poisson modules is equivalent to the module category of its Poisson enveloping algebra, where the Poisson enveloping algebra is an associative one. In this article, for a Poisson structure on a polynomial algebra S, we first construct a Poisson algebra R, then prove that the Poisson enveloping algebra of S is isomorphic to the specialization of the quantized universal enveloping algebra of R, and therefore, is a deformation quantization of R.