Displaying similar documents to “Projective abelian Hopf algebras over a field”

Generalized E-algebras via λ-calculus I

Rüdiger Göbel, Saharon Shelah (2006)

Fundamenta Mathematicae

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An R-algebra A is called an E(R)-algebra if the canonical homomorphism from A to the endomorphism algebra E n d R A of the R-module R A , taking any a ∈ A to the right multiplication a r E n d R A by a, is an isomorphism of algebras. In this case R A is called an E(R)-module. There is a proper class of examples constructed in [4]. E(R)-algebras arise naturally in various topics of algebra. So it is not surprising that they were investigated thoroughly in the last decades; see [3, 5, 7, 8, 10, 13, 14, 15, 18,...

Monomorphisms of coalgebras

A. L. Agore (2010)

Colloquium Mathematicae

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We prove new necessary and sufficient conditions for a morphism of coalgebras to be a monomorphism, different from the ones already available in the literature. More precisely, φ: C → D is a monomorphism of coalgebras if and only if the first cohomology groups of the coalgebras C and D coincide if and only if i I ε ( a i ) b i = i I a i ε ( b i ) for all i I a i b i C D C . In particular, necessary and sufficient conditions for a Hopf algebra map to be a monomorphism are given.

Generic representations of orthogonal groups: projective functors in the category q u a d

Christine Vespa (2008)

Fundamenta Mathematicae

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We continue the study of the category of functors q u a d , associated to ₂-vector spaces equipped with a nondegenerate quadratic form, initiated in J. Pure Appl. Algebra 212 (2008) and Algebr. Geom. Topology 7 (2007). We define a filtration of the standard projective objects in q u a d ; this refines to give a decomposition into indecomposable factors of the first two standard projective objects in q u a d : P H and P H . As an application of these two decompositions, we give a complete description of the polynomial...

Cartan-Eilenberg projective, injective and flat complexes

Xiaorui Zhai, Chunxia Zhang (2016)

Czechoslovak Mathematical Journal

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Let R be an associative ring with identity and a class of R -modules. In this article: we first give a detailed treatment of Cartan-Eilenberg complexes and extend the basic properties of the class to the class CE ( ). Secondly, we study and give some equivalent characterizations of Cartan-Eilenberg projective, injective and flat complexes which are similar to projective, injective and flat modules, respectively. As applications, we characterize some classical rings in terms of these...

Singularity categories of skewed-gentle algebras

Xinhong Chen, Ming Lu (2015)

Colloquium Mathematicae

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Let K be an algebraically closed field. Let (Q,Sp,I) be a skewed-gentle triple, and let ( Q s g , I s g ) and ( Q g , I g ) be the corresponding skewed-gentle pair and the associated gentle pair, respectively. We prove that the skewed-gentle algebra K Q s g / I s g is singularity equivalent to KQ/⟨I⟩. Moreover, we use (Q,Sp,I) to describe the singularity category of K Q g / I g . As a corollary, we find that g l d i m K Q s g / I s g < if and only if g l d i m K Q / I < if and only if g l d i m K Q g / I g < .

Hall algebras of two equivalent extriangulated categories

Shiquan Ruan, Li Wang, Haicheng Zhang (2024)

Czechoslovak Mathematical Journal

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For any positive integer n , let A n be a linearly oriented quiver of type A with n vertices. It is well-known that the quotient of an exact category by projective-injectives is an extriangulated category. We show that there exists an extriangulated equivalence between the extriangulated categories n + 1 and n , where n + 1 and n are the two extriangulated categories corresponding to the representation category of A n + 1 and the morphism category of projective representations of A n , respectively. As a...

A Note on Lax Projective Embeddings of Grassmann Spaces

Eva Ferrara Dentice (2018)

Rendiconto dell’Accademia delle Scienze Fisiche e Matematiche

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In the paper (Ferrara Dentice et al., 2004) a complete exposition of the state of the art for lax embeddings of polar spaces of finite rank 3 is presented. As a consequence, we have that if a Grassmann space G of dimension 3 and index 1 has a lax embedding in a projective space over a skew–field K , then G is the Klein quadric defined over a subfield of K . In this paper, I examine Grassmann spaces of arbitrary dimension d 3 and index h 1 having a lax embedding in a projective space.

Twisted group rings of strongly unbounded representation type

Leonid F. Barannyk, Dariusz Klein (2004)

Colloquium Mathematicae

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Let S be a commutative local ring of characteristic p, which is not a field, S* the multiplicative group of S, W a subgroup of S*, G a finite p-group, and S λ G a twisted group ring of the group G and of the ring S with a 2-cocycle λ ∈ Z²(G,S*). Denote by I n d m ( S λ G ) the set of isomorphism classes of indecomposable S λ G -modules of S-rank m. We exhibit rings S λ G for which there exists a function f λ : such that f λ ( n ) n and I n d f λ ( n ) ( S λ G ) is an infinite set for every natural n > 1. In special cases f λ ( ) contains every natural...

The duality theorem for twisted smash products of Hopf algebras and its applications

Zhongwei Wang, Liangyun Zhang (2015)

Colloquium Mathematicae

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Let A T H denote the twisted smash product of an arbitrary algebra A and a Hopf algebra H over a field. We present an analogue of the celebrated Blattner-Montgomery duality theorem for A T H , and as an application we establish the relationship between the homological dimensions of A T H and A if H and its dual H* are both semisimple.

Examples of polynomial identities distinguishing the Galois objects over finite-dimensional Hopf algebras

Christian Kassel (2013)

Annales mathématiques Blaise Pascal

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We define polynomial H -identities for comodule algebras over a Hopf algebra  H and establish general properties for the corresponding T -ideals. In the case  H is a Taft algebra or the Hopf algebra  E ( n ) , we exhibit a finite set of polynomial H -identities which distinguish the Galois objects over  H up to isomorphism.

Finite groups of OTP projective representation type over a complete discrete valuation domain of positive characteristic

Leonid F. Barannyk, Dariusz Klein (2012)

Colloquium Mathematicae

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Let S be a commutative complete discrete valuation domain of positive characteristic p, S* the unit group of S, Ω a subgroup of S* and G = G p × B a finite group, where G p is a p-group and B is a p’-group. Denote by S λ G the twisted group algebra of G over S with a 2-cocycle λ ∈ Z²(G,S*). For Ω satisfying a specific condition, we give necessary and sufficient conditions for G to be of OTP projective (S,Ω)-representation type, in the sense that there exists a cocycle λ ∈ Z²(G,Ω) such that every indecomposable...

Finite groups of OTP projective representation type

Leonid F. Barannyk (2012)

Colloquium Mathematicae

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Let K be a field of characteristic p > 0, K* the multiplicative group of K and G = G p × B a finite group, where G p is a p-group and B is a p’-group. Denote by K λ G a twisted group algebra of G over K with a 2-cocycle λ ∈ Z²(G,K*). We give necessary and sufficient conditions for G to be of OTP projective K-representation type, in the sense that there exists a cocycle λ ∈ Z²(G,K*) such that every indecomposable K λ G -module is isomorphic to the outer tensor product V W of an indecomposable K λ G p -module...

Bipartite coalgebras and a reduction functor for coradical square complete coalgebras

Justyna Kosakowska, Daniel Simson (2008)

Colloquium Mathematicae

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Let C be a coalgebra over an arbitrary field K. We show that the study of the category C-Comod of left C-comodules reduces to the study of the category of (co)representations of a certain bicomodule, in case C is a bipartite coalgebra or a coradical square complete coalgebra, that is, C = C₁, the second term of the coradical filtration of C. If C = C₁, we associate with C a K-linear functor C : C - C o m o d H C - C o m o d that restricts to a representation equivalence C : C - c o m o d H C - c o m o d s p , where H C is a coradical square complete hereditary...

A F -algebras and topology of mapping tori

Igor Nikolaev (2015)

Czechoslovak Mathematical Journal

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The paper studies applications of C * -algebras in geometric topology. Namely, a covariant functor from the category of mapping tori to a category of A F -algebras is constructed; the functor takes continuous maps between such manifolds to stable homomorphisms between the corresponding A F -algebras. We use this functor to develop an obstruction theory for the torus bundles of dimension 2 , 3 and 4 . In conclusion, we consider two numerical examples illustrating our main results.

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 of non-negative integers there are only a finite number of isomorphism classes of indecomposables X 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].