Measurings of Hopf algebroids and morphisms in cyclic (co)homology theories

In this paper, we consider measurings between Hopf algebroids and show that they induce morphisms on cyclic homology and cyclic cohomology. We also consider comodule measurings between SAYD modules over Hopf algebroids. These measurings induce morphisms on cyclic (co)homology of Hopf algebroids with SAYD coefficients. Finally, we obtain morphisms on cyclic homology induced by measurings of cyclic comp modules over operads with multiplication

Noncommutative supports, local cohomology and spectral sequences

The purpose of this paper is to study local cohomology in the noncommutative algebraic geometry framework of Artin and Zhang. The noncommutative spaces are obtained by base change of a Grothendieck category that is locally noetherian or strongly locally noetherian. Using what we call elementary objects and their injective hulls, we develop a theory of supports and associated primes in these categories. We apply our theory to study a general functorial setup that requires certain conditions on the injective hulls of elementary objects and gives us spectral sequences for derived functors associated to local cohomology objects, as well as generalized local cohomology and also generalized Nagata ideal transforms.Comment: Some parts shortened, some new results adde

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 $\mathfrak S$. 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 ${_A}\mathfrak S^H$ of relative $(A,H)$-Hopf modules in $\mathfrak S$, where $H$ is a Hopf algebra and $A$ is a right $H$-comodule algebra. We study the cohomological theory in ${_A}\mathfrak S^H$ by means of spectral sequences. Using coinduction functors and functors of coinvariants, we study torsion theories and how they relate to injective resolutions in ${_A}\mathfrak S^H$. 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 ${_A}\mathfrak S^H$.Comment: Minor update

Simple derivations on tensor product of polynomial algebras

Let A be an unique factorization domain containing a field k of characteristic zero and let A[X] and A[Y ] be two k-algebras. Let d1 and d2 be two generalized triangular k-derivations of A[X] and A[Y ], respectively. Denote the unique k-derivation d1 ? 1 + 1 ?d2 of A[X, Y ] by d1 ?d2. Then with some conditions on d1 and d2, it is shown that d1 ? d2 is a simple derivation of A[X, Y ] if and only if A[X] is d1-simple and A[Y ] is d2-simple. We also show that if d1 and d2 are positively homogeneous derivations and d2 is a generalized triangular derivation, then d1 ? d2 is simple derivation of A[X, Y ] if and only if d1 is a simple derivation of A[X] and d2 is a simple derivation of A[Y ].by Surjeet Kou

On nth class preserving automorphisms of n-isoclinism family

Let G be a finite group and M,N be two normal subgroups of G. Let AutMN(G) denote the group of all automorphisms of G which fix N element wise and act trivially on G/M. Let n be a positive integer. In this article we have shown that if G and H are two n-isoclinic groups, then there exists an isomorphism from Autγn+1(G)Zn(G)(G) to Autγn+1(H)Zn(H)(H), which maps the group of nth class preserving automorphisms of G to the group of nth class preserving automorphisms of H. Also, for a nilpotent group of class at most (n+1), with some suitable conditions on γn+1(G), we prove that Autγn+1(G)Zn(G)(G) is isomorphic to the group of inner automorphisms of a quotient group of G.by Surjeet Kou

On equality of certain automorphism groups

Let G=H×A be a finite group, where H is a purely non-abelian subgroup of G and A is a non-trivial abelian factor of G. Then, for n≥2, we show that there exists an isomorphism ϕ:Autγn(G)Z(G)(G)→Autγn(H)Z(H)(H) such that ϕ(Autn−1c(G))=Autn−1c(H). We also give some necessary and sufficient conditions on a finite p-group G such that Autcent(G)=Autn−1c(G) . Furthermore, for a finite non-abelian p-group G, we give some necessary and sufficient conditions for Autγn(G)Z(G)(G) to be equal to AutZ(G)γ2(G)(G).by Surjeet Kour and Vishakh

(A,delta)-modules, Hochschild homology and higher derivations

In this paper, we develop the theory of modules over (A,delta), where A is an algebra and delta :A?A is a derivation. Our approach is heavily influenced by Lie derivative operators in noncommutative geometry, which make the Hochschild homologies HH.(A) of A into a module over (A,delta). We also consider modules over (A,Delta), where Delta={Delta n}n >= 0 is a higher derivation on A. Further, we obtain a Cartan homotopy formula for an arbitrary higher derivation on A

On equality of certain automorphism groups

by Surjeet Kour and Vishakha Sharm