arXiv subjects

Surjeet Kour

Publications and source records attributed to Surjeet Kour.

At least 19 recordsLinked to original sources

arXiv2026

Gerstenhaber type structures on Davydov-Yetter cohomology with coefficients

We obtain Gerstenhaber type structures on Davydov-Yetter cohomology with coefficients in half-braidings for a monoidal functor. Our approach uses a formal analogy between half-braidings of a monoidal functor and the entwining of a coalgebra with an algebra. We show that the Davydov-Yetter complex with coefficients carries the structure of a weak comp algebra. In particular, it is equipped with two distinct cup product structures $\cup$ and $\sqcup$ which are related in a manner that replaces graded commutativity. By considering entwining structures in the centralizer of the monoidal functor, we introduce subcomplexes of the Davydov-Yetter complex with coefficients, which carry comp algebra structures. As a result, we obtain a graded commutative cup product and a graded Lie bracket on their cohomology which forms a Gerstenhaber algebra in the usual sense.

math.CT↗

arXiv2026

Hochschild theory of multiplicative sequences of algebras and coalgebra measurings

We study coalgebra measurings between multiplicative sequences of algebras and the maps induced by them on Hochschild homology. The Hochschild theory of multiplicative sequences is introduced as a functor taking values in graded algebras in the symmetric monoidal category of chain complexes, constructed with the help of the shuffle product. We develop the universal measuring coalgebra, or Sweedler Hom for multiplicative sequences, as well as study several other Sweedler operations in this context. In particular, we obtain an enrichment of multiplicative sequences over cocommutative coalgebras. Using an appropriate theory of bimodules over multiplicative sequences, we study maps induced by comodule measurings on the Hochschild theory with coefficients, as well as the corresponding enriched categories. Finally, we consider measurings and generalized Sweedler operations between multiplicative sequences induced by comultiplicative sequences of coalgebras, and also the maps in Hochschild theory obtained from them.

math.RA↗

arXiv2026

Coalgebra measurings, cyclic theory and homologies of matrix algebras

In this paper, we consider coalgebra measurings and the maps induced by them between Hochschild and cyclic homology of algebras. We show that these induced maps are well behaved with respect to the various structures appearing on Hochschild and cyclic homology, such as the $λ$-decomposition, the product structure, as well as the module structure of Hochschild homology over cyclic homology. Thereafter, we relate the maps between homology theories of algebras induced by a coalgebra measuring to those induced between homologies of matrix algebras. This is done in the following contexts: (a) cyclic homology and the primitive part of Lie algebra homology of the matrix algebra, (b) Hochschild homology and the primitive part of Leibniz homology of the matrix algebra, and (c) Dihedral homology of an involutive algebra and the primitive part of Lie algebra homology of symplectic or skew symmetric matrices.

math.RA↗

arXiv2026

Characterizations of higher derivations and higher differential torsion theories in Eilenberg-Moore categories of monads

Let $T$ be a monad on a category $\mathscr{C}$. In this paper, we introduce the notion of higher derivations on the monad $T$ and characterize them in terms of ordinary derivations on $T$. We also define higher derivations on modules over the monad $T$ in the Eilenberg-Moore category $EM_T$ and establish their characterization in a similar manner. We provide several examples that illustrate and support our results. Furthermore, we examine the conditions under which a torsion theory on $EM_T$ is higher differential, and show that this holds if and only if every higher derivation on a module $M \in EM_T$ extends uniquely to its module of quotients $Q_τ(M)$.

math.CT↗

arXiv2026

Categorification of modules and construction of schemes

We use categorification of monoid actions to study algebraic geometry over symmetric monoidal categories. This brings together the relative algebraic geometry over symmetric monoidal categories developed by Toën and Vaquié, along with the theory of actegories over monoidal categories. We obtain schemes over a datum $(\mathcal C,\mathcal M)$, where $(\mathcal C,\otimes,1)$ is a symmetric monoidal category and $\mathcal M$ is an actegory over $\mathcal C$. One of our main tools is using the datum $(\mathcal C,\mathcal M)$ to give a Grothendieck topology on the category of affine schemes over $(\mathcal C,\otimes,1)$ that we call the ``spectral $\mathcal M$-topology.'' This consists of ``fpqc $\mathcal M$-coverings'' with certain special properties. We provide a description of schemes over $(\mathcal C,\mathcal M)$ in terms of quotients of disjoint unions of affine schemes over a certain equivalence relation. These categories of schemes are closed under pullbacks and coproducts, and are equipped with change of base functors induced by symmetric monoidal adjuctions accompanied by lax $\mathcal C$-linear functors.

math.AG↗

arXiv2025

Eilenberg-Moore categories and quiver representations of monads and comonads

We consider representations of quivers taking values in monads or comonads over a Grothendieck category $\mathcal C$. We treat these as scheme like objects whose ``structure sheaf'' consists of monads or comonads. By using systems of adjoint functors between Eilenberg-Moore categories, we obtain a categorical framework of modules over monad quivers, and of comodules over comonad quivers. Our main objective is to give conditions for these to be Grothendieck categories, which play the role of noncommutative spaces. As with usual ringed spaces, we have to study two kinds of module categories over a monad quiver. The first behaves like a sheaf of modules over a ringed space. The second consists of modules that are cartesian, which resemble quasi-coherent sheaves. We also obtain an extension of the classical quasi-coherator construction to modules over a monad quiver with values in Eilenberg-Moore categories. We establish similar results for comodules over a comonad quiver. One of our key steps is finding a modulus like bound for an endofunctor $U:\mathcal C\longrightarrow \mathcal C$ in terms of $κ(G)$, where $G$ is a generator for $\mathcal C$ and $κ(G)$ is a cardinal such that $G$ is $κ(G)$-presentable. Another feature of our paper is that we study modules over a monad quiver in two different orientations, which we refer to as ``cis-modules'' and ``trans-modules.'' We conclude with rational pairings of a monad quiver with a comonad quiver, which relate comodules over a comonad quiver to coreflective subcategories of modules over monad quivers.

math.CT↗

arXiv2025

A note on measurings and higher order Hochschild homology of algebras

We know that coalgebra measurings behave like generalized maps between algebras. In this note, we show that coalgebra measurings between commutative algebras induce morphisms between higher order Hochschild homology groups of algebras. By higher order Hochschild homology, we mean the the Hochschild homology groups of a commutative algebra with respect to a simplicial set as introduced by Pirashvili.

math.RA↗

arXiv2025

Galois measurings for noncommutative base change of entwined contramodule and entwined comodule categories

We study the noncommutative base change of an entwining structure $(A,C,ψ)$ by a Grothendieck category $\mathfrak S$, using two module like categories. These are the categories of entwined comodule objects and entwined contramodule objects in $\mathfrak S$ over the entwining structure $(A,C,ψ)$. We consider criteria for maps between these noncommutative spaces, induced by generalized maps between entwining structures, known as measurings, to behave like Galois extensions. We also study conditions for extensions of these noncommutative spaces, understood as functors between module like categories, to have separability, Frobenius or Maschke type properties.

math.RA↗

arXiv2024

Isotropy group of Lotka-Volterra derivations

In this paper, we study the isotropy group of Lotka-Volterra derivations of $K[x_{1},\cdots,x_{n}]$, i.e., a derivation $d$ of the form $d(x_{i})=x_{i}(x_{i-1}-C_{i}x_{i+1})$. If $n=3$ or $n \geq 5$, we have shown that the isotropy group of $d$ is finite. However, for $n=4$, it is observed that the isotropy group of $d$ need not be finite. Indeed, for $C_{i}=-1$, we observed an infinite collection of automorphisms in the isotropy group of $d$. Moreover, for $n \geq 3, ~~\text{and}~~C_{i}=1$, we have shown that the isotropy group of $d$ is isomorphic to the dihedral group of order $2n$.

math.RA↗

arXiv2024

Entwined comodules and contramodules over coalgebras with several objects: Frobenius, separability and Maschke theorems

We study module like objects over categorical quotients of algebras by the action of coalgebras with several objects. These take the form of ``entwined comodules'' and ``entwined contramodules'' over a triple $(\mathscr C,A,ψ)$, where $A$ is an algebra, $\mathscr C$ is a coalgebra with several objects and $ψ$ is a collection of maps that ``entwines'' $\mathscr C$ with $A$. Our objective is to prove Frobenius, separability and Maschke type theorems for functors between categories of entwined comodules and entwined contramodules.

math.CT↗

arXiv2024

Differential torsion theories on Eilenberg-Moore categories of monads

Let $\mathcal C$ be a Grothendieck category and $U$ be a monad on $\mathcal C$ that is exact and preserves colimits. In this article, we prove that every hereditary torsion theory on the Eilenberg-Moore category of modules over a monad $U$ is differential. Further, if $δ:U\longrightarrow U$ denotes a derivation on a monad $U$, then we show that every $δ$-derivation on a $U$-module $M$ extends uniquely to a $δ$-derivation on the module of quotients of $M$.

math.CT↗

arXiv2024

Hilbert spaces over $C^*$-tensor categories, Fredholm modules and cyclic cohomology

We construct Fredholm modules over an algebra taking values in generalized Hilbert spaces over a rigid $C^*$-tensor category. Using methods of Connes, we obtain Chern characters taking values in cyclic cohomology. These Chern characters are well behaved with respect to the periodicity operator, and depend only on the homotopy class of the Fredholm module.

math.OA↗

arXiv2023

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.

math.CT↗

arXiv2023

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 give an enrichment of the global category of SAYD modules over comodules. These measurings also induce morphisms on cyclic (co)homology of Hopf algebroids with SAYD coefficients, which are compatible with Hopf-Galois maps. Finally, we consider non-$Σ$ operads with multiplication. We obtain an enrichment of cyclic unital comp modules over non-$Σ$ operads, as well as morphisms on cyclic homology induced by measurings of comp modules over operads with multiplication.

math.CT↗

arXiv2023

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 ${_A}\mathfrak S^H$.

math.RA↗

arXiv2022

On generalized cyclotomic derivations

In this article we study the field of rational constants and Darboux polynomials of a generalized cyclotomic $K$-derivation $d$ of $K[X]$. It is shown that $d$ is without Darboux polynomials if and only if $K(X)^d=K$. Result is also studied in the tensor product of polynomial algebras.

math.AC↗

arXiv2020

On measurings of algebras over operads and homology theories

The notion of a coalgebra measuring, introduced by Sweedler, is a kind of generalized ring map between algebras. We begin by studying maps on Hochschild homology induced by coalgebra measurings. We then introduce a notion of coalgebra measuring between Lie algebras and use it to obtain maps on Lie algebra homology. Further, these measurings between Lie algebras satisfy nice adjoint like properties with respect to universal enveloping algebras. More generally, we introduce and undertake a detailed study of the notion of coalgebra measuring between algebras over any operad $\mathcal O$. In case $\mathcal O$ is a binary and quadratic operad, we show that a measuring of $\mathcal O$-algebras leads to maps on operadic homology. In general, for any operad $\mathcal O$, we construct universal measuring coalgebras to show that the category of $\mathcal O$-algebras is enriched over coalgebras. We develop measuring comodules and universal measuring comodules for this theory. We also relate these to measurings of the universal enveloping algebra $U_{\mathcal O}(\mathscr A)$ of an $\mathcal O$-algebra $\mathscr A$ and the modules over it. Finally, we construct the Sweedler product $C\rhd \mathscr A$ of a coalgebra $C$ and an $\mathcal O$-algebra $\mathscr A$. The object $C\rhd \mathscr A$ is universal among $\mathcal O$-algebras that arise as targets of $C$-measurings starting from $\mathscr A$.

math.CT↗

arXiv2017

On $n^{th}$ class preserving automorphisms of $n$-isoclinism family

Let $G$ be a finite group and $M,N$ be two normal subgroups of $G$. Let $Aut_N^M(G)$ 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 $Aut_{Z_n(G)}^{γ_{n+1}(G)}(G)$ to $Aut_{Z_n(H)}^{γ_{n+1}(H)}(H)$, which maps the group of $n^{th}$ class preserving automorphisms of $G$ to the group of $n^{th}$ class preserving automorphisms of $H$. Also, for a nilpotent group of class at most $(n+1)$, with some suitable conditions on $γ_{n+1}(G)$, we prove that $Aut_{Z_n(G)}^{γ_{n+1}(G)}(G)$ is isomorphic to the group of inner automorphisms of a quotient group of $G$.

math.GR↗