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Constantin-Cosmin Todea

Publications and source records attributed to Constantin-Cosmin Todea.

13 recordsLinked to original sources
arXiv2026

Equivariant Hochschild cohomology of group algebras and relative $\operatorname{Ext}$

For a finite group $\Gamma$, acting on a finite group $G,$ we find necessary conditions for which the first $\Gamma_0$-equivariant Hochschild cohomology of the group algebra $kG$ is non-trivial, where $k$ is a field of characteristic $p$ dividing the order of $G$ and $\Gamma_0$ is the stabilizer subgroup in $\Gamma$ of some element in $G.$ For any field $k$ we show that the $\Gamma$-equivariant Hochschild cohomology of $\Gamma$-algebras with coefficients in a $\Gamma$-equivariant bimodule (Jensen, 1996) is isomorphic with some $k\Gamma$-relative $\operatorname{Ext},$ in the context of relative homological algebra.

math.KT↗
arXiv2026

Reduction theorems for a conjecture on basis in source algebras of blocks of finite groups

The aim of this short research note is to present some results about a conjecture of Barker and Gelvin claiming that any source algebra of a block of a finite group has the unit group containing a basis stabilised by the left and right actions of the defect group. We obtain some reduction theorems for the existence of stable unital basis in source algebras of block algebras. Along the way we investigate this problem for the blocks of some finite simple groups.

math.RT↗
arXiv2025

A reduction theorem for non-vanishing of Hochschild cohomology of block algebras and Happel's property

In this short research note we obtain a reduction theorem for the non-vanishing of the first Hochschild cohomology of block algebras of finite groups with non-trivial defect groups. Along the way we investigate this problem for the blocks of some simple finite group algebras. Mimicking the case of blocks of finite group algebras we find some examples of category algebras that satisfy Happel's property.

math.RT↗
arXiv2022

Stable unital basis, hyperfocal subalgebras and basic Morita equivalences

We investigate Conjecture 1.6 introduced by Barker and Gelvin in [3], which says that any source algebra of a p-block (p is a prime) of finite group has the unit group containing a basis stabilized by left and right action of the defect group. We will reduce this conjecture to a similar statement about basis of the hyperfocal subalgebras in the source algebra. We will also show that such unital basis of source algebras of two p-blocks, stabilized by left and right action of the defect group, are transported trough basic Morita equivalences.

math.GR↗
arXiv2021

Symmetric Hochschild cohomology of twisted group algebras

We show that there is an action of the symmetric group on the Hochschild cochain complex of a twisted group algebra with coefficients in a bimodule. This allows us to define the symmetric Hochschild cohomology of twisted group algebras, similarly to th construction of symmetric group cohomology due to Staic. We give explicit embeddings and connecting homomorphisms between the symmetric cohomology spaces and symmetric Hochschild cohomology of twisted group algebras.

math.KT↗
arXiv2019

Block extensions, local categories, and basic Morita equivalences

Let $(\mathcal{K},\mathcal{O},k)$ be a $p$-modular system with $k$ algebraically closed, let $b$ be a block of the normal subgroup $H$ of $G$ having defect pointed group $Q_δ$ in $H$ and $P_γ$ in $G$, and consider the block extension $b\mathcal{O}G$. One may attach to $b$ an extended local category $\mathcal{E}_{(b,H,G)}$, a group extension $L$ of $Z(Q)$ by $N_G(Q_δ)/C_H(Q)$ having $P$ as a Sylow $p$-subgroup, and a cohomology class $[α]\in H^2(N_G(Q_δ)/QC_H(Q),k^\times)$. We prove that these objects are invariant under the $G/H$-graded basic Morita equivalences. Along the way, we give alternative proofs of the results of Külshammer and Puig (1990), Puig and Zhou (2012) on extensions of nilpotent blocks. We also deduce by our methods a result of Zhou (2016) on $p'$-extensions of inertial blocks.

math.RT↗
arXiv2018

Fusions and Clifford extensions

We introduce $\bar G$-fusions of local pointed groups on a block extension $A=b\mathcal{O}G$, where $H$ is a normal subgroup of the finite group $G$, $\bar G=G/H$, and $b$ is a $G$-invariant block of $\mathcal{O}H$. We show that certain Clifford extensions associated to these pointed groups are invariant under group graded basic Morita equivalences.

math.RT↗
arXiv2018

Frobenius induction for algebras

Let $B\rightarrow A$ be a homomorphism of Hopf algebras and let $C$ be an algebra. We consider the induction from $B$ to $A$ of $C$ in two cases: when $C$ is a $B$-interior algebra and when $C$ is a $B$-module algebra. Our main results establish the connection between the two inductions. The inspiration comes from finite group representation theory, and some constructions work in even more general contexts.

math.RA↗
arXiv2016

On cohomology of saturated fusion systems and support varieties

In this short note we study the cohomology algebra of saturated fusion systems using finite groups which realize saturated fusion systems and Hochschild cohomology of group algebras. A similar result to a theorem of Alperin is proved for varieties of cohomology algebras of fusions systems associated to block algebras of finite groups.

math.GR↗
arXiv2016

Bockstein homomorphisms for Hochschild cohomology of group algebras and of block algebras of finite groups

We give an explicit approach for Bockstein homomorphisms of the Hochschild cohomology of a group algebra and of a block algebra of a finite group and we show some properties. To give explicit definitions for these maps we use an additive decomposition and a Product Formula for the Hochschild cohomology of group algebras given by Siegel and Witherspoon in 1999. For k an algebraically closed field of characteristic p and G a finite group we prove an additive decomposition and a Product Formula for the cohomology algebra of a defect group of a block ideal of kG with coefficients in the source algebra of this block, and we define similar Bockstein homomorphisms.

math.GR↗