Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2403.18049

Quillen (co)homology of divided power algebras over an operad

Ioannis Dokas·Martin Frankland·Sacha Ikonicoff
View original source ↗Read source document ↗Export metadata ↓

Abstract

Barr--Beck cohomology, put into the framework of model categories by Quillen, provides a cohomology theory for any algebraic structure, for example André--Quillen cohomology of commutative rings. Quillen cohomology has been studied notably for divided power algebras and restricted Lie algebras, both of which are instances of divided power algebras over an operad $P$: the commutative and Lie operad respectively. In this paper, we investigate the Quillen cohomology of divided power algebras over an operad $P$, identifying Beck modules, derivations, and Kähler differentials in that setup. We also compare the cohomology of divided power algebras over $P$ with that of $P$-algebras, and work out some examples.

Explore related subjects

math.RAmath.AT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Ioannis Dokas, Martin Frankland, Sacha Ikonicoff. 2025-01-10. Quillen (co)homology of divided power algebras over an operad. https://arxiv.org/abs/2403.18049

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Maximal tails,character fibres and induced modules for pullback Kumjian-Pask algebras

Let $f:\N^{k}\to\N^{\ell}$ be a surjective monoid homomorphism and let $Γ$ be a row-finite $\ell$-graph with no sources and finitely many vertices. We give an explicit graded isomorphism from the Kumjian--Pask algebra of the pullback $f^{*}Γ$ onto the tensor product of $\KP_{\K}(Γ)$ and the group algebra of the kernel of the group completion of $f$. When $Γ$ is strongly aperiodic, but need not be cofinal, every maximal tail $T$ and every maximal ideal $\mathfrak m$ of the kernel group algebra determine an explicit primitive ideal and primitive quotient. If, in addition, $\K$ is uncountable and algebraically closed, these ideals exhaust the primitive spectrum. We prove that the resulting parametrisation is a homeomorphism for the product of the maximal-tail and Zariski topologies. Each primitive ideal is realised as the annihilator of a simple module induced from the isotropy of a path which is cofinal in $T$, and the character fibres are algebraic tori. Two examples exhibit, respectively, a single character fibre and the non-Hausdorff gluing of two such fibres.

math.RA↗
arXiv2026

A classification of group gradings on incidence algebras over commutative rings

Let $R$ be a commutative ring with 1, $P$ a locally finite partially ordered set, and $G$ a group. We derive necessary and sufficient conditions for an $R$-algebra isomorphism between the incidence algebra $I(P,R)$ and the group algebra $RG$. Then, for an indecomposable ring $R$, a finite poset $P$ and an arbitrary group $G$, we classify the $G$-gradings of $I(P,R)$ up to graded isomorphism. The classification rests on a complete set of primitive orthogonal homogeneous idempotents. The corner algebras are split group algebras of finite abelian subgroups of $G$, and the off-diagonal Peirce blocks are multiplicity-free sums of bimodules induced from characters of double coset stabilizers. Graded isomorphisms are shown to have a rigid form, and a grading is determined up to graded isomorphism by the poset of idempotents, the corner groups, the types of the atomic bimodules and the structure constants of their multiplication. The data which occur are characterized by polynomial conditions, and over an algebraically closed field of characteristic zero only finitely many graded isomorphism classes share given partial invariants. An example shows that the structure constants cannot be omitted. Some previous results are extended and enhanced, while providing alternative proofs for some known facts.

math.RA↗