Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0810.1241

Koszulity of splitting algebras associated with cell complexes

Vladimir Retakh·Shirlei Serconek·Robert Lee Wilson
View original source ↗Read source document ↗Export metadata ↓

Abstract

We associate to a good cell decomposition of a manifold M a quadratic algebra and show that the Koszulity of the algebra implies a restriction on the Euler characteristic of M. For a two-dimensional manifold M the algebra is Koszul if and only if the Euler characteristic of M is two.

Explore related subjects

math.RAmath.AT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Vladimir Retakh, Shirlei Serconek, Robert Lee Wilson. 2008-11-06. Koszulity of splitting algebras associated with cell complexes. https://arxiv.org/abs/0810.1241

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

KEEP EXPLORING

Related papers

arXiv2026

Normal Quaternionic Matrices and Finitely Generated Witt Rings

We present a new approach to verify the Elementary Type Conjecture for abstract Witt rings with small number of square classes. To do so, we make use of an abstract analogue of the 2-torsion part of the Brauer group. We develop a description of the entire structure of an abstract Witt ring with $2^n$ square classes in terms of a unique $n\times n$ matrix satisfying a small additional condition that particularly holds for Witt rings of fields. Via computational search, we find all these matrices for $n$ up to $7$. This verifies that all Witt rings of fields with up to $128$ square classes are of elementary type.

math.RA↗
arXiv2026

Graded differential polynomial rings

We study differential polynomial rings $R[t;δ]$ over $Γ$-graded rings, where $Γ$ is an arbitrary group. We show that $R[t;δ]$ admits a $Γ$-grading compatible with that of $R$ if and only if $δ$ is a $γ$-derivation for some $γ\in C_Γ(Γ_R)$, and that this grading is unique once $°(t)=γ$ is fixed; if $δ\neq0$, then $γ$ is itself uniquely determined by $δ$. We characterize the resulting graded ring by a universal property. We prove a characteristic-free center criterion for gr-simplicity whenever $Z(R[t;δ])$ is a graded subring; in characteristic zero, gr-simplicity is equivalent to $δ$-gr-simplicity of $R$ and $γ$-outerness of $δ$, extending Jordan's simplicity criterion to the graded setting. We further show that $R[t;δ]$ is gr-prime if and only if $R$ is $δ$-gr-prime, and that gr-Noetherianity of $R$ passes to $R[t;δ]$, recovering a graded Hilbert basis theorem as a special case. When $Γ$ is abelian, gr-simplicity and gr-primality are shown to be invariants of homogeneous graded Morita equivalence, and every ring homogeneously graded equivalent to $R[t;δ]$ via a compatible idempotent is again a graded differential polynomial ring.

math.RA↗
arXiv2026

Affinization of algebraic structures: Poisson algebras

An affinization of the notion of a Poisson algebra is presented. This is termed a Poisson affgebra and consists of an affine space together with an associative bi-affine multiplication and a bi-affine Lie bracket that acts as an affine derivation for the associative product. The constructive relation between Poisson affgebras and Poisson algebras is described and several low-dimensional examples are studied in detail.

math.RA↗