Noncommutative Images of Commutative Spectra

非交换与n元Γ-半环的结构与谱理论

Structure and Spectral Theory of Non-Commutative and $n$-ary $Γ$-Semirings

本文针对非交换与n元Γ-半环，拓展既有交换三元框架，定义多类理想、构建根理论与谱拓扑，最终得到非交换Wedderburn-Artin型分解，统一不同元数情形的相关理论。

Non commutative finite dimensional manifolds II. Moduli space and structure of non commutative 3-spheres

This paper contains detailed proofs of our results on the moduli space and the structure of noncommutative 3-spheres. We develop the notion of central quadratic form for quadratic algebras, and a general theory which creates a bridge between noncommutative differential geometry and its purely algebraic counterpart. It allows to construct a morphism from an involutive quadratic algebras to a C*-algebra constructed from the characteristic variety and the hermitian line bundle associated to the central quadratic form. We apply the general theory in the case of noncommutative 3-spheres and show th…

非交换Novikov代数

Noncommutative Novikov algebras

本文研究非交换Novikov代数，提出从带导子的非交换结合代数构造一类更一般的线性代数，推广了经典Novikov代数的构造方法。

Noncommutative Complete Intersections

Several generalizations of a commutative ring that is a graded complete intersection are proposed for a noncommutative graded $k$-algebra; these notions are justified by examples from noncommutative invariant theory.

Commutative and Noncommutative Invariant Theory

The purpose of this survey paper is to bring to a large mathematical audience (containing also non-algebraists) some topics of invariant theory both in the classical commutative and the recent noncommutative case. We have included only several topics from the classical invariant theory -- the finite generating (the Endlichkeitssatz) and the finite presenting (the Basissatz) of the algebra of invariants, the Molien formula for its Hilbert series and the Shephard-Todd-Chevalley theorem for the invariants of a finite group generated by pseudo-reflections. Then we give analogues of these results f…

Reverse decomposition of unipotents over noncommutative rings I: General linear groups

Recently is has been proved that if $σ\in GL_n(R)$ where $R$ is an commutative ring and $n\geq 3$, then each of the elementary transvections $t_{kl}(σ_{ij})~(i\neq j,k\neq l)$ is a product of eight $E_n(R)$-conjugates of $σ$ and $σ^{-1}$. In this article we show that similar results hold true if $R$ is a (noncommutative) von Neumann regular ring, or a Banach algebra, or a ring satisfying a stable range condition, or a ring with Euclidean algorithm, or an almost commutative ring.

(Positive) Totally Ordered Noncommutative Monoids -- How Noncommutative Can They Be?

Commutative totally ordered monoids abound, number systems for example. When the monoid is not assumed commutative, one may be hard pressed to find an example. One suggested by Professor Orr Shalit are the countable ordinals with addition. In this note we attempt an introductory investigation of totally (also partially) ordered monoids, not assumed commutative (still writing them additively), and taking them as positive, i.e.\ every element is greater than the unit element. That, in the usual commutative cases, allows the ordering to be defined via the algebraic structure, namely, as divisibil…

Noncommutative matrix factorizations with an application to skew exterior algebras

Theory of matrix factorizations is useful to study hypersurfaces in commutative algebra. To study noncommutative hypersurfaces, which are important objects of study in noncommutative algebraic geometry, we introduce a notion of noncommutative matrix factorization for an arbitrary nonzero non-unit element of a ring. First we show that the category of noncommutative graded matrix factorizations is invariant under the operation called twist (this result is a generalization of the result by Cassidy-Conner-Kirkman-Moore). Then we give two category equivalences involving noncommutative matrix factor…

Noncommutative (crepant) desingularizations and the global spectrum of commutative rings

In this paper we study endomorphism rings of finite global dimension over not necessarily normal commutative rings. These objects have recently attracted attention as noncommutative (crepant) resolutions, or NC(C)Rs, of singularities. We propose a notion of a NCCR over any commutative ring that appears weaker but subsumes all previous notions. Our results yield strong necessary and sufficient conditions for the existence of such objects in many cases of interest. We also give new examples of NCRs of curve singularities, regular local rings and normal crossing singularities. Moreover, we introd…

VIC-modules over noncommutative rings

For a finite ring $R$, not necessarily commutative, we prove that the category of $\text{VIC}(R)$-modules over a left Noetherian ring $\mathbf{k}$ is locally Noetherian, generalizing a theorem of the authors that dealt with commutative $R$. As an application, we prove a very general twisted homology stability for $\text{GL}_n(R)$ with $R$ a finite noncommutative ring.

Rationality of Hilbert series in noncommutative invariant theory

It is a fundamental result in commutative algebra and invariant theory that a finitely generated graded module over a commutative finitely generated graded algebra has rational Hilbert series, and consequently the Hilbert series of the algebra of polynomial invariants of a group of linear transformations is rational, whenever this algebra is finitely generated. This basic principle is applied here to prove rationality of Hilbert series of algebras of invariants that are neither commutative nor finitely generated. Our main focus is on linear groups acting on certain factor algebras of the tenso…