A non-commutative topology on rep A

The Closed-Point Zariski Topology for Irreducible Representations

In previous work, the second author introduced a topology, for spaces of irreducible representations, that reduces to the classical Zariski topology over commutative rings but provides a proper refinement in various noncommutative settings. In this paper, a concise and elementary description of this refined Zariski topology is presented, under certain hypotheses, for the space of simple left modules over a ring R. Namely, if R is left noetherian (or satisfies the ascending chain condition for semiprimitive ideals), and if R is either a countable dimensional algebra (over a field) or a ring who…

Noetherianity of the Space of Irreducible Representations

Let R be an associative ring with identity. We study an elementary generalization of the classical Zariski topology, applied to the set of isomorphism classes of simple left R-modules (or, more generally, simple objects in a complete abelian category). Under this topology the points are closed, and when R is left noetherian the corresponding topological space is noetherian. If R is commutative (or PI, or FBN) the topology is equivalent to the Zariski topology, and when R is the first Weyl algebra (in characteristic zero) we obtain a one-dimensional irreducible noetherian topological space. Com…

Poset topology and homological invariants of algebras arising in algebraic combinatorics

We present a beautiful interplay between combinatorial topology and homological algebra for a class of monoids that arise naturally in algebraic combinatorics. We explore several applications of this interplay. For instance, we provide a new interpretation of the Leray number of a clique complex in terms of non-commutative algebra. Résumé. Nous présentons une magnifique interaction entre la topologie combinatoire et l'algèbre homologique d'une classe de monoïdes qui figurent naturellement dans la combinatoire algébrique. Nous explorons plusieurs applications de cette interaction. Par exemple, …

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。

A Zariski Topology for Bicomodules and Corings

In this paper we introduce and investigate top (bi)comodules} of corings, that can be considered as dual to top (bi)modules of rings. The fully coprime spectra of such (bi)comodules attains a Zariski topology, defined in a way dual to that of defining the Zariski topology on the prime spectra of (commutative rings. We restrict our attention in this paper to duo (bi)comodules (satisfying suitable conditions) and study the interplay between the coalgebraic properties of such (bi)comodules and the introduced Zariski topology. In particular, we apply our results to introduce a Zariski topology on …

Zariski-like Topologies for Lattices with Applications to Modules over Commutative Rings

We study Zariski-like topologies on a proper class $X\varsubsetneqq L$ of a complete lattice $\mathcal{L}=(L,\wedge ,\vee ,0,1)$. We consider $X$ with the so called classical Zariski topology $(X,τ^{cl})$ and study its topological properties (e.g. the separation axioms, the connectedness, the compactness) and provide sufficient conditions for it to be $\textit{spectral}$. We say that $\mathcal{L}$ is $X$\emph{-top} iff% \begin{equation*} τ:=\{X\backslash V(a)\mid a\in L\},\text{ where }V(a)=\{x\in L\mid a\leq x\} \end{equation*}% is a topology. We study the interplay between the \textit{algebr…

Non-commutative Hodge-to-de Rham degeneration via the method of Deligne-Illusie

We use a version of the method of Deligne-Illusie to prove that the Hodge-to-de Rham, a.k.a. Hochschild-to-cyclic spectral sequence degenerates for a large class of associative, not necessariyl commutative DG algebras. This proves, under some assumption, a conjecture by Kontsevich and Soibelman made in math.RA/0606241. The approach is similar to my earlier paper math.AG/0511665, but the proof is more straightforward, and the underlying algebraic topology notions are explicitly described. The paper is independent of math.AG/0511665 and in a sense, supercedes it.

sheaf理论与非交换Γ几何 over 非交换Γ谱

Sheaf Theory and Derived Gamma Geometry over the Non-Commutative Gamma Spectrum

本文提出非交换Γ几何框架，构建非交换Γ谱上的sheaf和导出理论，并证明非交换局部对偶定理及同调不变量的谱和动机解释。

非交换分解与自由代数的有限维表示

Non-commutative factorizations and finite-dimensional representations of free algebras

本文利用Leavitt代数将单变量经典理论推广至非交换变量多项式，通过将多项式关联到有限维模，为多项式分解、除法算法及最大公因数计算提供了新见解。

Non-Commutative Representations of Families of k^2 Commutative Polynomials in 2k^2 Commuting Variables

Given a collection P of k^2 commutative polynomials in 2k^2 commutative variables, the objective is to find a condensed representation of these polynomials in terms of a single non-commutative polynomial p(X,Y) in two k x k matrix variables X and Y. Algorithms that will generically determine whether the given family P has a non-commutative representation and that will produce such a representation are developed. These algorithms will determine a non-commutative representation for families P that admit a a non-commutative representation in an open, dense subset of the vector space of non-commut…

On the prime spectrum of an le-module

Here we continue to characterize a recently introduced notion, le-modules $_{R}M$ over a commutative ring $R$ with unity \cite{Bhuniya}. This article introduces and characterizes Zariski topology on the set $Spec(M)$ of all prime submodule elements of $M$. Thus we extend many results on Zariski topology for modules over a ring to le-modules. The topological space Spec(M) is connected if and only if $R/Ann(M)$ contains no idempotents other than $\overline{0}$ and $\overline{1}$. Open sets in the Zariski topology for the quotient ring $R/Ann(M)$ induces a base of quasi-compact open sets for the …

任意交换环上箭图的刚性积分表示

Rigid integral representations of quivers over arbitrary commutative rings

本文将作者早期关于主理想环上由有限生成自由模构成的箭图刚性表示分类结果，推广至任意交换环上由有限生成投射模构成的箭图表示。

Galois closures of non-commutative rings and an application to Hermitian representations

Galois closures of commutative rank n ring extensions were introduced by Bhargava and the second author. In this paper, we generalize the construction to the case of non-commutative rings. We show that non-commutative Galois closures commute with base change and satisfy a product formula. As an application, we give a uniform construction of many of the representations arising in arithmetic invariant theory, including many Vinberg representations.

Commutative algebras and representations of the category of finite sets

We prove that two finite-dimensional commutative algebras over an algebraically closed field are isomorphic if and only if they give rise to isomorphic representations of the category of finite sets and surjective maps.

奇偶对合交换剩余链的群表示

Group representation for even and odd involutive commutative residuated chains

本文用带附加结构的阿贝尔o-群直系统表示奇偶对合交换剩余链，推广了Dunn关于有限Sugihara幺半群的结果。