Related papers: Algebra retracts and Stanley-Reisner rings

Let $A \to B$ be a $G$-Galois extension of rings, or more generally of $\mathbb{E}_\infty$-ring spectra in the sense of Rognes. A basic question in algebraic $K$-theory asks how close the map $K(A) \to K(B)^{hG}$ is to being an equivalence,…

We consider the problem of classifying (possibly noncommutative) R-algebras of low rank over an arbitrary base ring R. We first classify algebras by their degree, and we relate the class of algebras of degree 2 to algebras with a standard…

Let $S = K[x_1, ..., x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $\deg x_i = 1$. Let $\Delta$ be a simplicial complex on $[n] = \{1, ..., n \}$ and $I_\Delta \subset S$ its Stanley--Reisner ideal. We write…

We provide $\mathbb{N}$-filtrations on the negative part $U_q(\mathfrak{n}^-)$ of the quantum group associated to a finite-dimensional simple Lie algebra $\mathfrak{g}$, such that the associated graded algebra is a skew-polynomial algebra…

We define a class of associative algebras generalizing 'clannish algebras', as introduced by the second author, but also incorporating semilinear structure, like a skew polynomial ring. Clannish algebras generalize the well known 'string…

The main results of this paper establish a partial correspondence between two previously-studied analogues of Groebner bases in the setting of algebras: namely, subalgebra (aka SAGBI) bases for quotients of polynomial rings and Khovanskii…

One of the main open questions in liaison theory is whether every homogeneous Cohen-Macaulay ideal in a polynomial ring is glicci, i.e. if it is in the G-liaison class of a complete intersection. We give an affirmative answer to this…

Stark-Heegner points are conjectural substitutes for Heegner points when the imaginary quadratic field of the theory of complex multiplication is replaced by a real quadratic field $K$. They are constructed analytically as local points on…

A conjecture due to Y. Han asks whether that Hochschild homology groups of a finite dimensional algebra vanish for sufficiently large degrees would imply that the algebra is of finite global dimension. We investigate this conjecture from…

Let $ K[x, y]$ be the polynomial algebra in two variables over a field $K$ of characteristic $0$. A subalgebra $R$ of $K[x, y]$ is called a retract if there is an idempotent homomorphism (a {\it retraction}, or {\it projection}) $\varphi:…

The loop algebra construction by Allison, Berman, Faulkner, and Pianzola, describes graded-central-simple algebras with split centroid in terms of central simple algebras graded by a quotient of the original grading group. Here the…

We generalise Koszul and D-Koszul algebras by introducing a class of graded algebras called (D,A)-stacked algebras. We give a characterisation of (D,A)-stacked algebras and show that their Ext algebra is finitely generated as an algebra in…

In this note we consider monoidal complexes and their associated algebras, called toric face rings. These rings generalize Stanley-Reisner rings and affine monoid algebras. We compute initial ideals of the presentation ideal of a toric face…

We consider generalized $\Lambda$-structures on algebras and schemes over the ring of integers $\mathit{O}_K$ of a number field $K$. When $K=\mathbb{Q}$, these agree with the $\lambda$-ring structures of algebraic K-theory. We then study…

The goal of this paper is to study Goldbach's conjecture for rings of regular functions of affine algebraic varieties over a field. Among our main results, we define the notion of Goldbach condition for Newton polytopes, and we prove in a…

We study polynomial Poisson algebras with some regularity conditions. Linear (Lie-Berezin-Kirillov) structures on dual spaces of semi-simple Lie algebras, quadratic Sklyanin elliptic algebras of \cite{FO1},\cite{FO2} as well as polynomial…

The aim of this article is to investigate interrelated structures lying among three notable problems in commutative algebra. These are Lifting problem, Ascent/descent along associated graded rings, and Matijevic-Roberts type problem.

As is known, every finite-dimensional algebra over a field is isomorphic to the centralizer algebra of \textbf{two} matrices. So it is fundamental to study first the centralizer algebra of a single matrix, called a centralizer matrix…

We give degree formulas for Grothendieck polynomials indexed by vexillary permutations and $1432$-avoiding permutations via tableau combinatorics. These formulas generalize a formula for degrees of symmetric Grothendieck polynomials which…

We introduce a notion of Milnor square of stable $\infty$-categories and prove a criterion under which algebraic K-theory sends such a square to a cartesian square of spectra. We apply this to prove Milnor excision and proper excision…