Related papers: Rouquier dimension versus global dimension

We extend the usual notion of fully commutative elements from the Coxeter groups to the complex reflection groups. Then we decompose the sets of fully commutative elements into natural subsets according to their combinatorial properties,…

We offer a solution to the long-standing problem of group completing within the context of rig categories (also known as bimonoidal categories). Given a rig category R we construct a natural additive group completion R' that retains the…

Let $R$ be a commutative noetherian ring, $\frak a$ be an ideal of $R$, $\cS$ be an arbitrary Serre subcategory of $R$-modules and let $\cN$ be the subcategory of finitely generated $R$-modules. In this paper, we study $\cN\cS$-$\frak…

Let $R$ be a commutative Noetherian local ring. We prove that the finiteness of the injective dimension of a finitely generated $R$-module $C$ is determined by the existence of a Cohen--Macaulay module $M$ that satisfies an inequality…

We provide various counter-examples to the long-standing so-called "Omnibus Conjecture" in Rational Homotopy Theory. That is, we show that a space with finite dimensional even-degree rational cohomology and finite dimensional spherical…

This article investigates strong generation within the module category of a commutative Noetherian ring. We establish a criterion for such rings to possess strong generators within their module category, addressing a question raised by…

We prove a rank-finiteness conjecture for modular categories: up to equivalence, there are only finitely many modular categories of any fixed rank. Our technical advance is a generalization of the Cauchy theorem in group theory to the…

It is proven that each commutative arithmetical ring $R$ has a finitistic weak dimension $\leq 2$. More precisely, this dimension is 0 if $R$ is locally IF, 1 if $R$ is locally semicoherent and not IF, and 2 in the other cases.

Let $R$ be a commutative Noetherian ring. Using the new concept of linkage of ideals over a module, we show that if $\mathfrak{a}$ is an ideal of $R$ which is linked by the ideal $I$, then $cd(\mathfrak{a},R) \in \{ grad \mathfrak{a},…

Let $\mathfrak{a}$ be an ideal of a commutative noetherian ring $R$ and $M, N$ two finitely generated $R$-modules. By using a spectral sequence argument, it is shown that if either $\mathrm{dim}_RM\leq2$ and $\mathrm{H}^{i}_\mathfrak{a}(N)$…

We study the topological variant of Rokhlin dimension for topological dynamical systems (X,{\alpha},Z^m) in the case where X is assumed to have finite covering dimension. Finite Rokhlin dimension in this sense is a property that implies…

We introduce higher-dimensional module factorizations associated to a regular sequence. They include higher-dimensional matrix factorizations, which are commutative cubes consisting of free modules with edges being classical matrix…

Global dimensions for fusion categories defined by a pair (G,k), where G is a Lie group and k a positive integer, are expressed in terms of Lie quantum superfactorial functions. The global dimension is defined as the square sum of quantum…

We prove that for a compact metric space the property of having finite covering dimension is equivalent to the existence of a total order with finite snake number.

In order to simultaneously generalize matrix rings and group graded crossed products, we introduce category crossed products. For such algebras we describe the center and the commutant of the coefficient ring. We also investigate the…

In this paper, we introduce a new notion, called the integral dimension, for noetherian rings. It can be regarded as the weak Briancon-Skoda numbers of rings. The point is that every noetherian local ring has finite integral dimension.

We find necessary and sufficient conditions for the finite separability of finitely generated commutative rings. Namely, we prove that every such ring is a finite extension of its torsion ideal $I_k$ where $k$ is square-free, and $I_k$ is a…

In this paper, we introduce and study the $q$-Krull dimension of a commutative ring via its $q$-operation. A new characterization of $\tau_q$-von Neumann regular rings is obtained, and some properties of rings $q$-Krull dimension 0 are…

We show that Iacob-Iyengar's answer to a question of Avromov-Foxby extends from Noetherian to coherent rings. In particular, a coherent ring R is regular if and only if the injective (resp. projective) dimension of each complex X of…

Starting from the notion of totally reflexive modules, we survey the theory of Gorenstein homological dimensions for modules over commutative rings. The account includes the theory's connections with relative homological algebra and with…