Narrow Results

The famous finitistic dimension conjecture says that every finite-dimensional 𝕂-algebra over a field 𝕂 should have finite finitistic dimension. This conjecture is equivalent to the following statement: If B is a subalgebra of a finite-dimensional 𝕂-algebra A such that the radical of B is a left ideal in A, and if A has finite finitistic dimension, then B has finite finitistic dimension. In the paper, we shall work with a more general setting of Artin algebras. Let B be a subalgebra of an Artin algebra A such that the radical of B is a left ideal in A. (1) If the category of all finitely generated (A, B)-projective A-modules is closed under taking A-syzygies, then fin.dim(B) ≤ fin.dim(A) + fin.dim(_BA) + 3, where fin.dim(A) denotes the finitistic dimension of A, and where fin.dim(_BA) stands for the supremum of the projective dimensions of those direct summands of _BA that have finite projective dimension. (2) If the extension B ⊆ A is n-hereditary for a non-negative integer n, then gl.dim(A) ≤ gl.dim(B) + n. Moreover, we show that the finitistic dimension of the trivially twisted extension of two algebras of finite finitistic dimension is again finite. Also, a new formulation of the finitistic dimension conjecture in terms of relative homological dimension is given. Our approach in this paper is completely different from the one in our earlier papers.

We investigate the module theory of a certain class of semilocal rings connected with nearly simple uniserial domains. For instance, we classify finitely presented and pure-projective modules over these rings and calculate their projective dimension.

We study a class of down–up algebras 𝒜(α, β, ϕ) defined over a polynomial base ring 𝕂[t₁,…,t_n] and establish several analogous results. We first construct a 𝕂-basis for the algebra 𝒜(α, β, ϕ). As an application, we completely determine the center of 𝒜(α, β, ϕ) when char 𝕂 = 0, and prove that the Gelfand–Kirillov dimension of 𝒜(α, β, ϕ) is n + 3. Then, we prove that 𝒜(α, β, ϕ) is a noetherian domain if and only if β ≠ 0, and 𝒜(α, β, ϕ) is Auslander-regular when β ≠ 0. We show that the global dimension of 𝒜(α, β, ϕ) is n + 3, and 𝒜(α, β, ϕ) is a prime ring except when α = β = ϕ = 0. Finally, we obtain some results on the Krull dimensions, isomorphisms and automorphisms of 𝒜(α, β, ϕ).

An $n$-dimensional quantum torus is defined as the $F$-algebra generated by variables $x_1,\dots ,x_n$ together with their inverses satisfying the relations $x_i{x}_j=q_{ij}{x}_j{x}_i$, where $q_{ij}\in F$. The Krull and global dimensions of this algebra are known to coincide and the common value is equal to the supremum of the rank of certain subgroups of $〈x_1,\dots ,x_n〉$ that can be associated with this algebra. In this paper we study how these dimensions behave with respect to taking tensor products of quantum tori over the base field. We derive a best possible upper bound for the dimension of such a tensor product and from this special cases in which the dimension is additive with respect to tensoring.

Let $\mathrm{\Lambda }$ be a finite dimensional Auslander algebra. For a $\mathrm{\Lambda }$-module $N$, we prove that the projective dimension of $N$ is at most one if and only if the projective dimension of its socle soc$N$ is at most one. As an application, we give a new characterization of Auslander algebras $\mathrm{\Lambda }$ and prove that a finite dimensional algebra $\mathrm{\Lambda }$ is an Auslander algebra provided its global dimension gl.d$\mathrm{\Lambda }\le 2$ and an injective $\mathrm{\Lambda }$-module is projective if and only if the projective dimension of its socle is at most one.

The polynomial rings over hereditary Noetherian prime rings have global dimension 2 and any two-sided ideal which is either left $v$-ideal or right $v$-ideal is left and right projective. By using the latter property, we define the concept of generalized hereditary Noetherian prime rings (G-HNP rings). We study the structure of projective ideals in G-HNP rings and some overrings of G-HNP rings. As it is shown in the examples, the class of G-HNP rings ranges from rings with global dimension 2 to rings with infinite global dimension and Noetherian prime rings with global dimension 2 are not necessarily G-HNP rings.

We show that a noetherian ring graded by an abelian group of finite rank satisfies the Auslander condition if and only if it satisfies the graded Auslander condition. In addition, we also study the injective dimension, the global dimension and the Cohen–Macaulay property from the same perspective as that for the Auslander condition. A key step of our approach is to establish homological relations between a graded ring $R$, its quotient ring modulo the ideal $\hslash R$ and its localization ring with respect to the Ore set ${\{{\hslash }^i\}}_{i\ge 0}$, where $\hslash$ is a homogeneous regular normal non-invertible element of $R$.

Let $R$ be a commutative ring and let $I\ne 0$ be a proper ideal of $R$. In this paper, we study some algebraic and homological properties of a family of rings $R{(I)}_{a,b}$, with $a,b\in R$, that are obtained as quotients of the Rees algebra associated with the ring $R$ and the ideal $I$. Specially, we study when $R{(I)}_{a,b}$ is a von Neumann regular ring, a semisimple ring and a Gaussian ring. Also, we study the classical global and weak global dimensions of $R{(I)}_{a,b}$. Finally, we investigate some homological properties of $R{(I)}_{a,b}$-modules and we show that $R$ and $I$ are Gorenstein projective $R{(I)}_{a,b}$-modules, provided some special conditions.

We prove that cellular Noetherian algebras with finite global dimension are split quasi-hereditary over a regular commutative Noetherian ring with finite Krull dimension and their quasi-hereditary structure is unique, up to equivalence. In the process, we establish that a split quasi-hereditary algebra is semi-perfect if and only if the ground ring is a local commutative Noetherian ring. We give a formula to determine the global dimension of a split quasi-hereditary algebra over a commutative regular Noetherian ring (with finite Krull dimension) in terms of the ground ring and finite-dimensional split quasi-hereditary algebras. For the general case, we give upper bounds for the finitistic dimension of split quasi-hereditary algebras over arbitrary commutative Noetherian rings. We apply these results to Schur algebras over regular Noetherian rings and to Schur algebras over quotients rings of the integers.

By associating a graph to the ring under discussion, we propose a graph approach to extend a well-known ring splitting theorem due to Zaks, and we then describe the global and finitistic dimensions of the ring in the language of the graph.

Let be a K-algebra defined by a finite Gröbner basis . In this paper, it is shown how to use the Ufnarovski graph and the graph of n-chains to calculate gl.dim G^ℕ(A) and , where G^ℕ(A), respectively , is the associated ℕ-graded algebra of A, respectively the Rees algebra of A with respect to the ℕ-filtration FA of A induced by a weight ℕ-grading filtration of K 〈X₁,…, X_n〉.

Let A be a hereditary Artin algebra and T a tilting A-module. The possibilities for the global dimension of the endomorphism algebra of a generator-cogenerator for the subcategory T^⊥ in A-mod are determined in terms of relative Auslander-Reiten orbits of indecomposable A-modules in T^⊥.

Let $(\mathcal{A},\mathcal{B},\mathcal{C})$ be a recollement of extriangulated categories. In this paper we introduce the global dimension and extension dimension of extriangulated categories, and give some upper bounds of global dimensions and extension dimensions of the categories involved in $(\mathcal{A},\mathcal{B},\mathcal{C})$, which give a simultaneous generalization of some results in recollements of abelian categories and triangulated categories.

Let $A$ be a finite dimensional $k$-algebra and $T$ be a support $\tau$-tilting right $A$-module. In this note, we give lower and upper bounds for the global dimension of the endomorphism algebra ${\mathsf{End}}_A(T)$ under some mild conditions. Finally, we give some examples to illustrate that both the upper and lower bounds can be reached.

The chapter focuses on global challenges and threats to national financial security in the context of geopolitical tensions and the COVID-19 pandemic. An urgent methodological task is to form a study of national financial security through its global dimension, namely, through the prism of analyzing the structural transformation of the global financial market into national financial security and the integration of countries with emerging markets into global financial security based on market instruments. The chapter provides a comprehensive review of financial flows. It allocates financial flows in the following areas: (1) analysis of the dynamics of the central bank in relation to financial risks and the onset of global financial crises and (2) the development of indicators of financial shocks (stresses) calculated to assess the instability of the financial system. The authors used analytical materials from the International Monetary Fund, the World Bank, the Central Bank of Russia, the Federal Financial Monitoring Service, other international organizations, and interdepartmental and intergovernmental organizations of the Russian Federation. Additionally, the research provides an overview of Guidelines for the Advancement of Information Security in the Financial Sector for 2019–2021 by the Central Bank of the Russian Federation.