No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
On the dimension of modules and algebras. III. Global dimension
Abstract
Let Λ be a ring with unit. If A is a left Λ -module, the dimension of A (notation: 1.dim Λ A ) is defined to be the least integer n for which there exists an exact sequence
0 → X n → … → X 0 → A → 0
where the left Λ -modules X 0 , …, X n are projective. If no such sequence exists for any n , then 1. dim A A = ∞. The left global dimension of Λ is
1. gl. dim Λ = sup 1. dim A A
where A ranges over all left Λ -modules, The condition 1. dim A A < n is equivalent with ( A, C ) = 0 for all left Λ -modules C . The condition 1.gl. dim Λ < n is equivalent with = 0. Similar definitions and theorems hold for right Λ -modules.
... For a complex P • , [1] denotes the shift functor, that is, P • [1] j = P j+1 and d j P • [1] = −d j+1 P • for any j ∈ Z, and τ ≤i (P • ) denotes a new complex defined by τ ≤i (P • ) j = P j for all j ≤ i, and zero otherwise. The category of complexes over C with chain maps is denoted by C(C). ...
... For a complex P • , [1] denotes the shift functor, that is, P • [1] j = P j+1 and d j P • [1] = −d j+1 P • for any j ∈ Z, and τ ≤i (P • ) denotes a new complex defined by τ ≤i (P • ) j = P j for all j ≤ i, and zero otherwise. The category of complexes over C with chain maps is denoted by C(C). ...
... Then 1 and Ω(M k λ,1 ) ∼ = M k+1 1,−λ for k = 1, . . . , n − 1, λ ∈ k and Ω(M n µ,λ ) = P n+1 for λ, µ ∈ k. ...
In this survey, we review the fundamental properties of the Igusa-Todorov functions, the ϕ-dimension, the ψ-dimension and their generalizations.
... The tensor product is a very effective research tool in representation theory of finite dimension algebras [1,8,23,24,29,30]. For n-, m-representation-finite algebras Λ, respectively Γ over perfect field K, under condition of l-homogeneity, Herschend and Iyama in [15] showed that tensor product Λ ⊗ K Γ is an (n + m)representation finite algebra which admits the (n+m)-APR tilting (Λ⊗ K Γ)-module associated with simple projective module. ...
... Proof. We only prove (1), the proof of (2) is dual to (1). For any Λ-module M ∈ mod Λ and Γ-module N ∈ mod Γ, suppose P M , P N are the projective cover of modules M , N respectively, then P M and P N are projective, hence by 3.3. ...
... Proof. We only prove (1), the proof of (2) is dual to (1). Suppose Ext i Λ (DΛ, M ) = 0 for any 0 ≤ i < n and Ext j Γ (DΓ, N ) = 0 for any 0 ≤ j < m. ...
The higher APR tilting modules and higher BB tilting modules were introduced and studied in higher Auslander-Reiten theory. Our objective is to consider these tilting modules by the corresponding simple modules, and show that the tensor product of higher APR (BB) tilting modules is a higher APR (BB) tilting module.
... Then X • has the derived double centraliser property (resp. is two-sided tilting) if and only if for any indecomposable projective A-module Ae, (1) there is a unique i ∈ Z such that Hom D b (A) (Ae[i], X • ) 0; ...
... . By Lemma 4.5, gl(A) < ∞. Now by [1,Theorem 16], gl(A ⊗ K B op ) = gl(A) + gl(B) < ∞. Hence we may assume X • is a bounded complex of projective A ⊗ K B op -modules. ...
... (i). Suppose that there exists a minimal left add X • -approximation sequence (1) Ae ...
In representation theory, the double centraliser property is an important property for a module (bimodule). It plays a fundamental role in many theories. In this paper, we extend this property to complexes in derived categories of finite dimensional algebras, under the name derived double centraliser property. Characterizations for complexes with the derived double centraliser property and (two-sided) tilting complexes in derived categories of hereditary algebras are given. In particular, all complexes with this property in the derived categories of lower triangular matrix algebras are classified.
... In (2) it remains to note that the algebra A ⊗ k A has finite global dimension since A and A are basic and have finite global dimension, see [Au55,Th. 16]. Now we use Lemma 2.4. ...
... Proof. Let C := A ⊗ k B. Since A, B are basic, C is also basic and has finite global dimension by [Au55,Th. 16]. Now the statement follows from isomorphisms ...
... Proof. The statement about global dimension follows from general theory and the fact that gldim(kΓ m ) = 1, see [Au55,Th. 16 ...
This is a companion paper of arXiv:1901.09461, where different notions of dimension for triangulated categories are discussed. Here we compute dimensions for some examples of triangulated categories and thus illustrate and motivate material from loc. cit. Our examples include path algebras of finite ordered quivers, orbifold projective lines, some tensor powers of path algebras in Dynkin quivers of type A and categories, generated by an exceptional pair.
... (3) Let J = J(A). In this case, S = A/J is perfect as a left A-module by part (2), so that A is finitely generated as an algebra and locally finite according to Lemma 2.3. ...
... Then S is perfect as both a left and a right A-module by Lemma 3.2(2). Next, A 0 is homologically smooth by Lemma 3.9 (2). Now Rickard's Theorem 3.6 implies that ...
... If (ii ′ ) holds, then by taking cohomology we obtain (ii). If (ii) holds, then since A is a perfect A e -module we obtain (ii ′ ) from Lemma 4.3 (2). ...
This is a general study of twisted Calabi-Yau algebras that are graded and locally finite-dimensional, with the following major results. We prove that a locally finite graded algebra is twisted Calabi-Yau if and only if it is separable modulo its radical and satisfies one of several suitable generalizations of the Artin-Schelter regularity property, adapted from the work of Martinez-Villa as well as Minamoto and Mori. We characterize twisted Calabi-Yau algebras of dimension~0 as separable $k$-algebras, and we similarly characterize graded twisted Calabi-Yau algebras of dimension~1 as tensor algebras of certain invertible bimodules over separable algebras. Finally, we prove that a graded twisted Calabi-Yau algebra of dimension~2 is noetherian if and only if it has finite GK dimension.
... right) global dimension is finite. Note that for noetherian rings the left and right global dimension coincide, see [2]. In this case we call smooth a left or right smooth k-algebra. ...
... right) global dimension is finite. For noetherian rings the left and right global dimensions coincide, see [2]. ...
Let $B\subset A$ be a bounded extension of finite dimensional algebras. We use the Jacobi-Zariski long nearly exact sequence to show that $B$ satisfies Han's conjecture if and only if $A$ does, regardless if the extension splits or not. We provide conditions ensuring that an extension by arrows and relations is bounded. Examples of non split bounded extensions are given and we obtain a structure result for the extensions of an algebra given by a quiver and admissible relations.
... Because A is finitely generated, the ring End R (A) is Noetherian. Thus the maximal length of a minimal projective resolution in the category of left and right End R (A)-modules is the same, see [Aus55]. Therefore, we focus attention on the right modules over End R (A). 5 In practice, it is easy to see how large is sufficiently large; see the proof of Proposition 4.15. ...
... Since End R (A) is a finitely generated module over a commutative Noetherian ring R, a theorem of Auslander [Aus55] implies the left and right global dimensions coincide, and a subsequent theorem of Bass [Bas68, III, Prop. 6.7] implies the following. ...
Let $R$ be the coordinate ring of an affine toric variety. We show that the endomorphism ring $End_R(\mathbb A),$ where $\mathbb A$ is the (finite) direct sum of all (isomorphism classes of) conic $R$-modules, has finite global dimension. Furthermore, we show that $End_R(\mathbb A)$ is a non-commutative crepant resolution if and only if the toric variety is simplicial. For toric varieties over a perfect field $k$ of prime characteristic, we show that the ring of differential operators $D_\mathsf{k}(R)$ has finite global dimension.
... For V = FpInj(A) we use the notation fp-id A for V -id A and, similarly, fp-silp(A) for V -silp(A) and fp-FID(A) for V -FID(A). The next corollary is an equivalent in Gorenstein homological algebra to [3,Thm. 4]. ...
Distinctive characteristics of Iwanaga--Gorenstein rings are typically understood through their intrinsic symmetry. We show that several of those that pertain to the Gorenstein global dimensions carry over to the one-sided situation, even without the noetherian hypothesis. Our results yield new relations among homological invariants related to the Gorenstein property, not only Gorenstein global dimensions but also the suprema of projective/injective dimensions of injective/projective modules and finitistic dimensions.
... (1 ⇒ 3) By Corollary 11 of [1], gldimA < ∞ if and only if pd(S) < ∞ for all S ∈ S A , so we can assume that there is a simple module S with pd(S) = ∞, then Q ∞ = ∅. ...
In this article, we prove that, under certain conditions, Morita context algebras that arise from Igusa-Todorov (LIT) algebras and have zero bimodule morphisms are also Igusa-Todorov (LIT). For a finite dimensional algebra $A$, we prove that the class $\phi_0^{-1}(A) = \{M: \phi(M)=0\}$ is a 0-Igusa-Todorov subcategory if and only if $A$ is selfinjective or gl$\dim l(A)< \infty$. As a consequence $A$ is an $(n,V, \phi_0^{-1}(A))$ algebra if and only if $A$ is selfinjective or gl$\dim(A)< \infty$. We also show that the opposite algebra of a LIT algebra is not LIT in general.
... The Igusa-Todorov φ-Dimension on Morita Context Algebras for i = 1, 2, 3, λ ∈ k, n ∈ N and a 4 = b 4 = c 0 M 0,λ,μ,n such that supp(M 0,λ,μ,n ) = b 1 c 0 ...
In this article we prove that, under certain hypotheses, Morita context algebras with zero bimodule morphisms have finite ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document}-dimension. For these algebras we also study the behaviour of the ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document}-dimension for an algebra and its opposite. In particular we show that the ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document}-dimension of an Artin algebra is not symmetric, i.e. there exists an Artin algebra A such that ϕdim(A)≠ϕdim(Aop)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi \dim (A) \not = \phi \dim (A^{op})$$\end{document}.
... In this way, the proposition follows from a result of Auslander [Aus55,Theorem 16]. ...
In 1989, D. Happel pointed out for a possible connection between the global dimension of a finite-dimensional algebra and its Hochschild cohomology: is it true that the vanishing of Hochschild cohomology higher groups is sufficient to deduce that the global dimension is finite? After the discovery of a counterexample, Y. Han proposed, in 2006, to reformulate this question to homology. In this survey, after introducing the concepts and results involved, I present the efforts made until now towards the comprehension of Han's conjecture; which includes: examples of algebras that have been proven to satisfy it and extensions that preserve it.
... Sometimes, the value of a dimension is unaffected by this choice, that is, the dimension is left-right symmetric. This is true for both the global dimension of a Noetherian ring [Aus55,Theorem 4] To prove Theorem A, we define a ring construction that takes a basic finite dimensional algebra A and produces a related algebraÃ. The finitistic dimension ofà bounds the finitistic dimension of A from above. ...
We show that the finitistic dimension conjecture holds for all finite dimensional algebras if and only if, for all finite dimensional algebras, the finitistic dimension of an algebra being finite implies that the finitistic dimension of its opposite algebra is also finite. We also prove the equivalent statement for injective generation.
... Hence A#B is also a semi-primary ring. By [Aus,Corollary 12], gld A#B = sup{n | Ext n A#B ((A#B)/J, (A#B)/J)}. The conclusion follows from Proposition 2.8. ...
We prove that the Yoneda Ext-algebra of a Takeuchi smash product is the graded Takeuchi smash product of the Yoneda Ext-algebras of the two algebras or modules involved. As an application, we prove that graded Takeuchi smash products preserve Artin-Schelter regularity, and describe the Nakayama automorphism of the product.
... As it is mentioned in the Introduction, the word smooth is originated in commutative algebra and is useful here for brevity. Note that for noetherian rings, the left and right global dimensions are equal, see [2]. Han's conjecture states that for A a finite dimensional algebra, H * (A, A) vanishes in large enough degrees if and only if A is smooth. ...
... If C is a k-linear category, then the category algebra kC has an approximate unit given by sums of the form Remark 3. By a result of Auslander [2], the left and right global dimensions of a Noetherian ring coincide. The proof can easily be adapted to rings with approximate unit via replacing the Noetherian condition by the approximately Noetherian condition that for every i, every left S-submodule of Se i and every right S-submodule of e i S is finitely generated as a left, resp. ...
We describe necessary and sufficient conditions for the hereditarity of the category algebra of an infinite EI category satisfying certain combinatorial assumptions. More generally, we discuss conditions such that the left global dimension of a category algebra equals the maximal left global dimension of the endomorphism algebras of its objects, and classify its projective modules in this case. As applications, we completely classify transporter categories, orbit categories, and Quillen categories with left hereditary category algebras over a field.
... • an arbitrary R-module is projective if and only if it is injective. If R is a quasi-Frobenius ring, any R-module has projective dimension either 0 or ∞; hence the global dimension of R is either 0 or ∞ (note that left and right global dimension of R coincide by [Aus55,Cor. 5]). ...
Consider the obvious functor from the unbounded derived category of all finitely generated modules over a left noetherian ring $R$ to the unbounded derived category of all modules. We answer the natural question whether this functor defines an equivalence onto the full subcategory of complexes with finitely generated cohomology modules in two special cases. If $R$ is a quasi-Frobenius ring of infinite global dimension, then this functor is not full. If $R$ has finite left global dimension, this functor is an equivalence. We also prove variants of the latter assertion for left coherent rings, for noetherian schemes and for locally noetherian Grothendieck categories.
... We obtain a bound for the global dimension of the ring Λ when this is right noetherian, because then the global dimensions of Λ and mod Λ coincide, cf. [1]. ...
Given a tilting object of the derived category of an abelian category of finite global dimension, we give (under suitable finiteness conditions) a bound for the global dimension of its endomorphism ring.
... As it is mentioned in the Introduction, the word smooth is originated in commutative algebra and is useful here for brevity. Note that for noetherian rings, the left and right global dimensions are equal, see [2]. Han's conjecture states that for A a finite dimensional algebra, H * (A, A) vanishes in large enough degrees if and only if A is smooth. ...
A main purpose of this paper is to prove that the class of finite dimensional algebras which verify Han's conjecture is closed under split bounded extensions.
... Proof. By [Aus55] it is sufficient to calculate the projective dimensions of all simple modules E ν corresponding to the vertices ν ...
We continue the combinatorial description of thick subcategories in hereditary categories started by Igusa-Schiffler-Thomas and Krause. We show that for a weighted projective line $\mathbb{X}$ there exists an order preserving bijection between the thick subcategories of $\text{coh}(\mathbb{X})$ generated by an exceptional sequence and the factorization poset of a Coxeter transformation $c$ in the Weyl group of a simply-laced generalized root system if the Hurwitz action is transitive on the reduced factorizations of $c$. By using combinatorial and group theoretical tools we show that this assumption on the transitivity of the Hurwitz action is fulfilled for a weighted projective line $\mathbb{X}$ of tubular type. In this case the factorization poset is given as the set of prefixes of a Coxeter transformation in certain elliptic Weyl groups described by Saito. As a byproduct we obtain a result on the Hurwitz action on non-reduced reflection factorizations in finite Coxeter groups which partially generalizes a result of Lewis-Reiner.
... Proposition 3.2 ([3].) Let M be a module over an algebra A, I a non-empty well-ordered set, {M i } i∈I be a family of submodules of M such that if i, j ∈ I and i ≤ j then M i ⊆ M j. ...
The aim of the paper is to prove two conjectures that the (left and right) global dimension of the algebra of polynomial integro-differential operators $\mathbb{I}_n$ and the Jacobian algebra $\mathbb{A}_n$ is equal to $n$ (over a field of characteristic zero). An analogue of Hilbert's Syzygy Theorem is proven for them. The algebras $\mathbb{I}_n$ and $\mathbb{A}_n$ are neither left nor right Noetherian. Furthermore, they contain infinite direct sums of nonzero left/right ideals and are not domains. It is proven that the global dimension of all prime factor algebras of the algebras $\mathbb{I}_n$ and $\mathbb{A}_n$ is $n$ and the weak global dimension of all the factor algebras of $\mathbb{I}_n$ and $\mathbb{I}_n$ is $n$.
Distinctive characteristics of Iwanaga–Gorenstein rings are typically understood through their intrinsic symmetry. We show that several of those that pertain to the Gorenstein global dimensions carry over to the one-sided situation, even without the noetherian hypothesis. Our results yield new relations among homological invariants related to the Gorenstein property, not only Gorenstein global dimensions but also the suprema of projective/injective dimensions of injective/projective modules and finitistic dimensions.
We show that the finitistic dimension conjecture for finite‐dimensional algebras is equivalent to the left–right symmetry of finite finitistic dimension for finite‐dimensional algebras. We also prove the equivalent statement for injective generation.
Let R be an Artinian ring with a dualizing complex D•. Then (1) if the injective modules generate the derived category \({\cal D}(R)\) as a triangulated category with arbitrary coproducts, then the finitistic dimension of R is finite; (2) if the injective modules generate the homotopy category \({\cal K}(R - {\rm{lnj}})\) as a triangulated category with arbitrary coproducts, then the global dimension of R is finite. The first result generalizes Theorem 4.3 in [Adv. Math., 2019, 354: 106735, 21 pp.].
In this article we prove that, under certain hypotheses, Morita context algebras that have zero bimodule morphisms have finite $\phi$-dimension. We also study the behaviour of the $\phi$-dimension for an algebra and its opposite. In particular we show that the $\phi$-dimension of an Artin algebra is not symmetric, i.e. there exists a finite dimensional algebra $A$ such that $\phi\dim (A) \not = \phi\dim (A^{op})$.
We describe necessary and sufficient conditions for the hereditarity of the category algebra of an infinite EI category satisfying certain combinatorial assumptions. More generally, we discuss conditions such that the left global dimension of a category algebra equals the maximal left global dimension of the endomorphism algebras of its objects. As applications, we completely classify orbit categories, Quillen categories and transporter categories with left hereditary category algebras over a field.
Let B⊂A be a left or right bounded extension of finite dimensional algebras. We use the Jacobi-Zariski long nearly exact sequence to show that B satisfies Han's conjecture if and only if A does, regardless if the extension splits or not. We provide conditions ensuring that an extension by arrows and relations is left or right bounded. Finally we give a structure result for extensions of an algebra given by a quiver and admissible relations, and examples of non split left or right bounded extensions.
This is a companion paper of [7], where different notions of dimension for triangulated categories are discussed. Here we compute dimensions for some examples of triangulated categories and thus illustrate and motivate material from [7]. Our examples include path algebras of finite ordered quivers, orbifold projective lines, some tensor powers of path algebras in Dynkin quivers of type A and categories, generated by an exceptional pair.
Our first result provides a new characterization of Auslander algebras using a property of hereditary torsion pairs. The second result shows an Auslander algebra Λ is left or right glued if and only if Λ is representation-finite. Finally, our third result shows the module category of any Auslander algebra contains a tilting module with a particular property, which we call the hereditary property. Applications of this property are investigated.
Consider the obvious functor from the unbounded derived category of all finitely generated modules over a left noetherian ring R to the unbounded derived category of all modules. We answer the natural question whether this functor defines an equivalence onto the full subcategory of complexes with finitely generated cohomology modules in two special cases. If R is a quasi-Frobenius ring of infinite global dimension, then this functor is not full. If R has finite left global dimension, this functor is an equivalence. We also prove variants of the latter assertion for left coherent rings, for noetherian schemes and for locally noetherian Grothendieck categories.
We introduce an analytic method that uses the global dimension function $\operatorname{gldim}$ to produce contractible flows on the space $\operatorname{Stab}\mathcal{D}$ of stability conditions on a triangulated category $\mathcal{D}$. In the case when $\mathcal{D}=\mathcal{D}(\mathbf{S}^\lambda)$ is the topological Fukaya category of a graded surface $\mathbf{S}^\lambda$, we show that $\operatorname{gldim}^{-1}(0,y)$ contracts to $\operatorname{gldim}^{-1}(0,x)$ for any $1\le x\le y$, provided $(x,y)$ does not contain `critical' values $\{1+w_\partial/m_\partial \mid w_\partial\ge0, \partial\in\partial\mathbf{S}^\lambda\}$, where the pair $(m_\partial,w_\partial)$ consists of the number $m_\partial$ of marked points and the winding number $w_\partial$ associated to a boundary component $\partial$ of $\mathbf{S}^\lambda$. One consequence is that the global dimension of $\mathcal{D}(\mathbf{S}^\lambda)$ must be one of these critical values. Besides, we remove the assumptions in Kikuta-Ouchi-Takahashi's classification result on triangulated categories with global dimension less than 1.
Let Λ be a 1-Auslander-Gorenstein algebra. We give a necessary and sufficient condition for Λ to be a tilted algebra.
The Gabriel–Roiter measure is used to give an alternative proof of the finiteness of the representation dimension for Artin algebras, a result established by Iyama in 2002. The concept of Gabriel–Roiter measure can be extended to abelian length categories and every such category has multiple Gabriel–Roiter measures. Using this notion, we prove the following broader statement: given any object $X$ and any Gabriel–Roiter measure $\mu $ in an abelian length category $\mathcal{A}$, there exists an object $X^{\prime}$ that depends on $X$ and $\mu $, such that $\Gamma =\operatorname{End}_{\mathcal{A}}(X\oplus X^{\prime}) $ has finite global dimension. Analogously to Iyama’s original results, our construction yields quasihereditary rings and fits into the theory of rejective chains.
Let C be a symmetrizable generalized Cartan matrix with symmetrizer D and orientation \(\Omega \). In Geiß et al. (Invent Math 209(1):61–158, 2017) we constructed for any field \(\mathbb {F}\) an \(\mathbb {F}\)-algebra \(H := H_\mathbb {F}(C,D,\Omega )\), defined in terms of a quiver with relations, such that the locally free H-modules behave in many aspects like representations of a hereditary algebra \({\widetilde{H}}\) of the corresponding type. We define a Noetherian algebra \({\widehat{H}}\) over a power series ring, which provides a direct link between the representation theory of H and of \({\widetilde{H}}\). We define and study a reduction and a localization functor relating the module categories of \({\widehat{H}}\), \({\widetilde{H}}\) and H. These are used to show that there are natural bijections between the sets of isoclasses of tilting modules over the three algebras \({\widehat{H}}\), \({\widetilde{H}}\) and H. We show that the indecomposable rigid locally free H-modules are parametrized, via their rank vectors, by the real Schur roots associated to \((C,\Omega )\). Moreover, the left finite bricks of H, in the sense of Asai, are parametrized, via their dimension vectors, by the real Schur roots associated to \((C^T,\Omega )\).
Let R be the coordinate ring of an affine toric variety. We prove, using direct elementary methods, that the endomorphism ring EndR(A), where A is the (finite) direct sum of all (isomorphism classes of) conic R-modules, has finite global dimension equal to the dimension of R. This gives a precise version, and an elementary proof, of a theorem of Spenko and Van den Bergh ˇ implying that EndR(A) has finite global dimension. Furthermore, we show that EndR(A) is a non-commutative crepant resolution if and only if the toric variety is simplicial. For toric varieties over a perfect field k of prime characteristic, we show that the ring of differential operators Dk(R) has finite global dimension.
The Gabriel-Roiter measure is used to give an alternative proof of the finiteness of the representation dimension for Artin algebras, a result established by Iyama in 2002. The concept of Gabriel-Roiter measure can be extended to abelian length categories and every such category has multiple Gabriel-Roiter measures. Using this notion, we prove the following broader statement: given any object $X$ and any Gabriel-Roiter measure $\mu$ in an abelian length category $\mathcal{A}$, there exists an object $X'$ which depends on $X$ and $\mu$, such that $\Gamma = \operatorname{End}_{\mathcal{A}}(X \oplus X')$ has finite global dimension. Analogously to Iyama's original results, our construction yields quasihereditary rings and fits into the theory of rejective chains.
Let $C$ be a symmetrizable generalized Cartan matrix with symmetrizer $D$ and orientation $\Omega$. In previous work we associated an algebra $H$ to this data, such that the locally free $H$-modules behave in many aspects like representations of a hereditary algebra $\tilde{H}$ of the corresponding type. We define a Noetherian algebra $\hat{H}$ over a power series ring, which provides a direct link between the representation theory of $H$ and of $\tilde{H}$. We define and study a reduction and a localization functor relating the module categories of these three types of algebras. These are used to show that there are natural bijections between the sets of isoclasses of tilting modules over $H$, $\hat{H}$ and $\tilde{H}$. We show that the indecomposable rigid locally free modules over $H$ and $\hat{H}$ are parametrized, via their rank vector, by the real Schur roots associated to $(C,\Omega)$. Moreover, the left finite bricks of $H$, in the sense of Asai, are parametrized, via their dimension vector, by the real Schur roots associated to the dual datum $(C^T,\Omega)$.
In the paper, Ding projective modules and Ding projective complexes are considered. In particular, it is proven that Ding projective complexes are precisely the complexes X for which each Xm is a Ding projective R-module for all m ∈ ℤ.
In this paper, we mainly investigate $\mathfrak{X}$-Gorenstein projective dimension of modules and the (left) $\mathfrak{X}$-Gorenstein global dimension of rings. Some properties of $\mathfrak{X}$-Gorenstein projective dimensions are obtained. Furthermore, we prove that the (left) $\mathfrak{X}$-Gorenstein global dimension of ring $R$ is equal to the supremum of the set of $\mathfrak{X}$-Gorenstein projective dimensions of all cyclic (left) $R$-modules. This result extends the well-known Auslander's theorem on the global dimension and its Gorenstein homological version.
We conjecture that the injective dimension of the Jacobson radical equals the global dimension for Artin algebras. We provide a proof of this conjecture in case the Artin algebra has finite global dimension and in some other cases.
We study the following generalization of singularity categories. Let X be a quasi-projective Gorenstein scheme with isolated singularities and A a non-commutative resolution of singularities of X in the sense of Van den Bergh. We introduce the relative singularity category as the Verdier quotient of the bounded derived category of coherent sheaves on A modulo the category of perfect complexes on X. We view it as a measure for the difference between X and A. The main results of this thesis are the following. (i) We prove an analogue of Orlov's localization result in our setup. If X has isolated singularities, then this reduces the study of the relative singularity categories to the affine case. (ii) We prove Hom-finiteness and idempotent completeness of the relative singularity categories in the complete local situation and determine its Grothendieck group. (iii) We give a complete and explicit description of the relative singularity categories when X has only nodal singularities and the resolution is given by a sheaf of Auslander algebras. (iv) We study relations between relative singularity categories and classical singularity categories. For a simple hypersurface singularity and its Auslander resolution, we show that these categories determine each other. (v) The developed technique leads to the following `purely commutative' application: a description of Iyama & Wemyss triangulated category for rational surface singularities in terms of the singularity category of the rational double point resolution. (vi) We give a description of singularity categories of gentle algebras.
We prove a version of the classical 'generic smoothness' theorem with smooth varieties replaced by non-commutative resolutions of singular varieties. This in particular implies a non-commutative version of the Bertini theorem.
Format:
LaTeX
MathJax
PDF
Abstract
A classification of simple weight modules over the Schrödinger algebra is given. The Krull and the global dimensions are found for the centralizer $C_{\mathcal{S}}(H)$ (and some of its prime factor algebras) of the Cartan element $H$ in the universal enveloping algebra $\mathcal{S}$ of the Schrödinger (Lie) algebra. The simple $C_{\mathcal{S}}(H)$-modules are classified. The Krull and the global dimensions are found for some (prime) factor algebras of the algebra $\mathcal{S}$ (over the centre). It is proved that some (prime) factor algebras of $\mathcal{S}$ and $C_{\mathcal{S}}(H)$ are tensor homological/Krull minimal.
It is shown that a knice subgroup with cardinality ℵ 1 {\aleph _1} , of a torsion-free completely decomposable abelian group, is again completely decomposable. Any torsion-free abelian k k -group of cardinality ℵ n {\aleph _n} has balanced projective dimension ≤ n \leq n .
This paper contains the proofs of the two following theorems: (1) Let (Mα)<y be a well-ordered decreasing system of submodules of the module M such that M=M0If M is strongly complete and strongly Hausdorff then (equation omitted) (2) Let R be a commutative ring having nonzero minimal idempotent ideals (Sα)α< y and let S,=∐<γ, Sa. An A-module is injective if and only if M=Annih S®M0where Annih S is injective and M0is strongly complete and Hausdorff in the topology introduced by annihilators of the direct sums of Sx.
The balanced-projective dimension of every abelian p-group is de- termined by means of a structural property that generalizes the third axiom of countability. As a corollary to our general structure theorem, we show for A = Wn that every pA-high subgroup of a p-group G has balanced-projective dimension exactly n whenever G has cardinality Nn but pAG *" O. Our characterization of balanced-projective dimension also leads to new classes of groups where different infinite dimensions are distinguished.
Let A be a finitely generated algebra over an absolutely flat commutative ring. Using sheaf-theoretic techniques, it is shown that the weak In the second section, it is shown that the natural analogue of the central simple algebra over a field is the central algebra over an absolutely flat ring which is a biregular ring in the sense of (5). Moreover, we show for every finitely gen- erated algebra A of finite Hochschild dimension that A modulo its Jacobson radical N is separable over R. This allows us to obtain the characterization theorem (6, Corollary 2 to Theorem 3, p. 3111 in case R is absolutely flat. In the final section, a structure theorem is obtained for algebras of Hochschild dimension 1 and their homomorphic images in terms of their idempotent structure. These results are a generalization of those of S. IU. Chase in (4).