Showing papers on "Global dimension published in 2017"
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TL;DR: In this paper, it was shown that every irreducible V G -module occurs in an irreducerible g-twisted V-module for some g ∈ G.
79 citations
TL;DR: In this paper, the notion of infinitely generated tilting objects of infinite homological dimension in an abelian category was defined and a one-to-one correspondence between ''infty''-tilting objects with an injective cogenerator and''infty$-cotilting objects in complete, cocomplete abelians with a projective generator was constructed.
Abstract: We define the notion of an infinitely generated tilting object of infinite homological dimension in an abelian category. A one-to-one correspondence between $\infty$-tilting objects in complete, cocomplete abelian categories with an injective cogenerator and $\infty$-cotilting objects in complete, cocomplete abelian categories with a projective generator is constructed. We also introduce $\infty$-tilting pairs, consisting of an $\infty$-tilting object and its $\infty$-tilting class, and obtain a bijective correspondence between $\infty$-tilting and $\infty$-cotilting pairs. Finally, we discuss the related derived equivalences and t-structures.
74 citations
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TL;DR: The notion of Krull dimension has an explicit computational content in the form of existence (or lack of existence) of some algebraic identities as mentioned in this paper, where abstract results about dimensions are used to show the existence of concrete elements.
Abstract: We present constructive versions of Krull's dimension theory for commutative rings and distributive lattices. The foundations of these constructive versions are due to Joyal, Espanol and the authors. We show that the notion of Krull dimension has an explicit computational content in the form of existence (or lack of existence) of some algebraic identities. We can then get an explicit computational content where abstract results about dimensions are used to show the existence of concrete elements. This can be seen as a partial realisation of Hilbert's program for classical abstract commutative algebra.
53 citations
TL;DR: Homological dimensions of algebras linked by recollements of derived module categories are studied, and a series of new upper bounds and relationships among their finitistic or global dimensions are established.
Abstract: In this paper, we study homological dimensions of algebras linked by recollements of derived module categories, and establish a series of new upper bounds and relationships among their finitistic or global dimensions. This is closely related to a longstanding conjecture, the finitistic dimension conjecture, in representation theory and homological algebra. Further, we apply our results to a series of situations of particular interest: exact contexts, ring extensions, trivial extensions, pullbacks of rings, and algebras induced from Auslander-Reiten sequences. In particular, we not only extend and amplify Happel’s reduction techniques for finitistic dimenson conjecture to more general contexts, but also generalise some recent results in the literature.
36 citations
TL;DR: For a ring R of weak global dimension at most one, the Telescope Conjecture holds for any commutative von Neumann regular ring R, and it holds precisely for those Prufer domains which are strongly discrete as mentioned in this paper.
30 citations
TL;DR: In this article, the authors study certain special tilting and cotilting modules for an algebra with positive dominant dimension, each of which is generated or cogenerated (and usually both) by projective-injectives.
Abstract: We study certain special tilting and cotilting modules for an algebra with positive dominant dimension, each of which is generated or cogenerated (and usually both) by projective-injectives. These modules have various interesting properties, for example that their endomorphism algebras always have global dimension at most that of the original algebra. We characterise minimal d-Auslander-Gorenstein algebras and d-Auslander algebras via the property that these special tilting and cotilting modules coincide. By the Morita-Tachikawa correspondence, any algebra of dominant dimension at least 2 may be expressed (essentially uniquely) as the endomorphism algebra of a generator-cogenerator for another algebra, and we also study our special tilting and cotilting modules from this point of view, via the theory of recollements and intermediate extension functors.
24 citations
TL;DR: In this article, it was shown that for any symmetric ring spectrum whose graded homotopy ring is concentrated in dimensions divisible by a natural number and has homological dimension at most three, it is equivalent to the derived category of the Johnson-Wilson spectrum.
Abstract: Using Franke's methods we construct new examples of exotic equivalences. We show that for any symmetric ring spectrum $R$ whose graded homotopy ring $\pi_*R$ is concentrated in dimensions divisible by a natural number $N \geq 5$ and has homological dimension at most three, the homotopy category of $R$-modules is equivalent to the derived category of $\pi_*R$. The Johnson-Wilson spectrum $E(3)$ and the truncated Brown-Peterson spectrum $BP\langle 2 \rangle$ for any prime $p \geq 5$ are our main examples. If additionally the homological dimension of $\pi_*R$ is equal to two, then the homotopy category of $R$-modules and the derived category of $\pi_*R$ are triangulated equivalent. Here the main examples are $E(2)$ and $BP \langle 1 \rangle$ at $p \geq 5$. The last part of the paper discusses a triangulated equivalence between the homotopy category of $E(1)$-local spectra at a prime $p \geq 5$ and the derived category of Franke's model. This is a theorem of Franke and we fill a gap in the proof.
12 citations
TL;DR: Using the resolution quiver for a connected Nakayama algebra, a fast algorithm was given to decide whether its global dimension is finite or not and whether it is Gorenstein or not as discussed by the authors.
Abstract: Using the resolution quiver for a connected Nakayama algebra, a fast algorithm is given to decide whether its global dimension is finite or not and whether it is Gorenstein or not. The latter strengthens a result of Ringel.
11 citations
TL;DR: In this paper, a higher zigzag algebra is defined as a twisted trivial extension of the Koszul dual of the original algebra of a quiver whose underlying graph is a tree.
Abstract: Given any Koszul algebra of finite global dimension one can define a new algebra, which we call a higher zigzag algebra, as a twisted trivial extension of the Koszul dual of our original algebra. If our original algebra is the path algebra of a quiver whose underlying graph is a tree, this construction recovers the zigzag algebras of Huerfano and Khovanov. We study examples of higher zigzag algebras coming from Iyama's iterative construction of type A higher representation finite algebras. We give presentations of these algebras by quivers and relations, and describe relations between spherical twists acting on their derived categories. We then make a connection to the McKay correspondence in higher dimensions: if G is a finite abelian subgroup of the special linear group acting on affine space, then the skew group algebra which controls the category of G-equivariant sheaves is Koszul dual to a higher zigzag algebra. Using this, we show that our relations between spherical twists appear naturally in examples from algebraic geometry.
9 citations
TL;DR: In this paper, necessary and sufficient conditions for a bi-amalgamation to inherit the arithmetical property were established, with applications on the weak global dimension and transfer of the semi-hereditary property.
Abstract: This paper establishes necessary and sufficient conditions for a bi-amalgamation to inherit the arithmetical property, with applications on the weak global dimension and transfer of the semihereditary property. The new results compare to previous works carried on various settings of duplications and amalgamations, and capitalize on recent results on bi-amalgamations. All results are backed with new and illustrative examples arising as bi-amalgamations.
9 citations
TL;DR: In this article, the projective dimension of Mackey functors and cohomological functors over a field of characteristic p has been examined, and it has been shown that the only functors of finite projective dimensions are projective.
Abstract: We examine the projective dimensions of Mackey functors and cohomological Mackey functors. We show over a field of characteristic p that cohomological Mackey functors are Gorenstein if and only if Sylow p-subgroups are cyclic or dihedral, and they have finite global dimension if and only if the group order is invertible or Sylow subgroups are cyclic of order 2. By contrast, we show that the only Mackey functors of finite projective dimension over a field are projective. This allows us to give a new proof of a theorem of Greenlees on the projective dimension of Mackey functors over a Dedekind domain. We conclude by completing work of Arnold on the global dimension of cohomological Mackey functors over ℤ.
TL;DR: In this article, the authors generalize these results to commutative noetherian -rings and show that the local cohomology functor associated with does not increase injective dimension.
Abstract: Let be a commutative noetherian ring, let be an ideal, and let be an injective -module. A basic result in the structure theory of injective modules states that the -module consisting of -torsion elements is also an injective -module. Recently, de Jong proved a dual result: If is a flat -module, then the -adic completion of is also a flat -module. In this paper we generalize these facts to commutative noetherian -rings: let be a commutative non-positive -ring such that is a noetherian ring and for each -module is finitely generated. Given an ideal , we show that the local cohomology functor associated with does not increase injective dimension. Dually, the derived -adic completion functor does not increase flat dimension.
TL;DR: In this article, the authors investigated the strong global dimension of a finite-dimensional piecewise hereditary algebra over an algebraically closed field, characterised in terms of the lengths of sequences of tilting mutations relating A to a hereditary abelian category.
TL;DR: In this paper, the equivalence of non-commutative projective schemes and cluster tilting modules was studied, and the relation between the two was shown to be equivalent.
Abstract: In this paper, we study the relationship between equivalences of noncommutative projective schemes and cluster tilting modules. In particular, we prove the following result. Let $A$ be an AS-Gorenstein algebra of dimension $d\geq 2$ and ${\mathsf{tails}\,} A$ the noncommutative projective scheme associated to $A$. If $\operatorname{gldim}({\mathsf{tails}\,} A)< \infty$ and $A$ has a $(d-1)$-cluster tilting module $X$ satisfying that its graded endomorphism algebra is $\mathbb N$-graded, then the graded endomorphism algebra $B$ of a basic $(d-1)$-cluster tilting submodule of $X$ is a two-sided noetherian $\mathbb N$-graded AS-regular algebra over $B_0$ of global dimension $d$ such that ${\mathsf{tails}\,} B$ is equivalent to ${\mathsf{tails}\,} A$.
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TL;DR: In this article, it was shown that the Noetherian dimension of an Artinian module with homogeneous uniserial dimension is less than or equal to the Krull dimension of these modules.
Abstract: In this article, we first show that non-Noetherian Artinian uniserial modules over commutative rings, duo rings, finite $R$-algebras and right Noetherian rings are $1$-atomic exactly like $Bbb Z_{p^{infty}}$. Consequently, we show that if $R$ is a right duo (or, a right Noetherian) ring, then the Noetherian dimension of an Artinian module with homogeneous uniserial dimension is less than or equal to $1$. In particular, if $A$ is a quotient finite dimensional $R$-module with homogeneous uniserial dimension, where $R$ is a locally Noetherian (or, a Noetherian duo) ring, then $n$-dim $Aleq 1$. We also show that the Krull dimension of Noetherian modules is bounded by the uniserial dimension of these modules. Moreover, we introduce the concept of qu-uniserial modules and by using this concept, we observe that if $A$ is an Artinian $R$-module, such that any of its submodules is qu-uniserial, where $R$ is a right duo (or, a right Noetherian) ring, then $n$-dim $Aleq 1$.
TL;DR: In this paper, it was shown that if $H$ and $L$ are two Hopf algebras such that their comodule categories are monoidal equivalent, then they are twisted Calabi-Yau Hopf Algebra (CYHopf) when it is homologically smooth.
Abstract: Let $H$ and $L$ be two Hopf algebras such that their comodule categories are monoidal equivalent. We prove that if $H$ is a twisted Calabi-Yau (CY) Hopf algebra, then $L$ is a twisted CY algebra when it is homologically smooth. Especially, if $H$ is a Noetherian twisted CY Hopf algebra and $L$ has finite global dimension, then $L$ is a twisted CY algebra.
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TL;DR: In this article, the affine algebraic varieties whose points correspond to associative algebras are introduced and the properties of the affines within a variety share many important homological properties.
Abstract: In this paper we introduce new affine algebraic varieties whose points correspond to associative algebras. We show that the algebras within a variety share many important homological properties. In particular, any two algebras in the same variety have the same dimension. The case of finite dimensional algebras as well as that of graded algebras arise as subvarieties of the varieties we define. As an application we show that for algebras of global dimension two over the complex numbers, any algebra in the variety continuously deforms to a monomial algebra.
TL;DR: In this article, a new proof is given of the description of the center of quadratic Sklyanin algebras of global dimension three and four and the centre of cubic Sklynin algesbras with global dimension 3 and 4.
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TL;DR: The existence of a tilting module is equivalent to the existence of an algebra having a dominant dimension at least $2, independent of its global dimension as discussed by the authors, which is the case for all algebras with tilting modules that are generated and cogenerated by projective-injective modules.
Abstract: We study which algebras have tilting modules that are both generated and cogenerated by projective-injective modules Crawley-Boevey and Sauter have shown that Auslander algebras have such tilting modules; and for algebras of global dimension $2$, Auslander algebras are classified by the existence of such tilting modules
In this paper, we show that the existence of such a tilting module is equivalent to the algebra having dominant dimension at least $2$, independent of its global dimension In general such a tilting module is not necessarily cotilting Here, we show that the algebras which have a tilting-cotilting module generated-cogenerated by projective-injective modules are precisely $1$-Auslander-Gorenstein algebras
When considering such a tilting module, without the assumption that it is cotilting, we study the global dimension of its endomorphism algebra, and discuss a connection with the Finitistic Dimension Conjecture Furthermore, as special cases, we show that triangular matrix algebras obtained from Auslander algebras and certain injective modules, have such a tilting module We also give a description of which Nakayama algebras have such a tilting module
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TL;DR: In this paper, Chen and Koenig showed that any algebra of dominant dimension at least two which has exactly one projective non-injective indecomposable module is a finitistic Auslander algebra.
Abstract: Recently, Chen and Koenig in \cite{CheKoe} and Iyama and Solberg in \cite{IyaSol} independently introduced and characterised algebras with dominant dimension coinciding with the Gorenstein dimension and both dimensions being larger than or equal to two. In \cite{IyaSol}, such algebras are named Auslander-Gorenstein algebras. Auslander-Gorenstein algebras generalise the well known class of higher Auslander algebras, where the dominant dimension additionally coincides with the global dimension. In this article we generalise Auslander-Gorenstein algebras further to algebras having the property that the dominant dimension coincides with the finitistic dimension and both dimension are at least two. We call such algebras finitistic Auslander algebras. As an application we can specialise to reobtain known results about Auslander-Gorenstein algebras and higher Auslander algebras such as the higher Auslander correspondence with a very short proof. We then give several conjectures and classes of examples for finitistic Auslander algebras. For a local Hopf algebra $A$ and an indecomposable non-projective $A$-module $M$, we show that $End_A(A \oplus M)$ is always a finitistic Auslander algebra of dominant dimension two. In particular this shows that $Ext_A^1(M,M)$ is always non-zero, which generalises a result of Tachikawa who proved that $Ext_A^1(M,M)
eq 0$ for indecomposable non-projective modules $M$ over group algebras of $p$-groups. We furthermore conjecture that every algebra of dominant dimension at least two which has exactly one projective non-injective indecomposable module is a finitistic Auslander algebra. We prove this conjecture for a large class of algebras which includes all representation-finite algebras.
TL;DR: In this article, the definition of piecewise Noetherian rings was extended to the noncommutative case, and various properties of such rings were investigated, including Gabriel dimension and Gabriel correspondence between prime ideals and injective modules.
Abstract: We extend the definition of a piecewise Noetherian ring to the noncommutative case, and investigate various properties of such rings. In particular, we show that a ring with Krull dimension is piecewise Noetherian. Certain fully bounded piecewise Noetherian rings have Gabriel dimension and exhibit the Gabriel correspondence between prime ideals and indecomposable injective modules.
TL;DR: In this paper, the authors introduce and study (pre)resolving subcategories of a triangulated category and the homological dimension relative to these sub-categories, and apply the obtained properties to relative Gorenstein categories.
Abstract: We introduce and study (pre)resolving subcategories of a triangulated category and the homological dimension relative to these subcategories. We apply the obtained properties to relative Gorenstein categories.
TL;DR: In this paper, the homological dimensions of the Leavitt path algebra L K (E ) of a finite graph E with coefficients in a commutative ring K were established.
TL;DR: In this article, the authors studied Krull dimension and unique factorization in the ring $R_H[x] and showed that it is a Krull domain if and only if $R$ is a unique domain, containing all rational numbers.
Abstract: Let $R$ be a commutative ring with identity, and let $R[x]$ be the collection of polynomials with coefficients in~$R$. We observe that there are many multiplications in $R[x]$ such that, together with the usual addition, $R[x]$ becomes a ring that contains $R$ as a subring. These multiplications belong to a class of functions $\lambda $ from $\mathbb {N}_0$ to $\mathbb {N}$. The trivial case when $\lambda (i) = 1$ for all $i$ gives the usual polynomial ring. Among nontrivial cases, there is an important one, namely, the case when $\lambda (i) = i!$ for all $i$. For this case, it gives the well-known Hurwitz polynomial ring $R_H[x]$. In this paper, we study Krull dimension and unique factorization in $R_H[x]$. We show in general that $\dim R \leq \dim R_H[x] \leq 2\dim R +1$. When the ring $R$ is Noetherian we prove that $\dim R \leq \dim R_H[x] \leq \dim R+1$. A condition for the ring $R$ is also given in order to determine whether $\dim R_H[x] = \dim R$ or $\dim R_H[x] = \dim R +1$ in this case. We show that $R_H[x]$ is a unique factorization domain, respectively, a Krull domain, if and only if $R$ is a unique factorization domain, respectively, a Krull domain, containing all of the rational numbers.
TL;DR: In this paper, the authors consider several homological dimensions of crossed products AG, where A is a left Noetherian ring and G is a finite group, and the global dimension and finitistic dimension of AG are classified.
Abstract: In this paper we consider several homological dimensions of crossed products AG, where A is a left Noetherian ring and G is a finite group. We revisit the induction and restriction functors in derived categories, generaliz- ing a few classical results for separable extensions. The global dimension and finitistic dimension of AG are classified: global dimension of AG is either infinity or equal to that of A, and finitistic dimension of AG coincides with that of A. A criterion for skew group rings to have finite global dimensions is deduced. Under the hypothesis that A is a semiprimary algebra containing a complete set of primitive orthogonal idempotents closed under the action of a Sylow p-subgroup S 6 G, we show that A and AG share the same ho- mological dimensions under extra assumptions, extending the main results in (15, 16).
TL;DR: In this article, the projective dimension of a finite dimensional Auslander algebra Λ is shown to be at most one if and only if the socle dimension of its socle socN is at most 1.
Abstract: Let Λ be a finite dimensional Auslander algebra. For a Λ-module N, we prove that the projective dimension of N is at most one if and only if the projective dimension of its socle socN is at most one. As an application, we give a new characterization of Auslander algebras Λ and prove that a finite dimensional algebra Λ is an Auslander algebra provided its global dimension gl.dΛ ≤ 2 and an injective Λ-module is projective if and only if the projective dimension of its socle is at most one.
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TL;DR: In this paper, it was shown that the global dimension of the complex algebra of the monoid of all partial functions on an n-set is $n-1$ for all $n\geq 1$.
Abstract: We prove that the global dimension of the complex algebra of the monoid of all partial functions on an n-set is $n-1$ for all $n\geq 1$. This is also the global dimension of the complex algebra of the category of all epimorphisms between subsets of an $n$-set. In our proof we use standard homological methods as well as combinatorial techniques associated to the representation theory of the symmetric group. As part of the proof, we obtain a partial description of the Cartan matrix of these algebras.
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TL;DR: In this paper, it was shown that there exists a right noetherian AS-regular algebra over a smooth quadric surface in the sense of Smith and Van den Bergh's path algebra of the 2-Kronecker quiver.
Abstract: Let $R$ be a finite dimensional algebra of finite global dimension over a field $k$. In this paper, we will characterize a $k$-linear abelian category $\mathscr C$ such that $\mathscr C\cong \operatorname {tails} A$ for some graded right coherent AS-regular algebra $A$ over $R$. As an application, we will prove that if $\mathscr C$ is a smooth quadric surface in a quantum $\mathbb P^3$ in the sense of Smith and Van den Bergh, then there exists a right noetherian AS-regular algebra $A$ over $kK_2$ of dimension 3 and of Gorenstein parameter 2 such that $\mathscr C\cong \operatorname {tails} A$ where $kK_2$ is the path algebra of the 2-Kronecker quiver.
TL;DR: Gorenstein coresolving categories as discussed by the authors unify the following notions: Gorenstein injective modules [8], FP-injective modules, AC-injection modules, and so on.
Abstract: Let 𝒜 be an abelian category. A subcategory 𝒳 of 𝒜 is called coresolving if 𝒳 is closed under extensions and cokernels of monomorphisms and contains all injective objects of 𝒜. In this paper, we introduce and study Gorenstein coresolving categories, which unify the following notions: Gorenstein injective modules [8], Gorenstein FP-injective modules [20], Gorenstein AC-injective modules [3], and so on. Then we define a resolution dimension relative to the Gorenstein coresolving category 𝒢ℐ𝒳(𝒜). We investigate the properties of the homological dimension and unify some important properties possessed by some known homological dimensions. In addition, we study stability of the Gorenstein coresolving category 𝒢ℐ𝒳(𝒜) and apply the obtained properties to special subcategories and in particular to module categories.
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TL;DR: In this paper, the authors established the specialization method on characteristic ideals for finitely generated torsion modules over a complete local normal domain R that is module-finite over $O[[x_1,..., x_d]], where O is the ring of integers of a finite extension of the field of p-adic integers $Q_p.
Abstract: The aim of this article is to establish the specialization method on characteristic ideals for finitely generated torsion modules over a complete local normal domain R that is module-finite over $O[[x_1, ..., x_d]]$, where $O$ is the ring of integers of a finite extension of the field of p-adic integers $Q_p$. The specialization method is a technique that recovers the information on the characteristic ideal $char_R(M)$ from $char_{R/I}(M/IM)$, where I varies in a certain family of nonzero principal ideals of R. As applications, we prove Euler system bound over Cohen-Macaulay normal domains by combining the main results in an earlier article of the first named author and then we prove one of divisibilities of the Iwasawa main conjecture for two-variable Hida deformations generalizing the main theorem obtained in an article of the first named author.