Finite Global Dimension Research Articles

Algebras with irreducible module varieties I

We call a finite-dimensional K-algebra A geometrically irreducible if for all d≥0 all connected components of the affine scheme of d-dimensional A-modules are irreducible. We show that the geometrically irreducible algebras without loops (this includes all algebras of finite global dimension) are the hereditary algebras. We also show that truncated polynomial rings are the only geometrically irreducible local algebras. Finally, we formulate a conjectural classification of all geometrically irreducible algebras.

Azumaya loci and discriminant ideals of PI algebras

We prove that, under mild assumptions, for all positive integers ℓ, the zero set of the discriminant ideal Dℓ(R/Z(R),tr) of a prime affine polynomial identity (PI) algebra R coincides with the zero set of the modified discriminant ideal MDℓ(R/Z(R),tr) of R, and give an explicit description of this set in terms of the dimensions of the irreducible representations of R. Furthermore, we prove that, when ℓ is the square of the PI-degree of R, this zero set is precisely the complement of the Azumaya locus of R. This description is used to classify the Azumaya loci of the multiparameter quantized Weyl algebras at roots of unity. As another application, we prove that the zero set of the top discriminant ideal of a prime affine PI algebra R coincides with the singular locus of the center of R, provided that the discriminant ideal has height at least 2, R has finite global dimension and R is a Cohen–Macaulay module over its center.

Contractible stability spaces and faithful braid group actions

We prove that any `finite-type' component of a stability space of a triangulated category is contractible. The motivating example of such a component is the stability space of the Calabi--Yau-$N$ category $\mathcal{D}(\Gamma_N Q)$ associated to an ADE Dynkin quiver. In addition to showing that this is contractible we prove that the braid group $\operatorname{Br}(Q)$ acts freely upon it by spherical twists, in particular that the spherical twist group $\operatorname{Br}(\Gamma_N Q)$ is isomorphic to $\operatorname{Br}(Q)$. This generalises Brav-Thomas' result for the $N=2$ case. Other classes of triangulated categories with finite-type components in their stability spaces include locally-finite triangulated categories with finite rank Grothendieck group and discrete derived categories of finite global dimension.

Triangle equivalences involving Gorenstein FP-injective modules

ABSTRACTRecently, Hu et al. [16] introduced the notion of Gorenstein FP-injective modules and investigated a notion of Gorenstein FP-injective dimension for complexes. Let R be a left coherent ring and Db(R-Mod) the bounded derived category of left R-modules. It is proved that the quotient triangulated category of the subcategory of Db(R-Mod) consisting of complexes with both finite Gorenstein FP-injective dimension and FP-projective dimension by the bounded homotopy category of FP-projective-injective left R-modules is triangle-equivalent to the stable category of the Frobenius category of all Gorenstein FP-injective and FP-projective left R-modules. Note that a similar argument is also valid for the case of Gorenstein injective left R-modules. We extend a triangle equivalence established by Beligiannis involving Gorenstein injective left R-modules from rings with finite left Gorenstein global dimension to arbitrary rings.

Comparisons between singularity categories and relative stable categories of finite groups

We consider the relationship between the relative stable category of and the usual singularity category for group algebras with coefficients in a commutative noetherian ring. When the coefficient ring is self-injective we show that these categories share a common, relatively large, Verdier quotient. At the other extreme, when the coefficient ring has finite global dimension, there is a semi-orthogonal decomposition, due to Poulton, relating the two categories. We prove that this decomposition is partially compatible with the monoidal structure and study the morphism it induces on spectra.

Noncommutative products of Euclidean spaces

We present natural families of coordinate algebras of noncommutative products of Euclidean spaces. These coordinate algebras are quadratic ones associated with an R-matrix which is involutive and satisfies the Yang-Baxter equations. As a consequence they enjoy a list of nice properties, being regular of finite global dimension. Notably, we have eight-dimensional noncommutative euclidean spaces which are particularly well behaved and are deformations parametrised by a two-dimensional sphere. Quotients include noncommutative seven-spheres as well as noncommutative "quaternionic tori". There is invariance for an action of $SU(2) \times SU(2)$ in parallel with the action of $U(1) \times U(1)$ on a "complex" noncommutative torus which allows one to construct quaternionic toric noncommutative manifolds. Additional classes of solutions are disjoint from the classical case.

Two problems from the Polishchuk and Positselski book on quadratic algebras

In the book Quadratic algebras by Polishchuk and Positselski [23], algebras with a small number of generators (n=2,3) are considered. For some number r of relations possible Hilbert series are listed, and those appearing as series of Koszul algebras are specified. The first case, where it was not possible to do, namely the case of three generators n=3 and six relations r=6 is formulated as an open problem. We give here a complete answer to this question, namely for quadratic algebras with \mathrm {dim} A_1=\mathrm {dim} A_2=3 , we list all possible Hilbert series, and find out which of them can come from Koszul algebras, and which can not. As a consequence of this classification, we found an algebra, which serves as a counterexample to another problem from the same book [23, Chapter 7, Sec. 1, Conjecture 2], saying that Koszul algebra of finite global homological dimension d has \mathrm {dim} A_1 \geq d . Namely, the 3-generated algebra A given by relations xx+yx=xz=zy=0 is Koszul and its Koszul dual algebra A^! has Hilbert series of degree 4: H_{A^!}(t)= 1+3t+3t^2+2t^3+t^4 , hence A has global homological dimension 4.

Duality and Serre functor in homotopy categories

ABSTRACTFor a (right and left) coherent ring A, we show that there exists a duality between homotopy categories and 𝕂b(mod-A). If A = Λ is an artin algebra of finite global dimension, this duality induces a duality between their subcategories of acyclic complexes, and As a result, it will be shown that, in this case, admits a Serre functor and hence has Auslander–Reiten triangles.

Проективнi гратки черепичних порядкiв

Tiled orders over discrete valuation ring have been studied since the 1970s by many mathematicians, in particular, by Yategaonkar V.A., Tarsy R.B., Roggenkamp K.W, Simson D., Drozd Y.A., Zavadsky A.G. and Kirichenko V.V. Yategaonkar V.A. proved that for every n > 2, there is, up to an isomorphism, a finite number of tiled orders over a discrete valuation ring O of finite global dimension which lie in $M_n(K)$ where K is a field of fractions of a commutatively discrete valuation ring O. The articles by R.B. Tarsy, V.A. Yategaonkar, H. Fujita, W. Rump and others are devoted to the study of the global dimension of tiled orders. H. Fujita described the reduced tiled orders in Mn(D) of finite global dimension for n = 4; 5. V.M. Zhuravlev and D.V. Zhuravlev described reduced tiled orders in Mn(D) of finite global dimension for n = 6: This paper examines the necessary condition for the finiteness of the global dimension of the tile order. Let A be a tiled order. The kernel of the projective resolvent of an irreducible lattice has the form M1f1 +M2f2 + ::: +Msfs, where Mi is irreducible lattice, fi is some vector. If the tile order has a finite global dimension, then there is a projective lattice that is the intersection of projective lattices. This condition is the one explored in the paper.

Invariants of Noncommutative Projective Schemes

In this note we compute several invariants (e.g. algebraic K-theory, cyclic homology and topological Hochschild homology) of the noncommutative projective schemes associated to Koszul algebras of finite global dimension.

Derived representation type and cleaving functors

ABSTRACTWe introduce the notation of weakly derived tameness, and establish the equivalence of derived tameness and weakly derived tameness for algebras of finite global dimension. Moreover, we observe the relation between derived representation type and cleaving functors, and obtain a method to judge an algebra to be derived wild. As an application, we determine the derived representation type of self-injective Nakayama algebras.

Affine cellularity of affine Yokonuma–Hecke algebras

We establish an explicit algebra isomorphism between the affine Yokonuma–Hecke algebra Yˆr,n(q) and a direct sum of matrix algebras with coefficients in tensor products of affine Hecke algebras of type A. As an application of this result, we show that Yˆr,n(q) is affine cellular in the sense of Koenig and Xi, and further prove that it has finite global dimension when the parameter q is not a root of the Poincaré polynomial. As another application, we also recover the modular representation theory of Yˆr,n(q) previously obtained in [7].

Calabi–Yau property under monoidal Morita–Takeuchi equivalence

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.

On the Projective Dimensions of Mackey Functors

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.

Minimal tiled orders of finite global dimension

A directed graph is associated to any basic tiled order, and it turns out that the graph is connected for all known examples of tiled orders of finite global dimension. It is proved that the minimal connected tiled orders of finite global dimension in a fixed algebra are of global dimension two, and that up to isomorphism, these minimal orders are characterized by their unoriented graph which is a tree. Their irreducible representations are in one-to-one correspondence with the possible orientations of this tree.

Anick resolution and Koszul algebras of finite global dimension

ABSTRACTWe show how to study a certain associative algebra recently discovered by Iyudu and Shkarin using the Anick resolution. This algebra is a counterexample to the conjecture of Positselski on Koszul algebras of finite global dimension.

Embeddings of algebras in derived categories of surfaces

By a result of Orlov there always exists an embedding of the derived category of a finite-dimensional algebra of finite global dimension into the derived category of a high-dimensional smooth projective variety. In this article we give some restrictions on those algebras whose derived categories can be embedded into the bounded derived category of a smooth projective surface. This is then applied to obtain explicit results for hereditary algebras.

Recollements of self-injective algebras, and classification of self-injective diagram algebras

Diagram algebras, in particular Brauer algebras, Birman–Murakami-Wenzl algebras and partition algebras, are used in representation theory and invariant theory of orthogonal and symplectic groups, in knot theory, in mathematical physics and elsewhere. Classifications are known when such algebras are semisimple, of finite global dimension or quasi-hereditary. We obtain a characterisation of the self-injective case, which is shown to coincide with the (previously also unknown) symmetric case. The main tool is to show that indecomposable self-injective algebras in general are derived simple, that is, their bounded derived module categories admit trivial recollements only. As a consequence, self-injective algebras are seen to satisfy a derived Jordan–Holder theorem.

A generalization of the matrix transpose map and its relationship to the twist of the polynomial ring by an automorphism

A generalization of the notion of symmetric matrix was introduced by Cassidy and Vancliff in 2010 and used by them in a construction that produces quadratic regular algebras of finite global dimension that are generalizations of graded Clifford algebras. In this article, we further their ideas by introducing a generalization of the matrix transpose map and use it to generalize the notion of skew-symmetric matrix. With these definitions, an analogue of the result that every n × n matrix is a sum of a symmetric matrix and a skew-symmetric matrix holds. We also prove an analogue of the result that the transpose map is an antiautomorphism of the algebra of n × n matrices, and show that the antiautomorphism property of our generalized transpose map is related to the notion of twisting the polynomial ring on n variables by an automorphism.

Some generalizations of preprojective algebras and their properties

In this note we consider a notion of relative Frobenius pairs of commutative rings S/R. To such a pair, we associate an N-graded R-algebra ΠR(S) which has a simple description and coincides with the preprojective algebra of a quiver with a single central node and several outgoing edges in the split case. If the rank of S over R is 4 and R is Noetherian, we prove that ΠR(S) is itself Noetherian and finite over its center and that each ΠR(S)d is finitely generated projective. We also prove that ΠR(S) is of finite global dimension if R and S are regular.