Noncommutative Projective Scheme Research Articles

Representation Theory of Geigle-Lenzing Complete Intersections

Weighted projective lines, introduced by Geigle and Lenzing in 1987, are important objects in representation theory. They have tilting bundles, whose endomorphism algebras are the canonical algebras introduced by Ringel. The aim of this paper is to study their higher dimensional analogs. First, we introduce a certain class of commutative Gorenstein rings R R graded by abelian groups L \mathbb {L} of rank 1 1 , which we call Geigle-Lenzing complete intersections. We study the stable category C M _ L R \underline {\mathsf {CM}}^{\mathbb {L}}R of Cohen-Macaulay representations, which coincides with the singularity category D s g L ( R ) \mathsf {D}^{\mathbb {L}}_{\mathrm {sg}}(R) . We show that C M _ L R \underline {\mathsf {CM}}^{\mathbb {L}}R is triangle equivalent to D b ( m o d A C M ) \mathsf {D}^{\mathrm {b}}(\mathsf {mod} A^{\mathrm {CM}}) for a finite dimensional algebra A C M A^{\mathrm {CM}} , which we call the CM-canonical algebra. As an application, we classify the ( R , L ) (R,\mathbb {L}) that are Cohen-Macaulay finite. We also give sufficient conditions for ( R , L ) (R,\mathbb {L}) to be d d -Cohen-Macaulay finite in the sense of higher Auslander-Reiten theory. Secondly, we study a new class of non-commutative projective schemes in the sense of Artin-Zhang, i.e. the category c o h X = m o d L R / m o d 0 L R \mathsf {coh}\mathbb {X}=\mathsf {mod}^{\mathbb {L}}R/\mathsf {mod}^{\mathbb {L}}_0R of coherent sheaves on the Geigle-Lenzing projective space X \mathbb {X} . Geometrically this is the quotient stack X = [ X / G ] \mathbb {X}=[X/G] for X = S p e c R ∖ { R + } X={\mathrm {Spec}}\,R\setminus \{R_+\} and G = S p e c k [ L ] G={\mathrm {Spec}}\,k[\mathbb {L}] . We show that D b ( c o h X ) \mathsf {D}^{\mathrm {b}}(\mathsf {coh}\mathbb {X}) is triangle equivalent to D b ( m o d A c a ) \mathsf {D}^{\mathrm {b}}(\mathsf {mod} A^{ca}) for a finite dimensional algebra A c a A^{\mathrm {ca}} , which we call a d d -canonical algebra. We study when X \mathbb {X} is d d -vector bundle finite, and when X \mathbb {X} is derived equivale’nt to a d d -representation infinite algebra in the sense of higher Auslander-Reiten theory. Our d d -canonical algebras provide a rich source of d d -Fano and d d -anti-Fano algebras in non-commutative algebraic geometry. We also observe Orlov-type semiorthogonal decompositions of D s g L ( R ) \mathsf {D}_{\mathrm {sg}}^{\mathbb {L}}(R) and D b ( c o h X ) \mathsf {D}^{b}(\mathsf {coh}\mathbb {X}) .

Combinatorial classification of (±1)-skew projective spaces

AbstractThe non-commutative projective scheme $\operatorname{\mathsf{Proj_{nc}}} S$ of a $(\pm 1)$-skew polynomial algebra S in n variables is considered to be a $(\pm 1)$-skew projective space of dimension n − 1. In this paper, using combinatorial methods, we give a classification theorem for $(\pm 1)$-skew projective spaces. Specifically, among other equivalences, we prove that $(\pm 1)$-skew projective spaces $\operatorname{\mathsf{Proj_{nc}}} S$ and $\operatorname{\mathsf{Proj_{nc}}} S^{\prime}$ are isomorphic if and only if certain graphs associated with S and Sʹ are switching (or mutation) equivalent. We also discuss invariants of $(\pm 1)$-skew projective spaces from a combinatorial point of view.

Frobenius-Perron theory for projective schemes

The Frobenius-Perron theory of an endofunctor of a k \Bbbk -linear category (recently introduced in Chen et al. [Algebra Number Theory 13 (2019), pp. 2005–2055]) provides new invariants for abelian and triangulated categories. Here we study Frobenius-Perron type invariants for derived categories of commutative and noncommutative projective schemes. In particular, we calculate the Frobenius-Perron dimension for domestic and tubular weighted projective lines, define Frobenius-Perron generalizations of Calabi-Yau and Kodaira dimensions, and provide examples. We apply this theory to the derived categories associated to certain Artin-Schelter regular and finite-dimensional algebras.

Kernels for noncommutative projective schemes

We give a noncommutative geometric description of the internal Hom dg-category in the homotopy category of dg-categories between two noncommutative projective schemes in the style of Artin-Zhang. As an immediate application, we give a noncommutative projective derived Morita statement along lines of Rickard and Orlov.

Hochschild cohomology related to graded down-up algebras with weights (1,n)

Let [Formula: see text] be a graded down-up algebra with [Formula: see text] and [Formula: see text], and let [Formula: see text] be the Beilinson algebra of [Formula: see text]. If [Formula: see text], then a description of the Hochschild cohomology group of [Formula: see text] is known. In this paper, we calculate the Hochschild cohomology group of [Formula: see text] for the case [Formula: see text]. As an application, we see that the structure of the bounded derived category of the noncommutative projective scheme of [Formula: see text] is different depending on whether (10) [Formula: see text] [Formula: see text] is zero or not. Moreover, it turns out that there is a difference between the cases [Formula: see text] and [Formula: see text] in the context of Grothendieck groups.

The wonderful compactification for quantum groups

In this paper, we introduce a quantum version of the wonderful compactification of a group as a certain noncommutative projective scheme. Our approach stems from the fact that the wonderful compactification encodes the asymptotics of matrix coefficients, and from its realization as a GIT quotient of the Vinberg semigroup. In order to define the wonderful compactification for a quantum group, we adopt a generalized formalism of $\mathsf{Proj}$ categories in the spirit of Artin and Zhang. Key to our construction is a quantum version of the Vinberg semigroup, which we define as a $q$-deformation of a certain Rees algebra, compatible with a standard Poisson structure. Furthermore, we discuss quantum analogues of the stratification of the wonderful compactification by orbits for a certain group action, and provide explicit computations in the case of $\mathrm{SL}_2$.

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.

Moduli of non-commutative polarized schemes

We construct, using geometric invariant theory, a quasi-projective Deligne–Mumford stack of stable graded algebras. We also construct a derived enhancement, which classifies twisted bundles of stable graded A_infty -algebras. The tangent complex of the derived scheme is given by graded Hochschild cohomology, which we relate to ordinary Hochschild cohomology. We obtain a version of Hilbert stability for non-commutative projective schemes.

Cluster tilting modules and noncommutative projective schemes

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 ${\mathsf{tails}\,} A$.

Non-commutative algebraic geometry of semi-graded rings

In this paper, we introduce the semi-graded rings, which extend graded rings and skew Poincaré–Birkhoff–Witt (PBW) extensions. For this new type of non-commutative rings, we will discuss some basic problems of non-commutative algebraic geometry. In particular, we will prove some elementary properties of the generalized Hilbert series, Hilbert polynomial and Gelfand–Kirillov dimension. We will extend the notion of non-commutative projective scheme to the case of semi-graded rings and we generalize the Serre–Artin–Zhang–Verevkin theorem. Some examples are included at the end of the paper.

On the Tensor Product of Linear Sites and Grothendieck Categories

We define a tensor product of linear sites, and a resulting tensor product of Grothendieck categories based upon their representations as categories of linear sheaves. We show that our tensor product is a special case of the tensor product of locally presentable linear categories, and that the tensor product of locally coherent Grothendieck categories is locally coherent if and only if the Deligne tensor product of their abelian categories of finitely presented objects exists. We describe the tensor product of non-commutative projective schemes in terms of Z-algebras, and show that for projective schemes our tensor product corresponds to the usual product scheme.

Hopf dense Galois extensions with applications

Let H be a finite dimensional Hopf algebra, and let A be a left H-module algebra. Motivated by the study of the isolated singularities of AH and the endomorphism ring EndAH(A), we introduce the concept of Hopf dense Galois extensions in this paper. Hopf dense Galois extensions yield certain equivalences between the quotient categories over A and AH. A special class of Hopf dense Galois extensions consists of the so-called densely group graded algebras, which are weaker versions of strongly graded algebras. A weaker version of Dade's Theorem holds for densely group graded algebras. As applications, we recover the classical equivalence of the noncommutative projective scheme over a noetherian N-graded algebra A and its d-th Veronese subalgebra A(d) respectively. Hopf dense Galois extensions are also applied to the study of noncommutative graded isolated singularities.

Ampleness of two-sided tilting complexes

In this paper we define the notion of ampleness for two-sided tilting complexes over finite dimensional algebras and prove its basic properties. We call a finite dimensional k-algebra A of finite global dimension Fano if (A∗[−d])−1 is ample for some d ≥ 0. For example geometric algebras in the sense of Bondal-Polishchuk are Fano. We give a characterization of representation type of a quiver from a noncommutative algebro-geometric view point, that is, a finite acyclic quiver has finite representation type if and only if its path algebra is fractional Calabi-Yau, and a finite acyclic quiver has infinite representation type if and only if its path algebra is Fano. 0 Introduction Let X be a nonsingular projective variety over a field k and let ωX be its canonical bundle. Then the functor SX := −⊗X ωX [dimX] : D(cohX) −→ D(cohX) is the Serre functor ,i.e., HomX(G ·,F ·)∗ is functorially isomorphic to HomX(F ·, SX(G ·)) for F ·,G · ∈ D(cohX). By this fact, from a noncommutative (or categorical) algebro-geometric view point, one thinks of a triangulated category T as the derived category of coherent sheaves of some ”space” X and of the Serre functor ST of T (if exists) as the derived tensor product of ” dimX”-shifted ”canonical bundle” ωX . From this view point, the notion of Calabi-Yau algebra ( and Calabi-Yau category ) is defined and studied extensively by many researchers. In this paper we introduce the notion of ampleness for two-sided tilting complexes over finite dimensional k-algebras. Let A be a finite dimensional k-algebra of finite global dimension. Definition 0.1 (Definition 2.6). A two-sided tilting complex σ over A is called very ample if H(σ) = 0 for i ≥ 1 and σ is pure for n A 0. σ is called ample if σ is pure for n A 0. In Section 2, we justify this definition by using the theory of noncommutative projective schemes due to Artin-Zhang [AZ] and Polishchuk [Po]. In the theory of noncommutative projective schemes , for a graded coherent ring R over k we attach an imaginary geometric object projR = ( cohprojR, R, (1) ) . An abelian category cohprojR is considered as the category of coherent sheaves on projR. (See Section 1.) In Section 2 we show that the following facts hold. If σ is a very ample tilting complex over A, then the tensor algebra T := TA(H (σ)) of H(σ) over A is a graded connected coherent ring over A and there is a t-structure D defined by σ in Perf A and its heart H is equivalent to cohprojT . Moreover the following Theorem holds. Theorem 0.2 (Theorem 2.8). Let A be a finite dimensional k-algebra of finite global dimension and let σ be a very ample two-sided tilting complex. Then there is a natural equivalence of triangulated categories D (mod-A) ∼ −→ D (cohprojT ) . where T := TA(H (σ)) is the tensor algebra of H(σ) over A.

Generalized Serre duality

We introduce a notion of generalized Serre duality on a Hom-finite Krull–Schmidt triangulated category T . This duality induces the generalized Serre functor on T , which is a linear triangle equivalence between two thick triangulated subcategories of T . Moreover, the domain of the generalized Serre functor is the smallest additive subcategory of T containing all the indecomposable objects which appear as the third term of an Auslander–Reiten triangle in T ; dually, the range of the generalized Serre functor is the smallest additive subcategory of T containing all the indecomposable objects which appear as the first term of an Auslander–Reiten triangle in T . We compute explicitly the generalized Serre duality on the bounded derived categories of artin algebras and of certain noncommutative projective schemes in the sense of Artin and Zhang. We obtain a characterization of Gorenstein algebras: an artin algebra A is Gorenstein if and only if the bounded homotopy category of finitely generated projective A-modules has Serre duality in the sense of Bondal and Kapranov.

Geometric algebras on projective surfaces

Let X be a projective surface, let \sigma be an automorphism of X, and let L be a \sigma-ample invertible sheaf on X. We study the properties of a family of subrings, parameterized by geometric data, of the twisted homogeneous coordinate ring B(X, L, \sigma). In particular, we find necessary and sufficient conditions for these subrings to be noetherian. We also study their homological properties, their associated noncommutative projective schemes, and when they are maximal orders. In the process, we produce new examples of maximal orders; these are graded and have the property that no Veronese subring is generated in degree 1. Our results are used in a companion paper to give defining data for a large class of noncommutative projective surfaces.

Arakelov theory of noncommutative arithmetic surfaces

The purpose of the present article is to initiate Arakelov theory of noncommutative arithmetic surfaces. Roughly speaking, a noncommutative arithmetic surface is a noncommutative projective scheme of cohomological dimension 1 of finite type over Spec ℤ. An important example is the category of coherent right 𝒪-modules, where 𝒪 is a coherent sheaf of 𝒪 X -algebras and 𝒪 X is the structure sheaf of a commutative arithmetic surface X . Since smooth hermitian metrics are not available in our noncommutative setting, we have to adapt the definition of arithmetic vector bundles on noncommutative arithmetic surfaces. Namely, we consider pairs (ℰ, β ) consisting of a coherent sheaf ℰ and an automorphism β of the real sheaf ℰ ℝ induced by ℰ. We define the intersection of two such objects using the determinant of the cohomology and prove a Riemann-Roch theorem for arithmetic line bundles on noncommutative arithmetic surfaces.

A noncommutative version of Beilinson's theorem

We prove that the category of representations of the N-Kronecker quiver is derived equivalent to the category qgr ( k 〈 X 1 , … , X N 〉 / ( ∑ i = 1 N X i 2 ) ) which is considered as the category of coherent sheaves on the noncommutative projective scheme associated to k 〈 X 1 , … , X N 〉 / ( ∑ i = 1 N X i 2 ) . This theorem is easily proved by applying Orlov's theorem. On the other hand, in our proof, a method of noncommutative projective geometry is used, and the quadratic relation ∑ i = 1 N X i 2 naturally arises from Auslander–Reiten theory.

Finite Cohen–Macaulay Type and Smooth Non-Commutative Schemes

AbstractA commutative local Cohen–Macaulay ring R of finite Cohen–Macaulay type is known to be an isolated singularity; that is, Spec(R) \ ﹛m﹜ is smooth. This paper proves a non-commutative analogue. Namely, if A is a (non-commutative) graded Artin–Schelter Cohen–Macaulay algebra which is fully bounded Noetherian and has finite Cohen–Macaulay type, then the non-commutative projective scheme determined by A is smooth.

Finite Cohen--Macaulay Type and Smooth Non-Commutative Schemes

A commutative local Cohen-Macaulay ring R of finite Cohen-Macaulay type is known to be an isolated singularity; that is, Spec(R)-m is smooth. This paper proves a non-commutative analogue. Namely, if A is a (non-commutative) graded AS Cohen-Macaulay algebra which is FBN and has finite Cohen-Macaulay type, then the non-commutative projective scheme determined by A is smooth.

Idealizer rings and noncommutative projective geometry

We study noetherian graded idealizer rings which have very different behavior on the right and left sides. In particular, we construct noetherian graded algebras T over an algebraically closed field k with the following properties: T is left but not right strongly noetherian; T⊗ k T is left but not right noetherian and T⊗ k T op is noetherian; the left noncommutative projective scheme T-Proj is different from the right noncommutative projective scheme Proj- T; and T satisfies left χ d for some d⩾2 yet fails right χ 1.