Idealizer Rings and Noncommutative Projective Geometry

We study some properties of graded idealizer rings with an emphasis on applications to the theory of noncommutative projective geometry. In particular we give examples of rings for which the $\chi$-conditions of Artin and Zhang and the strong noetherian property have very different behavior on the left and right sides.Comment: 17 Pages, revised version: significant changes--introduction rewritten, new section on tensor products added, main theorem restated at en

Generic Noncommutative Surfaces

We study a class of noncommutative surfaces and their higher dimensional analogues which provide answers to several open questions in noncommutative projective geometry. Specifically, we give the first known graded algebras which are noetherian but not strongly noetherian, answering a question of Artin, Small, and Zhang. In addition, these examples are maximal orders and satisfy the $\chi_1$ condition but not $\chi_i$ for $i \geq 2$, answering a questions of Stafford and Zhang and a question of Stafford and Van den Bergh. Finally, we show that these algebras have finite cohomological dimension.Comment: 43 pages, Latex, to appear in Advances in Math. Result on finite global dimension added, other minor change

Artin-Schelter Regular Algebras

Artin-Schelter regular algebras can be thought of as noncommutative versions of commutative polynomial rings, modeled after the special homological properties polynomial rings have as graded rings. First defined by Artin and Schelter in 1987, their introduction formed the beginning of the subject of noncommutative projective geometry. Artin-Schelter regular algebras have continued to play a large role in that subject, since geometrically they represent noncommutative (weighted) projective spaces. This is a survey of Artin-Schelter regular algebras, based on a talk at the 2022 meeting "Recent Advances and New Directions in the Interplay of Noncommutative Algebra and Geometry" at the University of Washington, in honor of the 65th birthday of S. Paul Smith. We review the earliest foundational results in the subject, and then describe some of the major themes of the last 30 years of research.Comment: 48 page

Skew Calabi-Yau triangulated categories and Frobenius Ext-algebras

We investigate the conditions that are sufficient to make the Ext-algebra of an object in a (triangulated) category into a Frobenius algebra and compute the corresponding Nakayama automorphism. As an application, we prove the conjecture that hdet($\mu_A$) = 1 for any noetherian Artin-Schelter regular (hence skew Calabi-Yau) algebra A.Comment: 31 page

Skew Calabi-Yau Algebras and Homological Identities

A skew Calabi-Yau algebra is a generalization of a Calabi-Yau algebra which allows for a non-trivial Nakayama automorphism. We prove three homological identities about the Nakayama automorphism and give several applications. The identities we prove show (i) how the Nakayama automorphism of a smash product algebra A # H is related to the Nakayama automorphisms of a graded skew Calabi-Yau algebra A and a finite-dimensional Hopf algebra H that acts on it; (ii) how the Nakayama automorphism of a graded twist of A is related to the Nakayama automorphism of A; and (iii) that Nakayama automorphism of a skew Calabi-Yau algebra A has trivial homological determinant in case A is noetherian, connected graded, and Koszul.Comment: 39 pages; minor changes, mostly in the Introductio

Growth of Graded Twisted Calabi-Yau Algebras

We initiate a study of the growth and matrix-valued Hilbert series of non-negatively graded twisted Calabi-Yau algebras that are homomorphic images of path algebras of weighted quivers, generalizing techniques previously used to investigate Artin-Schelter regular algebras and graded Calabi-Yau algebras. Several results are proved without imposing any assumptions on the degrees of generators or relations of the algebras. We give particular attention to twisted Calabi-Yau algebras of dimension d at most 3, giving precise descriptions of their matrix-valued Hilbert series and partial results describing which underlying quivers yield algebras of finite GK-dimension. For d = 2, we show that these are algebras with mesh relations. For d = 3, we show that the resulting algebras are a kind of derivation-quotient algebra arising from an element that is similar to a twisted superpotential.Comment: 49 page

Graded twisted Calabi-Yau algebras are generalized Artin-Schelter regular

This is a general study of twisted Calabi-Yau algebras that are $\mathbb{N}$-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 graded 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.Comment: 54 pages. Title has been changed (formerly titled "A twisted Calabi-Yau toolkit"). Revisions to the writing throughou