Daniel Rogalski

Skew Calabi–Yau algebras and homological identities

Abstract 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 the Nakayama automorphism of a skew Calabi–Yau algebra A has trivial homological determinant in case A is noetherian, connected graded, and Koszul.

Naïve noncommutative blowing up

Let B(X,\mathscr{L},σ) be the twisted homogeneous coordinate ring of an irreducible variety X over an algebraically closed field k with dim X ≥ 2. Assume that c \in X and σ \in Aut(X) are in sufficiently general position. We show that if one follows the commutative prescription for blowing up X at c, but in this noncommutative setting, one obtains a noncommutative ring R = R(X,c,

Generic noncommutative surfaces

\mathscr{L}

A class of noncommutative projective surfaces

,σ) with surprising properties. (1) R is always Noetherian but never strongly Noetherian (2) If R is generated in degree one, then the images of the R-point modules in qgr-R are naturally in one-to-one correspondence with the closed points of X. However, in both qgr-R and gr-R, the R-point modules are not parametrized by a projective scheme. (3) While qgr-R has finite cohomological dimension dim_k H^1 ( \mathscr{O} ) = ∞.

Skew Calabi-Yau triangulated categories and Frobenius Ext-algebras

Abstract We study a class of noncommutative surfaces, and their higher dimensional analogs, which come from generic subalgebras of twisted homogeneous coordinate rings of projective space. Such rings provide answers to several open questions in noncommutative projective geometry. Specifically, these rings R are the first known graded algebras over a field k which are noetherian but not strongly noetherian: in other words, R ⊗ k B is not noetherian for some choice of commutative noetherian extension ring B . This answers a question of Artin, Small, and Zhang. The rings R are also maximal orders, but they do not satisfy all of the χ conditions of Artin and Zhang. In particular, they satisfy the χ 1 condition but not χ i for i ⩾2, answering a question of Stafford and Zhang and a question of Stafford and Van den Bergh. Finally, we show that the noncommutative scheme R -proj has finite global dimension.

Some projective surfaces of GK-dimension 4

Let A=k+A_1+A_2.... be a connected graded, noetherian k-algebra that is generated in degree one over an algebraically closed field k. Suppose that the graded quotient ring Q(A) has the form Q(A)=k(Y)[t,t^{-1},sigma], where sigma is an automorphism of the integral projective surface Y. Then we prove that A can be written as a naive blowup algebra of a projective surface X birational to Y. This enables one to obtain a deep understanding of the structure of these algebras; for example, generically they are not strongly noetherian and their point modules are not parametrized by a projective scheme. This is despite the fact that the simple objects in the quotient category qgr A will always be in (1-1) correspondence with the closed points of the scheme X.

GK-dimension of birationally commutative surfaces

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(

Noncommutative blowups of elliptic algebras

\mu_A

Classifying orders in the Sklyanin algebra

) = 1 for any noetherian Artin-Schelter regular (hence skew Calabi-Yau) algebra A.

Algebras in which every subalgebra is noetherian

We construct an interesting family of connected graded domains of Gel’fand–Kirillov dimension 4, and show that the general member of this family is noetherian. The algebras we construct are Koszul and have global dimension 4. They fail to be Artin–Schelter Gorenstein, however, showing that a theorem of Zhang and Stephenson for dimension 3 algebras does not extend to dimension 4. The Auslander–Buchsbaum formula also fails to hold for these algebras. The algebras we construct are birational to ℙ 2 , and their existence disproves a conjecture of the first author and Stafford. The algebras can be obtained as global sections of a certain quasicoherent graded sheaf on ℙ 1 ×ℙ 1 , and our key technique is to work with this sheaf. In contrast to all previously known examples of birationally commutative graded domains, the graded pieces of the sheaf fail to be ample in the sense of Van den Bergh. Our results thus require significantly new techniques.