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We investigate conditions that are sufficient to make the Extalgebra 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(μA) = 1 for any noetherian Artin-Schelter regular (hence skew Calabi-Yau) algebra A.

Let S denote the 3-dimensional Sklyanin algebra over an algebraically closed field k and assume that S is not a finite module over its centre. (This algebra corresponds to a generic noncommutative ℙ2.) Let A=⊕i≥0 Ai be any connected graded k-algebra that is contained in and has the same quotient ring as a Veronese ring S(3n). Then we give a reasonably complete description of the structure of A. This is most satisfactory when A is a maximal order, in which case we prove, subject to a minor technical condition, that A is a noncommutative blowup of S(3n) at a (possibly noneffective) divisor on the associated elliptic curve E. It follows that A has surprisingly pleasant properties; for example, it is automatically noetherian, indeed strongly noetherian, and has a dualising complex.

We show that the twisted homogeneous coordinate rings of elliptic curves by infinite order automorphisms have the curious property that every subalgebra is both finitely generated and noetherian. As a consequence, we show that a localisation of a generic Skylanin algebra has the same property.

We study the existence of free subalgebras in division algebras, and prove the following general result: if A is a noetherian domain which is countably generated over an uncountable algebraically closed field k of characteristic 0, then either the quotient division algebra of A contains a free algebra on two generators, or it is left algebraic over every maximal subfield. As an application, we prove that if k is an uncountable algebraically closed field and A is a finitely generated k-algebra that is a domain of GK-dimension strictly less than 3, then either A satisfies a polynomial identity, or the quotient division algebra of A contains a free k-algebra on two generators. © 2014 Springer-Verlag Berlin Heidelberg.