1,137 research outputs found
Projectively simple rings
We introduce the notion of a projectively simple ring, which is an
infinite-dimensional graded k-algebra A such that every 2-sided ideal has
finite codimension in A (over the base field k). Under some (relatively mild)
additional assumptions on A, we reduce the problem of classifying such rings
(in the sense explained in the paper) to the following geometric question,
which we believe to be of independent interest.
Let X is a smooth irreducible projective variety. An automorphism f: X -> X
is called wild if it X has no proper f-invariant subvarieties. We conjecture
that if X admits a wild automorphism then X is an abelian variety. We prove
several results in support of this conjecture; in particular, we show that the
conjecture is true if X is a curve or a surface. In the case where X is an
abelian variety, we describe all wild automorphisms of X.
In the last two sections we show that if A is projectively simple and admits
a balanced dualizing complex, then Proj(A) is Cohen-Macaulay and Gorenstein.Comment: Some new material has been added in Section 1; to appear in Advances
in Mathematic
Noncommutative Blowups of Elliptic Algebras
We develop a ring-theoretic approach for blowing up many noncommutative
projective surfaces. Let T be an elliptic algebra (meaning that, for some
central element g of degree 1, T/gT is a twisted homogeneous coordinate ring of
an elliptic curve E at an infinite order automorphism). Given an effective
divisor d on E whose degree is not too big, we construct a blowup T(d) of T at
d and show that it is also an elliptic algebra. Consequently it has many good
properties: for example, it is strongly noetherian, Auslander-Gorenstein, and
has a balanced dualizing complex. We also show that the ideal structure of T(d)
is quite rigid. Our results generalise those of the first author. In the
companion paper "Classifying Orders in the Sklyanin Algebra", we apply our
results to classify orders in (a Veronese subalgebra of) a generic cubic or
quadratic Sklyanin algebra.Comment: 39 pages. Minor changes from previous version. The final publication
is available from Springer via http://dx.doi.org/10.1007/s10468-014-9506-
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() = 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
Naive Noncommutative Blowing Up
Let B(X,L,s) be the twisted homogeneous coordinate ring of an irreducible
variety X over an algebraically closed field k with dim X > 1. Assume that c in
X and s 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,L,s) with
surprising properties. In particular:
(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 (1-1) correspondence with the closed points of X.
However, both in qgr(R) and in gr(R), the R-point modules are not parametrized
by a projective scheme.
(3) qgr R has finite cohomological dimension yet H^1(R) is infinite
dimensional.
This gives a more geometric approach to results of the second author who
proved similar results for X=P^n by algebraic methods.Comment: Latex, 42 page
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.Comment: 39 pages; v2 results largely unchanged, but notation describing
algebras revised significantly. As a result details of many proofs have
changed, and statements of some results. To appear in Journal of Algebr
Algebras in which every subalgebra is noetherian
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.Comment: 5 pages; comments welcome; v2 only minor changes, most suggested by
refere
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