arXiv subjects

Dennis S. Keeler

Publications and source records attributed to Dennis S. Keeler.

7 recordsLinked to original sources

arXiv2018

Ample filters of invertible sheaves

Let $X$ be a scheme, proper over a commutative noetherian ring $A$. We introduce the concept of an ample filter of invertible sheaves on $X$ and generalize the most important equivalent criteria for ampleness of an invertible sheaf. We also prove the Theorem of the Base for $X$ and generalize Serre's Vanishing Theorem. We then generalize results for twisted homogeneous coordinate rings which were previously known only when $X$ was projective over an algebraically closed field. Specifically, we show that the concepts of left and right $σ$-ampleness are equivalent and that the associated twisted homogeneous coordinate ring must be noetherian.

math.AG↗

arXiv2018

Ample filters and Frobenius amplitude

Let $X$ be a projective scheme over a field. We show that the vanishing cohomology of any sequence of coherent sheaves is closely related to vanishing under pullbacks by the Frobenius morphism. We also compare various definitions of ample locally free sheaf and show that the vanishing given by the Frobenius morphism is, in a certain sense, the strongest possible. Our work can be viewed as various generalizations of the Serre Vanishing Theorem.

math.AG↗

arXiv2018

Fujita's Conjecture and Frobenius amplitude

We prove a version of Fujita's Conjecture in arbitrary characteristic, generalizing results of K.E. Smith. Our methods use the Frobenius morphism, but avoid tight closure theory. We also obtain versions of Fujita's Conjecture for coherent sheaves with certain ampleness properties.

math.AG↗

arXiv2002

Noncommutative ampleness for multiple divisors

The twisted homogeneous coordinate ring is one of the basic constructions of the noncommutative projective geometry of Artin, Van den Bergh, and others. Chan generalized this construction to the multi-homogeneous case, using a concept of right ampleness for a finite collection of invertible sheaves and automorphisms of a projective scheme. From this he derives that certain multi-homogeneous rings, such as tensor products of twisted homogeneous coordinate rings, are right noetherian. We show that right and left ampleness are equivalent and that there is a simple criterion for such ampleness. Thus we find under natural hypotheses that multi-homogeneous coordinate rings are noetherian and have integer GK-dimension.

math.AG↗

arXiv2002

Frobenius amplitude and strong vanishing theorems for vector bundles

The primary goal of this paper is to systematically exploit the method of Deligne-Illusie to obtain Kodaira type vanishing theorems for vector bundles and more generally coherent sheaves on algebraic varieties. The key idea is to introduce a number which provides a cohomological measure of the positivity of a coherent sheaf called the Frobenius or F-amplitude. As the terminology indicates, the definition is most natural over a field of positive characteristic, however it can be forced into characteristic 0 by standard tricks. The F-amplitude enters into the statement of the basic vanishing theorem, and this leads to the problem of calculating, or at least estimating, this number. Most of the work in this paper is devoted to doing this various situations. The key result in this direction is a bound on the F-amplitude of an ample vector bundle. When combined with the vanishing theorem, this has some reasonably down to earth corollaries (both old and new).

math.AG↗

arXiv2002

The rings of noncommutative projective geometry

In the past 15 years a study of ``noncommutative projective geometry'' has flourished. By using and generalizing techniques of commutative projective geometry, one can study certain noncommutative graded rings and obtain results for which no purely algebraic proof is known. For instance, noncommutative graded domains of quadratic growth, or ``noncommutative curves,'' have now been classified by geometric data and these rings must be noetherian. Rings of cubic growth, or ``noncommutative surfaces,'' are not yet classified, but a rich theory is currently forming. In this survey, we describe some of these results and examine the question of which rings should be included in noncommutative projective geometry.

math.RA↗

arXiv2000

Criteria for σ-ampleness

In the noncommutative geometry of Artin, Van den Bergh, and others, the twisted homogeneous coordinate ring is one of the basic constructions. Such a ring is defined by a $σ$-ample divisor, where $σ$ is an automorphism of a projective scheme X. Many open questions regarding $σ$-ample divisors have remained. We derive a relatively simple necessary and sufficient condition for a divisor on X to be $σ$-ample. As a consequence, we show right and left $σ$-ampleness are equivalent and any associated noncommutative homogeneous coordinate ring must be noetherian and have finite, integral GK-dimension. We also characterize which automorphisms $σ$ yield a $σ$-ample divisor.

math.AG↗