33 research outputs found
The Frobenius Structure of Local Cohomology
Given a local ring of positive prime characteristic there is a natural
Frobenius action on its local cohomology modules with support at its maximal
ideal. In this paper we study the local rings for which the local cohomology
modules have only finitely many submodules invariant under the Frobenius
action. In particular we prove that F-pure Gorenstein local rings as well as
the face ring of a finite simplicial complex localized or completed at its
homogeneous maximal ideal have this property. We also introduce the notion of
an anti-nilpotent Frobenius action on an Artinian module over a local ring and
use it to study those rings for which the lattice of submodules of the local
cohomology that are invariant under Frobenius satisfies the Ascending Chain
Condition.Comment: 35 pages. Section 3 was revised to emphasize Theorem 3.1, and some
minor corrections/changes were performed. To appear in Algebra and Number
Theor
Comparison of symbolic and ordinary powers of ideals
In this paper we generalize the theorem of Ein-Lazarsfeld-Smith (concerning
the behavior of symbolic powers of prime ideals in regular rings finitely
generated over a field of characteristic 0) to arbitrary regular rings
containing a field. The basic theorem states that in such rings, if P is a
prime ideal of height c, then for all n, the symbolic (cn)th power of P is
contained in the nth power of P. Results are also given in the non-regular
case: one must correct by a power of the Jacobian ideal in rings where the
Jacobian ideal is defined
Ideals Generated by Quadratic Polynomials
Let be a polynomial ring in variables over an arbitrary field and
let be an ideal of generated by polynomials of degree at most 2. We
show that there is a bound on the projective dimension of that depends
only on , and not on . The proof depends on showing that if is
infinite and is a positive integer, there exists a positive integer C(n),
independent of , such that any forms of degree at most 2 in are
contained in a subring of generated over by at most forms
of degree 1 or 2 such that is a
regular sequence in . C(n) is asymptotic to