Generic bounds for Frobenius closure and tight closure

Details Generic bounds for Frobenius closure and tight closure Generic bounds for Frobenius closure and tight closure

2008-10-24

arxiv.org/abs/0810.4518v3

Mathematics

Commutative Algebra

23 pages, 1 figure. v2: Minor changes to layout and phrasing; v3: Added more detail, changed two references

Scientific paper

We use geometric and cohomological methods to show that given a degree bound for membership in ideals of a fixed degree type in the polynomial ring P=k[x_0,..., x_d], one obtains a good generic degree bound for membership in the tight closure of an ideal of that degree type in any standard-graded k-algebra R of dimension d+1. This indicates that the tight closure of an ideal behaves more uniformly than the ideal itself. Moreover, if R is normal, one obtains a generic bound for membership in the Frobenius closure. If d is at most 2, then the bound for ideal membership in P can be computed from the known cases of the Froeberg conjecture and yields explicit generic tight closure bounds.

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