Mathematics – Commutative Algebra
Scientific paper
2009-07-16
Mathematics
Commutative Algebra
This revised version corrects a small number of misprints and is to appear in the Transactions of the American Mathematical So
Scientific paper
In two recent papers, the author has developed a theory of graded annihilators of left modules over the Frobenius skew polynomial ring over a commutative Noetherian ring $R$ of prime characteristic $p$, and has shown that this theory is relevant to the theory of test elements in tight closure theory. One result of that work was that, if $R$ is local and the $R$-module structure on the injective envelope $E$ of the simple $R$-module can be extended to a structure as a torsion-free left module over the Frobenius skew polynomial ring, then $R$ is $F$-pure and has a tight closure test element. One of the central results of this paper is the converse, namely that, if $R$ is $F$-pure, then $E$ has a structure as a torsion-free left module over the Frobenius skew polynomial ring; a corollary is that every $F$-pure local ring of prime characteristic, even if it is not excellent, has a tight closure test element. These results are then used, along with embedding theorems for modules over the Frobenius skew polynomial ring, to show that every excellent (not necessarily local) $F$-pure ring of characteristic $p$ must have a so-called `big' test element.
No associations
LandOfFree
An excellent F-pure ring of prime characteristic has a big tight closure test element does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with An excellent F-pure ring of prime characteristic has a big tight closure test element, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and An excellent F-pure ring of prime characteristic has a big tight closure test element will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-622319