AN EXCELLENT F-PURE RING OF PRIME CHARACTERISTIC HAS A BIG TIGHT CLOSURE TEST ELEMENT

作者：Sharp Rodney Y^*

来源：Transactions of the American Mathematical Society, 2010, 362(10): 5455-5481.

DOI：10.1090/S0002-9947-10-05166-4

• Commutative Noetherian ring
• prime characteristic
• Frobenius homomorphism
• tight closure
• test element
• F-pure ring
• Frobenius skew polynomial ring

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 prune 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

• 出版日期2010-10