arXiv 2103.064522021-03math.AC

测试元素、优环和内容功能

Test elements, excellent rings, and content functions

Neil Epstein

备注：: Propositions 2.6 and 4.6 are incorrect. Without them, I don't have proofs for the main theorems. I plan to incorporate many of the remaining valid statements into future joint work

摘要：

扩大现有的结果在文献中更广泛的类的环，我们证明：1。特征$p$的优良正则环的约化阶容许大的检验元;局部优良正则环中主理想的F-跳数集是$\mathbb Q$的离散子集，3.若$R$是一个局部优良正则素正则环的商，则$R$的剩余域的内射壳的Hartshorne-Speiser-Lyubeznik数存在一致上界.为此，我们发展了Ohm-Rush和交平坦代数的并行理论。证明了这两个性质在诺特环的平坦映射中都是局部可检验的。我们表明，相交平坦承认一个内容理论平行的Ohm-Rush内容的Ohm-Rush代数。我们开发这些属性的下降结果。利用交平坦性的下降结果，得到了诺特环的忠实平坦映射必交平坦的一个局部条件。交平坦的局部条件使我们可以得出结论：诺特环上的忠实平坦代数是交平坦的。将交平坦性的局部条件与Kunz和Radu的结果相结合，得到了与素特征的局部优良正则环相关联的Frobenius自同态是交平坦的结论，从而回答了Sharp的一个问题.后一个结果的应用包括上面列举的三个结果。我们也得到了紧闭包和参数测试理想的应用。

英文摘要：

Broadening existing results in the literature to much wider classes of rings, we prove among other things: 1. Reduced quotients of excellent regular rings of characteristic $p$ admit big test elements, 2. The set of F-jumping numbers of a principal ideal in a locally excellent regular ring is a discrete subset of $\mathbb Q$, and 3. If $R$ is a quotient of a locally excellent regular ring of prime characteristic, then there is a uniform upper bound on the Hartshorne-Speiser-Lyubeznik numbers of the injective hulls of the residue fields of $R$. To do so, we develop the parallel theories of Ohm-Rush and intersection flat algebras. We show that both properties can be checked locally in flat maps of Noetherian rings. We show that intersection-flatness admits a content theory parallel to that of Ohm-Rush content for Ohm-Rush algebras. We develop descent results for these properties. Using the descent result for intersection flatness, we obtain a local condition under which a faithfully flat map of Noetherian rings must be intersection-flat. The local condition for intersection-flatness allows us to conclude that finitely generated faithfully flat algebras over a Noetherian ring are intersection-flat. Combining the local condition for intersection flatness with results of Kunz and Radu yields the conclusion that the Frobenius endomorphism associated to a locally excellent regular ring of prime characteristic is intersection-flat, thus answering a question of Sharp. Applications of the latter result include the three enumerated results above. We also get applications to tight closure and parameter test ideals.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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