F-purity and rational singularity
F-purity and rational singularity
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F-纯粹性和有理奇点
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发表时间：
1983
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通讯作者：
R. Fedder
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作者：
R. Fedder
我们研究的奇异性是F-纯（分别为F-纯型）。一个特征为p的环R是F-纯的，如果对每个R-模M，0 M？R M？R是正合的，其中1 R表示通过Frobenius映射在R上导出的R-代数结构（如果r E R和s E 1 R，则rs = rPs in 1 R）。F-纯类型通过简化为characteristicp在特征0中定义。证明了当R = S/I是正则局部环S的商时，R在素理想Q上是F-纯的当且仅当（I[PI：I）？Q[P]。这里，J[P]表示理想{aP I a E J}。几个定理的结果从这个标准。如果f是具有权（r1，...，r）和原点处的孤立奇点：（1）17 1 ri> I蕴涵K[X 1，. X]/（f）在m =（X1，. X）。(2)1 n I ri < I意味着K [X_（1.Xn）]/（f）在m处不具有F-纯型。(3)1%= 1 ri = I仍然没有解决，但确实与数论家研究多年的问题有关。这个定理与关于有理奇点的已知结果相似。证明了对完备交理想的F-纯奇点的分类可归结为对超曲面的F-纯奇点的分类，并证明了K [X 1，.其中K是特征为P的理想域，是Zebrski开的.一个重要的猜想是：R/fR是F-纯（型）的，当R是Cohen-Macauley正规局部环时，R是F-纯（型）的.证明了Ext 1（1 R，R）= 0是一个充分条件，但不是必要条件。特征为p的局部环（R，m）是F-内射的，如果Frobenius映射诱导局部上同调模H'（R）H'（1 R）上的内射.构造了一个F-内射但不是F-纯的例子。由此构造了关于R/fR是F-纯的猜想的一个反例.然而，它不是一个域，更不正常。此外，它并没有导致一个反例的特征0版本的猜想。0.导论.设R是特征为p的环，R ′表示通过Frobenius映射F（r）= rP被看作R-模的环R。R是F-纯的，如果对每个R-模M，0 -R？M-4'R？M是精确的。F-纯类型的概念则通过将特征归约为特征p而在特征0中定义。F-纯环与不变量理论相联系，并出现在关于线性可约仿射线性群作用于正则环的不变量环是Cohen-Macaulay的证明中[3]。它也被证明，F-纯度措施良好的奇异性，在这个意义上，它意味着大量的简化计算的局部上同调[1]。编辑于1982年3月30日收到，修订版于1982年8月3日收到。1980年数学学科分类第13 H99 ? 1983年美国数学学会0002-9947/82/0000-0829/$05.25
We investigate singularities which are F-pure (respectively F-pure type). A ring R of characteristic p is F-pure if for every R-module M, 0 M ? R M ? 'R is exact where 1R denotes the R-algebra structure induced on R via the Frobenius map (if r E R and s E 1R, then r s = rPs in 1R). F-pure type is defined in characteristic 0 by reducing to characteristicp. It is proven that when R = S/I is the quotient of a regular local ring S, R is F-pure at the prime ideal Q if and only if (I[PI: I) ? Q[P]. Here, J[P] denotes the ideal {aP I a E J}. Several theorems result from this criterion. If f is a quasihomogeneous hypersurface having weights (r1,...,r) and an isolated singularity at the origin: (1) 17 1ri > I implies K[X1,. X]/(f) has F-pure type at m = (X1,. X). (2) 1n I ri < I implies K [ X1 .Xn ]/(f) does not have F-pure type at m. (3) 1%= 1ri = I remains unsolved, but does connect with a problem that number theorists have studied for many years. This theorem parallels known results about rational singularities. It is also proven that classifying F-pure singularities for complete intersection ideals can be reduced to classifying such singularities for hypersurfaces, and that the F-pure locus in the maximal spectrum of K [ X1, . Xn ]/I, where K is a perfect field of characteristic P, is Zariski open. An important conjecture is that R/fR is F-pure (type) should imply R is F-pure (type) whenever R is a Cohen-Macauley, normal local ring. It is proven that Ext1( 1R, R) = 0 is a sufficient, though not necessary, condition. A local ring (R, m) of characteristic p is F-injective if the Frobenius map induces an injection on the local cohomology modules H' (R) H' (1R). An example is constructed which is F-injective but not F-pure. From this a counterexample to the conjecture that R/fR is F-pure implies R is F-pure is constructed. However, it is not a domain, much less normal. Moreover, it does not lead to a counterexample to the characteristic 0 version of the conjecture. 0. Introduction. Let R be a ring of characteristic p and let 'R denote the ring R viewed as an R-module via the Frobenius map F(r) = rP. R is F-pure if for every R-module M, 0 -R ? M -4'R ? M is exact. A notion of F-pure type is then defined in characteristic 0 by reduction to characteristicp. F-pure rings are connected with invariant theory and appear in the proof that the ring of invariants of a linearly reductive affine linear group acting on a regular ring is Cohen-Macaulay [3]. It has also been demonstrated that F-purity measures good singularities in the sense that it implies a great deal of simplification in the computation of local cohomology [1]. Received by the editors March 30, 1982 and, in revised form, August 3, 1982. 1980 Mathematics Subject Classification. Primary 13H99. ?1983 American Mathematical Society 0002-9947/82/0000-0829/$05.25