The purity of the Frobenius and local cohomology
The purity of the Frobenius and local cohomology
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DOI：
10.1016/0001-8708(76)90073-6
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发表时间：
1976-08
影响因子：
1.7
通讯作者：
M. Hochster;Joel Roberts
中科院分区：
文献类型：
--
作者：
M. Hochster;Joel Roberts
1_ INTR~ uUCTI~ NRZ环同态（所有环都是交换的，有单位元，同态保持单位元）R-fS称为纯的，如果对每个R-模M，M-fMRRS via m-+ m@1是内射的.非常一般地，如果I是分次环R的齐次理想且rad（1）= rad（（fo，...，f12）R），其中f0，...，fn是R的形式，则局部上同调模H，i（R）可以表示为Koszul上同调的直接极限：f3，yf-q= limH + Y.（f '; R））（包括符号在内的细节在第2节中给出），并且模H，'（R）获得了与形式f= f”，.，..我们的第一个目的（第2节）是证明在特征p> 0中，在一定条件下（包括Frobenius同态的纯性），局部上同调的计算可以惊人地简化，许多分次片消失（参见第2节）。定理1.1，下面的命题2.4）。在第3节中发展了关于局部上同调的一般行为的一些先决条件之后，我们继续证明（第4节），对于特征为0的某些环类，通过传递到特征p>，0，可以得到类似的结果（参见第4节）。命题4.7和定理4.8）。在第5节中，我们考虑一些适合我们的类的环族的例子。在某些情况下（见例5.30），隶属度取决于算术级数中是否存在无穷多个素数。在第5节中，还制定了这些类别的成员资格的各种标准。的点
1_ INTR~ uUCTI~ N rZ ring homomorphism (all rings are commutative, with identity, and homomorphisms preserve the identity) R--f S is called pure if for every R-module M, M---f M RR S via m-+ m@ 1 is injective. Quite generally, if I is a homogeneous ideal of a graded ring R and rad (1)= rad ((fo,..., fi2) R), where f0,..., fn are forms of R, then the local cohomology modules H, i (R) can be expressed as direct limits of Koszul cohomology: f3, yf-q= lim H+ Y.(f’; R))(details, including notation, are given in Section 2) and the modules H,‘(R) acquire a Z-grading independent of the choice of the forms f= f”,...,. f,,.Our first objective here (Section 2) is to show that in characteristic p> 0, under certain conditions (including the purity of the Frobenius homomorphism) on R, the computation of local cohomology can be amazingly simplified and many graded pieces vanish (cf. Theorem 1.1, Proposition 2.4 below). After developing some prerequisites concerning the generic behavior of local cohomology in Section 3, we proceed to show (Section 4) that for certain classes of rings in characteristic 0 one may obtain analogous results by passing to characteristic p>, 0 (cf. Proposition 4.7 and Theorem 4.8). In Section 5, we consider a number of examples of families of rings that fit into one of our classes. In some cases (see Example 5.30) membership depends on the existence of infinitely many primes in an arithmetic progression. Also in Section 5, various criteria for membership in these classes are developed. A point that