Regular ring

Abstract: 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

Abstract: Let G be a compact, locally L-analytic group, where L is a finite extension of Qp. Let K be a discretely valued extension field of L. We study the algebra D(G,K) of K-valued locally analytic distributions on G, and apply our results to the locally analytic representation theory of G in vector spaces over K. Our objective is to lay a useful and powerful foundation for the further study of such representations. We show that the noncommutative, nonnoetherian ring D(G,K) "behaves" like the ring of functions on a rigid Stein space, and that (at least when G is Qp-analytic) it is a faithfully flat extension of its subring K\otimes Zp[[G]], where Zp[[G]] is the completed group ring of G. We use this point of view to describe an abelian subcategory of D(G,K) modules that we call coadmissible. We say that a locally analytic representation V of G is admissible if its strong dual is coadmissible as D(G,K)-module. For noncompact G, we say V is admissible if its strong dual is coadmissible as D(H,K) module for some compact open subgroup H. In this way we obtain an abelian category of admissible locally analytic representations. These methods allow us to answer a number of questions raised in our earlier papers on p-adic representations; for example we show the existence of analytic vectors in the admissible Banach space representations of G that we studied in "Banach space representations ...", Israel J. Math. 127, 359-380 (2002). Finally we construct a dimension theory for D(G,K), which behaves for coadmissible modules like a regular ring, and show that smooth admissible representations are zero dimensional.

Abstract: All given rings in this paper are commutative, associative with identity, and Noetherian. Recently, L. Ein, R. Lazarsfeld, and K. Smith [ELS] discovered a remarkable and surprising fact about the behavior of symbolic powers of ideals in affine regular rings of equal characteristic 0: if h is the largest height of an associated prime of I , then I (hn) ⊆ I n for all n ≥ 0. Here, if W is the complement of the union of the associated primes of I , I (t) denotes the contraction of I t RW to R, where RW is the localization of R at the multiplicative system W . Their proof depends on the theory of multiplier ideals, including an asymptotic version, and, in particular, requires resolution of singularities as well as vanishing theorems. We want to acknowledge that without their generosity and quickness in sharing their research this manuscript would not exist. Our objective here is to give stronger results that can be proved by methods that are, in some ways, more elementary. Our results are valid in both equal characteristic 0 and in positive prime characteristic p, but depend on reduction to characteristic p. We use tight closure methods and, in consequence, we need neither resolution of singularities nor vanishing theorems that may fail in positive characteristic. For the most basic form of the result, all that we need from tight closure theory is the definition of tight closure and the fact that in a regular ring, every ideal is tightly closed. We note that the main argument here is closely related to a proof given in [Hu, 5.14–16, p. 45] that regular local rings in characteristic p are UFDs,

Abstract: Let G be a compact, locally L-analytic group, where L is a finite extension of Qp. Let K be a discretely valued extension field of L. We study the algebra D(G,K) of K-valued locally analytic distributions on G, and apply our results to the locally analytic representation theory of G in vector spaces over K. Our objective is to lay a useful and powerful foundation for the further study of such representations. We show that the noncommutative, nonnoetherian ring D(G,K) "behaves" like the ring of functions on a rigid Stein space, and that (at least when G is Qp-analytic) it is a faithfully flat extension of its subring K\otimes Zp[[G]], where Zp[[G]] is the completed group ring of G. We use this point of view to describe an abelian subcategory of D(G,K) modules that we call coadmissible. We say that a locally analytic representation V of G is admissible if its strong dual is coadmissible as D(G,K)-module. For noncompact G, we say V is admissible if its strong dual is coadmissible as D(H,K) module for some compact open subgroup H. In this way we obtain an abelian category of admissible locally analytic representations. These methods allow us to answer a number of questions raised in our earlier papers on p-adic representations; for example we show the existence of analytic vectors in the admissible Banach space representations of G that we studied in "Banach space representations ...", Israel J. Math. 127, 359-380 (2002). Finally we construct a dimension theory for D(G,K), which behaves for coadmissible modules like a regular ring, and show that smooth admissible representations are zero dimensional.