Dimension, Multiplicity, Holonomic Modules, and an Analogue of the Inequality of Bernstein for Rings of Differential Operators in Prime Characteristic
نویسنده
چکیده
Let K be an arbitrary field of characteristic p > 0 and D(Pn) the ring of differential operators on a polynomial algebra Pn in n variables. A long anticipated analogue of the inequality of Bernstein is proved for the ring D(Pn). In fact, three different proofs are given of this inequality (two of which are essentially characteristic free): the first one is based on the concept of the filter dimension, the second, on the concept of a set of holonomic subalgebras with multiplicity, and the third works only for finitely presented modules and follows from a description of these modules (obtained in the paper). On the way, analogues of the concepts of (Gelfand-Kirillov) dimension, multiplicity, holonomic modules are found in prime characteristic (giving answers to old questions of how to find such analogs). The idea is very simple to find characteristic free generalizations (and proofs) which in characteristic zero give known results, and in prime characteristic, generalizations. An analogue of Quillen’s Lemma is proved for simple finitely presented D(Pn)-modules. Moreover, for each such module L, EndD(Pn)(L) is a finite separable field extension of K and dimK(EndD(Pn)(L)) is equal to the multiplicity e(L) of L. In contrast to the characteristic zero case where the Gelfand-Kirillov dimension of a nonzero finitely generated D(Pn)-module M can be any natural number from the interval [n, 2n], in the prime characteristic, the (new) dimension Dim(M) can be any real number from the interval [n, 2n]. It is proved that every holonomic module has finite length, but in contrast to the characteristic zero case it is not true that neither a nonzero finitely generated module of dimension n is holonomic nor that a holonomic module is finitely presented. Some of the surprising results are: (i) each simple finitely presented D(Pn)-module M is holonomic having the multiplicity which is a natural number (in characteristic zero rather the opposite is true, i.e. GK (M) = 2n − 1, as a rule), (ii) the dimension Dim(M) of a nonzero finitely presented D(Pn)-module M can be any natural number from the interval [n, 2n], (iii) the multiplicity e(M) exists for each finitely presented D(Pn)-module M and e(M) ∈ Q, the multiplicity e(M) is a natural number if Dim(M) = n, and can be an arbitrarily small rational number if Dim(M) > n. Received by the editors February 27, 2008. 2000 Mathematics Subject Classification. Primary 13N10, 16S32, 16P90, 16D30, 16W70. c ©2009 American Mathematical Society
منابع مشابه
analogue of the inequality of Bernstein for rings of differential operators in prime characteristic
Let K be an arbitrary field of characteristic p > 0 and D(Pn) be the ring of differential operators on a polynomial algebra Pn in n variables. A long anticipated analogue of the inequality of Bernstein is proved for the ring D(Pn). In fact, three different proofs are given of this inequality (two of which are essentially characteristic free): the first one is based on the concept of the filter ...
متن کاملGENERALIZED PRINCIPAL IDEAL THEOREM FOR MODULES
The Generalized Principal Ideal Theorem is one of the cornerstones of dimension theory for Noetherian rings. For an R-module M, we identify certain submodules of M that play a role analogous to that of prime ideals in the ring R. Using this definition, we extend the Generalized Principal Ideal Theorem to modules.
متن کاملAN INTEGRAL DEPENDENCE IN MODULES OVER COMMUTATIVE RINGS
In this paper, we give a generalization of the integral dependence from rings to modules. We study the stability of the integral closure with respect to various module theoretic constructions. Moreover, we introduce the notion of integral extension of a module and prove the Lying over, Going up and Going down theorems for modules.
متن کاملCohen-macaulay Modules and Holonomic Modules over Filtered Rings
We study Gorenstein dimension and grade of a module M over a filtered ring whose assosiated graded ring is a commutative Noetherian ring. An equality or an inequality between these invariants of a filtered module and its associated graded module is the most valuable property for an investigation of filtered rings. We prove an inequality G-dimM ≤ G-dimgrM and an equality gradeM = grade grM , whe...
متن کاملResults on Generalization of Burch’s Inequality and the Depth of Rees Algebra and Associated Graded Rings of an Ideal with Respect to a Cohen-Macaulay Module
Let be a local Cohen-Macaulay ring with infinite residue field, an Cohen - Macaulay module and an ideal of Consider and , respectively, the Rees Algebra and associated graded ring of , and denote by the analytic spread of Burch’s inequality says that and equality holds if is Cohen-Macaulay. Thus, in that case one can compute the depth of associated graded ring of as In this paper we ...
متن کامل