analogue of the inequality of Bernstein for rings of differential operators in prime characteristic

نویسنده

  • V. V. Bavula
چکیده

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 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 finding such analogs). An 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 the 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 Geland-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 neither that 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 arbitrary small rational number if Dim(M) > n.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Completeness results for metrized rings and lattices

The Boolean ring $B$ of measurable subsets of the unit interval, modulo sets of measure zero, has proper radical ideals (for example, ${0})$ that are closed under the natural metric, but has no prime ideal closed under that metric; hence closed radical ideals are not, in general, intersections of closed prime ideals. Moreover, $B$ is known to be complete in its metric. Togethe...

متن کامل

Dimension, Multiplicity, Holonomic Modules, and an 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) 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 dim...

متن کامل

D-modules over Rings with Finite F-representation Type

Smith and Van den Bergh [29] introduced the notion of finite F-representation type as a characteristic p analogue of the notion of finite representation type. In this paper, we prove two finiteness properties of rings with finite F-representation type. The first property states that if R = L n≥0 Rn is a Noetherian graded ring with finite (graded) F-representation type, then for every non-zerodi...

متن کامل

On the approximation by Chlodowsky type generalization of (p,q)-Bernstein operators

In the present article, we introduce Chlodowsky variant of $(p,q)$-Bernstein operators and compute the moments for these operators which are used in proving our main results. Further, we study some approximation properties of these new operators, which include the rate of convergence using usual modulus of continuity and also the rate of convergence when the function $f$ belongs to the class Li...

متن کامل

Some commutativity theorems for $*$-prime rings with $(sigma,tau)$-derivation

‎Let $R$ be a $*$-prime ring with center‎ ‎$Z(R)$‎, ‎$d$ a non-zero $(sigma,tau)$-derivation of $R$ with associated‎ ‎automorphisms $sigma$ and $tau$ of $R$‎, ‎such that $sigma$‎, ‎$tau$‎ ‎and $d$ commute with $'*'$‎. ‎Suppose that $U$ is an ideal of $R$ such that $U^*=U$‎, ‎and $C_{sigma,tau}={cin‎ ‎R~|~csigma(x)=tau(x)c~mbox{for~all}~xin R}.$ In the present paper‎, ‎it is shown that if charac...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2006