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Bjoerk J.E. - Rings of differential operators

Название: Rings of differential operators
Автор: Bjoerk J.E.
Категория: Математика
Тип: Книга
Дата: 04.01.2009 12:04:36
Скачано: 34
Описание: The results in Sections 1-6 are contained in I. N. Bernstein's work [1]. The Weyl algebra An(K) is introduced in Section 1 and it is defined as the algebra of K-linear differential operators on the polynomial ring K[Xi ■ ■ ■ xn]. Here n is some positive integer. An{K) is a non-commutative ring but we can consider the Bernstein filtration У = {^„}t?= о where &v = £ Kx* 6»: | a | + | /31 *S v. See Section 2 for the details. By definition each 5~„ is a finite dimensional vector space over К and the direct sum ,T0 ф ^~x /&~0 ф • • • has a natural structure as a ring which we denote by gr{An(K)) and in Proposition 2.2 we prove that gr(A„(K)) is a polynomial ring in In variables with coefficients in K. This is used to measure the size of finitely generated left or right /4„(K)-modules. In the final part of Section 2 we define good filiations {Г„} on finitely generated /l„(K)-modules M and construct the associated gr(/ln(K))-modules grr(M). In Section 3 we have recalled a famous result, due to D. Hilbert, which in the present situation shows that if {Г„} is a good filtration on a finitely generated A„(K)-mod\ile M, then the integer-valued function dimK(ry) = ad vd -I----+ a0 for all sufficiently large v. The polynomial ad ud + • • ■ + a0 is called a Hilbert polynomial. Of course, a finitely generated /l„(K)-module M can be equipped with different good nitrations but we can prove (see Corollary 3.5) that if {Г„} and {Qv} are two good nitrations on M, then there exists an integer w such that Г„ с Qv+W and Qv с Г„ + „ for all v. This implies easily that the Hilbert polynomials attached to Г and Q have the same degree and the same leading coefficient ad, where (rf!)^ is a positive integer which we denote by e(M). In this way we arrive at the two invariants d = d(M) and e(M). d(M) is called the Bernstein dimension of M and e(M) its multiplicity. Proposition 3.6 shows that these two invariants satisfy equations in exact sequences. In particular we prove that if 0 -* M, -* M2 -+ M3 -* 0 is an exact sequence of finitely generated /4„(K)-modules and if d(Mj) = d(M2) = d(M3), then e(M2) = e(M,) + e(M3). This addition formula for the multiplicity is an important tool for the proofs in Sections 4 and 5. Theorem 4.1 is the crucial result in this chapter. It asserts that d{M) > n for every non-zero and finitely generated /l„(K)-module M. This is called Bernstein's inequality and suggests the definition of the Bernstein class @t„ which consists of all finitely generated 4„(K)-modules M satisfying d(M) = n. Using the addition formula for multiplicities we prove that if M e 39„, then M has a finite length as an /4„(K)-module in Proposition 5.3.
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