Journal of Pure and Applied Algebra 149 (2000) 205–212
Injective dimension of D-modules:
a characteristic-free approach
Department of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
Received 10 November 1997; received in revised form 28 September 1998
Communicated by C.A. Weibel
We give a characteristic-free proof of the fact that if A is a ring of formal power series
in a ÿnite number of variables over a ÿeld k and M is any module over the ring of k-linear
dierential operators of A, then in the category of A-modules, the injective dimension of M is
bounded above by the dimension of the support of M . This is applied to give a characteristic-free
proof of the same inequality between the injective dimension and the dimension of the support
for local cohomology modules H
(R) where R is any regular Noetherian ring containing a ÿeld
and I ⊂ R is any ideal. This result for local cohomology modules had been proven before in
characteristic 0 and characteristic p¿0 by two methods that were completely dierent from
MSC: 13C99; 13D45; 13E99; 13N10
Throughout this paper R denotes a commutative ring with 1. If M is an R-module,
M denotes the injective dimension of M in the category of R-modules and
M denotes the dimension of the support of M in Spec R.IfI ⊂ R is an
(M ) denotes the ith local cohomology module of M with support in I.
If k ⊂ R is a subring, D(R; k) denotes the ring of k-linear dierential operators of R.
By a D(R; k)-module we always mean a left D(R; k)-module.
This paper is devoted to proving the following theorem and corollaries.
E-mail address: email@example.com (G. Lyubeznik)
Supported by the National Science Foundation.
0022-4049/00/$ - see front matter
2000 Elsevier Science B.V. All rights reserved.