Mathematical Analysis

Download Analytic D-Modules and Applications by Jan-Erik Björk PDF

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DRx(M) is a quasi-isomorphism over Xa . 5 is proved we establish a result about the de Rham complex of the filtered Vx-module M. Let us put :FE = r2j; 0 0 M v + p . Then {:F:} form an increasing sequence of subcomplexes of DRx(M). 6 Lemma. Let Xa

7) preserve coherence and yield an equivalence of categories between coh('Dx) and coh('D);-). From now on we mostly discuss left 'Dx-modules and remark only that similar results hold for right 'Dx-modules. We have the positive filtration {'Dx(v)} and begin to discuss filtered 'Dx-modules. 1 Definition. A good filtration on a coherent left 'Dx-module M consists of {M v} such that every Xo E X has an open neighborhood U where Mv = 'Dx(v - kdmi + ... ·. , ms C M(U) and k l , ... , ks are integers.

Put L(M) = 'Homcx(M , M) If cp E L(M) and f E Ox we get fcp E L(M) defined by m L(M) is an Ox-Module. Now we consider the sheaf 'Homo (ex , L(M)) f-> f· cp(m). Hence THE SHEAF 'Dx AND ITS MODULES 19 where the index X in Ox is dropped to simplify the notations. 10 Definition. A global section \7 of'Homo(e x , L(M)) is called an integrable connection on M if the following hold: (1) \76(fm)=8(f)·m+I·\76(m) (2) \7[6,6'1 = [\76, \76'] where [\7 5, \7 5'] is the commutator in the C-algebra L(M). 11 The category A.

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