# On the definition of regularity

In the litterature on D-modules, there are many definitions of regularity of holonomic D-modules.

(1) Bernstein first defines regularity on a curve then says a holonomic D-module is regular if its restriction to any curve is regular

(2) Mebkhout defines the irregularity complexes of a complex of D-modules along an hypersurface. The complex is then regular if its irregularity complexes are 0 along any hypersurface.

(3) Kashiwara defines a D-module as regular if it admits a good filtration $F_*M$ such that $Ann(Gr^F M)$ is a radical ideal of $Gr^F D_X = \pi_*O_{T^*X}$.

I think there are other definitions (in Deligne for example)

Where can I find proofs that all these definitions are equivalent? Thanks.

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Another definition, I believe from Björk (though I can't check it, because the author came by my office and "borrowed" the book "to check a few things" four months ago, and hasn't given it back), is that a complex of analytic holonomic $D_X$-modules $M^\bullet$ is regular if $RHom_{D_{X,x}}(M^\bullet_x,\widehat{O}_{X,x}/O_{X,x})=0$ for every $x\in X$; for the algebraic case one takes the direct image into a good compactification and checks the analytic localisation. –  Ketil Tveiten Jun 28 '12 at 8:54
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## 1 Answer

There are some comparison results in Chapter 5 of Bjork's `Analytic D-modules and Applications'. Also see Chapter 8. In particular, I think Thm. 8.7.3 combined with Thm. 5.6.5 (almost) gives (1) iff (3). Further, I think Prop. 5.6.22 gives the equivalence with (2). There are also results in there comparing Deligne's description.

I must admit though that I find Bjork quite notationally dense and am not particularly familiar with it, so I may be quite off with the references above. I am interested in this question also, so please comment/post if you find better references.

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