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I'm looking for references for the various characterizations of regular holonomic D-modules, in particular proofs of their equivalence. For instance, some characterizations I've seen (in the analytic category) are:

  1. $\chi(M_x,\mathcal O_x) =\chi(M_x,\mathcal {\hat O}_x)$ for all $x$ source for notation (Though they give a source for this, I was unable to find it there)
  2. $R\operatorname{Hom}_D(M_x, \mathcal O_x) \cong R\operatorname{Hom}_D(M_x,\hat {\mathcal O}_x)$ via the natural map for all $x$
  3. A characterization having to do with V-filtrations near smooth points of subvarieties
  4. The D-module inverse image along any morphism from a smooth curve is regular in the sense of Fuchs
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  • $\begingroup$ I think that the equivalence of 1,2, and 3, for D-modules on $\mathbb{A}^1$, is sketched in the first chapter of "D-modules coherents et holonomes" by Maisonobe and Sabbah. $\endgroup$ May 2, 2015 at 23:35
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    $\begingroup$ Have you looked in Bjork's book? I think most of this stuff is there. $\endgroup$ May 3, 2015 at 1:41
  • $\begingroup$ @SamGunningham Dang. It seems my library doesn't have Björk, and I can't afford my own copy at the moment. Well, I guess I'll be saving up my money. From what's available on Google books, it looks like what I'm looking for. $\endgroup$ May 3, 2015 at 4:38
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    $\begingroup$ Two suggestions: (1) Interlibrary loan, or (2) ask the obvious person if you can borrow his copy. $\endgroup$ May 3, 2015 at 6:25

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