# Proof of Kashiwara's constructibility theorem for algebraic D-modules

I am trying to understand the proof of Kashiwara's constructibility theorem for algebraic D-modules, following either the book "D-modules, Perverse sheaves, and representation theory" by Hotta, Takeuchi and Tanisaki, or Bernstein's lecture notes. My question is about theorem 4.7.7 of the D-module book, or theorem C a) on page 39 of Bernstein's note. These theorems say that if $$M$$ is a holonomic D-module on a smooth algebraic variety $$X$$ over $$\mathbb{C}$$, then the de Rham complex $$DR_X(M)$$ has constructible cohomologies with respect to a stratification by algebraic subsets.

Let us look at the proof of theorem 4.7.7. The proof begins with choosing an open $$U\subset X$$ on which $$M$$ is an integrable connection, and let $$Z$$ be an irreducible component of $$X \setminus U$$. The aim is to show $$DR_X(M)$$ has locally constant cohomologies on some dense Zariski open subsets of $$Z$$. Now one can find subsets $$V \subset X$$ and $$V' \subset \mathbb{A}^n$$, and an étale morphism $$f: V \to V'$$ such that $$f^{-1}(V'\cap \mathbb{A}^{n-k}) = Z \cap V = (X \setminus U) \cap V$$. Here $$n$$ and $$n-k$$ are the dimensions of $$X$$ and $$Z$$ respectively.

The book proceeds by saying, $$DR_V(M\vert_V)\in D^b_c(V)$$ if and only if $$f_*(DR_V(M\vert_V))\in D^b_c(V')$$ because $$f$$ is étale, and $$f_*(DR_V(M\vert_V)) = DR_{V'} (\int_f^0(M\vert_V))$$ by proposition 4.7.5. I don't understand both assertions. Can anyone explain to me why the étale condition implies the first if and only if statement, and why $$DR$$ commutes with $$f_*$$ when $$f$$ is not a priori proper, and why do the authors write $$f_*$$ and $$\int_f^0$$ instead of $$Rf_*$$ and $$\int_f$$, indicating there is no higher cohomologies?

I understand that the purpose of these steps is to change to the easier situation $$X = Z \times \mathbb{P}^k$$ and $$Z = Z \times pt$$, as Bernstein's note says. But Bernstein's note provides even less details than the D-module book I am using, and simply says that by a base change from $$\mathbb{A}^k$$ to $$W$$ (where his $$\mathbb{A}^k$$ is our $$\mathbb{A}^{n-k}$$ and his $$W$$ is our $$Z$$) we can assume $$X = W \times \mathbb{P}$$ and so on.

Another question of mine is that, How does Bernstein's method work? Does he mean the same thing as authors of the D-module book, or something different?

I am sorry for such a lengthy question, but I can't find a good way to put less words. Thank you very much in ahead of time for reading these paragraphs!

• I believe you’ll find more detail in Borel’s papers in (1984, 1987). – Francois Ziegler Jul 9 at 11:39
• @FrancoisZiegler Thank you very much for directing me to Borel's books! I find the geometric construction explained in Borel 1987 chapter 8 section 19.5 and section 20.1 (page 324 to page 344) quite relevant to my question. The method used there seems to have modified and elaborated the succinct arguments given in Bernstein's note. But I will need more time to understand his whole arguments. – sesame Jul 9 at 16:44
• @user2520938 Thank you for your comment! I am not very good at commutative algebra. I checked appendix A.5, but cannot recognise why we can assume the morphism $f$ is finite? Can you explain to me in a bit more detail? – sesame Jul 9 at 16:50
• @sesame are you sure you need the properness? I mean, for a smooth morphism $f$ we have $\int_f(M)=Rf_*(DR_f \otimes M)$ with some induced $D_Y$-module structure. In the etale case $DR_f$ is trivial and we get just $Rf_*M$. Then it seams that the comparison between the two sides, term-wise in the DeRahm complex, is just the projection formula $Rf_*(f^*\Omega^j_Y\otimes M)\cong \Omega^j_X \otimes Rf_*(M)$ which is true since $\Omega_Y^j$ is locally free. – S. carmeli Jul 9 at 21:22
• oops, of course the RHS should have $\Omega_Y^j$ in it. – S. carmeli Jul 9 at 22:14