##
**On the finiteness theorem of the \(p\)-adic cohomology of a non-singular affine variety.
(Sur le théorème de finitude de la cohomologie \(p\)-adique d’une variété affine non singulière.)**
*(French)*
Zbl 0926.14007

The aim of this basic paper is to prove finiteness of Monsky-Washnitzer cohomology for non singular affine varieties over a finite field \( k \) of characteristic \( p\). The proof relies on Dwork’s exponential modules [B. Dwork, “A deformation theory for singular hypersurfaces” in: Algebr. Geom., Bombay Colloq. 1968, 85-92 (1969; Zbl 0218.14014)] and runs along the method Monsky used for non singular varieties over a field of characteristic zero [P. Monsky, “Finiteness of De Rham cohomology”, Am. J. Math. 94, 237-245 (1972; Zbl 0241.14010)]. By means of De Jong alterations, Berthelot gave an another proof of the finiteness theorem [P. Berthelot, Invent. Math. 128, No. 2, 329-377 (1997; Zbl 0908.14005)] ; the method used here is much more explicit and then, unlike Berthelot’s one, seems adaptable for a possible extension to non constant coefficients.

The main idea is to reduce the general situation to the one dimensional case. Namely the general finiteness theorem is deduced from the existence of an index for a class of ordinary \( p\)-adic differential equations \( M_{f,n,m} \) where \( f \) is a polynomial in \( n \) variables whose coefficients are Teichmüller representatives (in \( {\mathbb C}_{p}\)) of elements of \( k \) and \( m \) is an integer prime to \( p \) and bigger than the total degree of \( f\). These differential equations are shown to have only two singularities: a regular one at infinity and an irregular one in zero. Moreover, because of their geometric origin, they are endowed with a Frobenius structure hence are known to have an index [G. Christol and Z. Mebkhout, “Sur le théorème de l’indice des équations différentielles \( p\)-adiques”. III, Ann. Math. (to appear)]. Actually the \( M_{f,n,m} \) build up a large set of examples and counter-examples of \( p\)-adic differential equations.

Each step in the reducing process needs auxiliary results. Most of them are interesting by themselves. So are the Bertini theorem in characteristic \( p\), the \( p\)-adic Gysin exact sequence for all codimensions, the “magic bound” (whose proof is a beautiful application of the Fourier transform) and the basic comparison theorem between algebraic and analytic cohomologies for exponential modules. Hence the various sides of the article justify more than enough the effort needed by its numerous notations and involved computations.

The main idea is to reduce the general situation to the one dimensional case. Namely the general finiteness theorem is deduced from the existence of an index for a class of ordinary \( p\)-adic differential equations \( M_{f,n,m} \) where \( f \) is a polynomial in \( n \) variables whose coefficients are Teichmüller representatives (in \( {\mathbb C}_{p}\)) of elements of \( k \) and \( m \) is an integer prime to \( p \) and bigger than the total degree of \( f\). These differential equations are shown to have only two singularities: a regular one at infinity and an irregular one in zero. Moreover, because of their geometric origin, they are endowed with a Frobenius structure hence are known to have an index [G. Christol and Z. Mebkhout, “Sur le théorème de l’indice des équations différentielles \( p\)-adiques”. III, Ann. Math. (to appear)]. Actually the \( M_{f,n,m} \) build up a large set of examples and counter-examples of \( p\)-adic differential equations.

Each step in the reducing process needs auxiliary results. Most of them are interesting by themselves. So are the Bertini theorem in characteristic \( p\), the \( p\)-adic Gysin exact sequence for all codimensions, the “magic bound” (whose proof is a beautiful application of the Fourier transform) and the basic comparison theorem between algebraic and analytic cohomologies for exponential modules. Hence the various sides of the article justify more than enough the effort needed by its numerous notations and involved computations.

Reviewer: G.Christol (Paris)

### MSC:

14F30 | \(p\)-adic cohomology, crystalline cohomology |

14F17 | Vanishing theorems in algebraic geometry |

12H25 | \(p\)-adic differential equations |