Rigid cohomology.

*(English)*Zbl 1131.14001
Cambridge Tracts in Mathematics 172. Cambridge: Cambridge University Press (ISBN 978-0-521-87524-0/hbk). xv, 319 p. (2007).

This is the first textbook on the foundations of rigid cohomology as developed by P. Berthelot in the 1980’s and early 1990’s. Beside his articles [P. Berthelot, Mém Soc. Math. France (N.S.) 23, 7–32 (1986; Zbl 0606.14017); Introductions aux cohomologies \(p\)-adiques, Luminy (1984)], the commonon reference for the basic definitions and constructions was the inventor’s preprints [P. Berthelot, Cohomologie rigide et cohomologie rigide supports propres, Première partie (version provisoire 1991), Prépublication IRMAR 96–03, Université de Rennes (1996)], both in French language. In particular, the present book is the first text in English on this important subject.

Let \(k\) be a field of characteristic \(p>0\), and let \(V\) be a complete discrete valuation ring with residue field \(k\) and fraction field \(K\) of characteristic \(0\). Rigid cohomology is a cohomology theory for \(k\)-varieties with values in \(K\)-vector spaces. It unifies two older theories. One the one hand, the rigid cohomology (with constant coefficients, say) of a smooth and proper \(k\)-variety is canonically isomorphic to its crystalline cohomology (tensored with \(K\)). On the other hand, the rigid cohomology of a smooth and affine \(k\)-variety is canonically isomorphic to its Monsky Washnitzer cohomology. More generally, suppose now that for a separated \(k\)-scheme \(X\) of finite type we are given an embedding of \(X\) into the special fibre of a smooth proper formal \(V\)-scheme \({\mathfrak Y}\). An overconvergent isocrystal on \(X\) over \(K\) is a module with an integrable connection living on a ‘strict neighbourhood’ the tubular neighbourhood of \(X\) in \({\mathfrak Y}\), i.e. on a ’strict neighbourhood’ of the \(K\)-rigid analytic space which is the preimage of \(X\) under the specialization map from the generic fibre (as a \(K\)-rigid space) of \({\mathfrak Y}\) to its special fibre. If in addition we are given a lift of the Frobenius endomorphism of \(X\) to this module with connection one speaks of an overconvergent \(F\)-isocrystal. Berthelot showed that the categories of overconvergent isocrystals on \(X\) and of overconvergent \(F\)-isocrystals on \(X\) are independent of the choice of \({\mathfrak Y}\) (and the embedding of \(X\)). The category of overconvergent \(F\)-isocrystals should be viewed as the \(p\)-adic coefficient category analoguous to the categories of lisse \(\ell\)-adic sheaves (\(\ell\neq p\)). For example, for an overconvergent isocrystal \(E\) on \(X\) one can ask for its (absolute) cohomology groups \(H_{\text{rig}}^*(X,E)\) (and its (absolute) cohomology groups with compact supports \(H_{\text{rig},c}^*(X,E)\)). These are \(K\)-vector spaces, the (absolute) ‘rigid cohomology’ of \(E\). By now they are now known to be finite dimensional: Berthelot proved this in the case where \(E\) is the constant overconvergent isocrystal and \(X\) is smooth; the reviewer then proved it for general \(X\) (still with \(E\) constant); for general \(E\) the finiteness theorem is due to Kedlaya. Moreover, rigid cohomology satisfies all the properties required of a Bloch-Ogus cohomology theory. Its particular interest (against \(\ell\)-adic cohomology (\(\ell\neq p\))) lies in its potential to estimate the \(p\)-adic valuations of the zeroes and poles of Zeta-functions of \(k\)-varieties (if \(k\) is finite). Moreover, in recent years it turned out that rigid cohomology is in many circumstances more suitable (than \(\ell\)-adic cohomology (\(\ell\neq p\))) to actually compute these Zeta-functions, i.e. many effective algorithms based on rigid cohomology have been worked out. In one word, it is clear that rigid cohomology is an important subject in current arithmetic geometry.

The present book grew out of a course the author had given at Peking University in 2005. Its aim is to present all the basic objects and constructions needed to define rigid cohomology and to develop its first principles. Further topics include dagger modules and weakly complete algebras, radius of convergence, Robba rings. Its intention is not to lead the reader to the current front line of research, but to rather pave him the way to it. For example, neither the finiteness proofs nor the algorithmic aspects mentioned above are included. However, the last chapter (‘Conclusions’, see below) gives (without proofs) some indications on where the theory is developing. The book is written in a student friendly style. Very diligently the necessary ingredients are assembled in an exhaustive manner. Proofs (in the main part of the book, i.e. excluding the first and the last chapter, see below) are complete to the details (many of them not written in the literature so far). It is assumed that besides algebraic geometry the reader has some basic knowledge of rigid analytic geometry. For example, the notions of tubes and their strict neighbourhoods, of modules with connection, differential modules, de Rham cohomology etc. are all explained. An emphasis is given on the (formal and rigid) geometry underlying the whole theory. As illustrations, several prominent examples are thoroughly worked out (e.g. Kummer- and Dwork isocrystals, Legendre family, hypergeometric family). As the theory develops, they are repeatedly discussed in their respective new aspects.

The chapters are as follows: 1. Introduction (p. 1), 2. Tubes (p. 12), 3. strict neighbourhoods (p. 35), 4. Calculus (p. 74), 5. Overconvergent sheaves (p. 125), 6. Overconvergent calculus (p. 177), 7. Overconvergent isocrystals (p. 230), 8. Rigid cohomology (p. 264), 9. Conclusions (p. 299–307). Chapter 1 contains motivations, and like chapter 9, it contains no proofs.

Let \(k\) be a field of characteristic \(p>0\), and let \(V\) be a complete discrete valuation ring with residue field \(k\) and fraction field \(K\) of characteristic \(0\). Rigid cohomology is a cohomology theory for \(k\)-varieties with values in \(K\)-vector spaces. It unifies two older theories. One the one hand, the rigid cohomology (with constant coefficients, say) of a smooth and proper \(k\)-variety is canonically isomorphic to its crystalline cohomology (tensored with \(K\)). On the other hand, the rigid cohomology of a smooth and affine \(k\)-variety is canonically isomorphic to its Monsky Washnitzer cohomology. More generally, suppose now that for a separated \(k\)-scheme \(X\) of finite type we are given an embedding of \(X\) into the special fibre of a smooth proper formal \(V\)-scheme \({\mathfrak Y}\). An overconvergent isocrystal on \(X\) over \(K\) is a module with an integrable connection living on a ‘strict neighbourhood’ the tubular neighbourhood of \(X\) in \({\mathfrak Y}\), i.e. on a ’strict neighbourhood’ of the \(K\)-rigid analytic space which is the preimage of \(X\) under the specialization map from the generic fibre (as a \(K\)-rigid space) of \({\mathfrak Y}\) to its special fibre. If in addition we are given a lift of the Frobenius endomorphism of \(X\) to this module with connection one speaks of an overconvergent \(F\)-isocrystal. Berthelot showed that the categories of overconvergent isocrystals on \(X\) and of overconvergent \(F\)-isocrystals on \(X\) are independent of the choice of \({\mathfrak Y}\) (and the embedding of \(X\)). The category of overconvergent \(F\)-isocrystals should be viewed as the \(p\)-adic coefficient category analoguous to the categories of lisse \(\ell\)-adic sheaves (\(\ell\neq p\)). For example, for an overconvergent isocrystal \(E\) on \(X\) one can ask for its (absolute) cohomology groups \(H_{\text{rig}}^*(X,E)\) (and its (absolute) cohomology groups with compact supports \(H_{\text{rig},c}^*(X,E)\)). These are \(K\)-vector spaces, the (absolute) ‘rigid cohomology’ of \(E\). By now they are now known to be finite dimensional: Berthelot proved this in the case where \(E\) is the constant overconvergent isocrystal and \(X\) is smooth; the reviewer then proved it for general \(X\) (still with \(E\) constant); for general \(E\) the finiteness theorem is due to Kedlaya. Moreover, rigid cohomology satisfies all the properties required of a Bloch-Ogus cohomology theory. Its particular interest (against \(\ell\)-adic cohomology (\(\ell\neq p\))) lies in its potential to estimate the \(p\)-adic valuations of the zeroes and poles of Zeta-functions of \(k\)-varieties (if \(k\) is finite). Moreover, in recent years it turned out that rigid cohomology is in many circumstances more suitable (than \(\ell\)-adic cohomology (\(\ell\neq p\))) to actually compute these Zeta-functions, i.e. many effective algorithms based on rigid cohomology have been worked out. In one word, it is clear that rigid cohomology is an important subject in current arithmetic geometry.

The present book grew out of a course the author had given at Peking University in 2005. Its aim is to present all the basic objects and constructions needed to define rigid cohomology and to develop its first principles. Further topics include dagger modules and weakly complete algebras, radius of convergence, Robba rings. Its intention is not to lead the reader to the current front line of research, but to rather pave him the way to it. For example, neither the finiteness proofs nor the algorithmic aspects mentioned above are included. However, the last chapter (‘Conclusions’, see below) gives (without proofs) some indications on where the theory is developing. The book is written in a student friendly style. Very diligently the necessary ingredients are assembled in an exhaustive manner. Proofs (in the main part of the book, i.e. excluding the first and the last chapter, see below) are complete to the details (many of them not written in the literature so far). It is assumed that besides algebraic geometry the reader has some basic knowledge of rigid analytic geometry. For example, the notions of tubes and their strict neighbourhoods, of modules with connection, differential modules, de Rham cohomology etc. are all explained. An emphasis is given on the (formal and rigid) geometry underlying the whole theory. As illustrations, several prominent examples are thoroughly worked out (e.g. Kummer- and Dwork isocrystals, Legendre family, hypergeometric family). As the theory develops, they are repeatedly discussed in their respective new aspects.

The chapters are as follows: 1. Introduction (p. 1), 2. Tubes (p. 12), 3. strict neighbourhoods (p. 35), 4. Calculus (p. 74), 5. Overconvergent sheaves (p. 125), 6. Overconvergent calculus (p. 177), 7. Overconvergent isocrystals (p. 230), 8. Rigid cohomology (p. 264), 9. Conclusions (p. 299–307). Chapter 1 contains motivations, and like chapter 9, it contains no proofs.

Reviewer: Elmar Große-Klönne (Berlin)