The first seven chapters of these notes reproduce the greatest part of a seminar held by the first author at Princeton University during the spring semester, 1974. The seminar was meant to provide the auditors with the basic tools used in the study of crystalline cohomology of algebraic varieties in positive characteristic, and did not cover all known results on this topic. These notes have the same limited purpose, and should really be"considered only as an introduction to the subject. In Chapter I, we draw a rapid picture of the various cohomology theories for algebraic varieties in characteristic p, and try to explain the specific need for a p-adic cohomology, as well as the motivations of the technical definition of the crystalline site; this chapter is purely introductory, and contains no proof. The second chapter introduces some basic notions of differential calculus, and in particular various presentations of the notions of connection and stratification. Actually, these notions, under this particular form, are relevant to crystalline cohomolog
only in characteristic zero; but it seemed more convenient to give first the algebraic presentation of ordinary differential calculus, which is more or less familiar to the reader, and then to explain how it has to be modified to yield a good formalism ?n characteristic p. To do this, we introduce in Chapter III the notion of. a divide power ideal. The main result in this chapter is the construction of the divided power envelope of an ideal in an arbitrary commutative ring, which is of constant use in what follows. Chapter IV reviews then the notions of Chapter II with the modifications necessary to work in characteristic p. With chapter V begins the theory of the crystalline topos. Once we have defined the crystalline site, and described the sheaves on this site, we establish the functoriality of the corresponding topos, and show in particular that if X is an 3-scheme, and (I,y) a divided power ideal in S which extends to X, the crystalline cohomology of X relatively to C,I,Y) depends only upon the reduction of X modulo I. The chapter ends with a discussion of the relations between the crystalline and the Zariski topoi, which will be used in Chapter VII to relate crystalline and de Rham cohomologies. Chapter VI is devoted to the notion of crystal. First we define crystals and show how they can be interpreted as modules on a suitable scheme endowed with a quasi-nilpotent integrable connection. We then associate to a complex K' of differential operators of order 1 on a smooth S-scheme У a complex of crystals, ; with linear differential, on the crystalline site of any Y-scheme X. In the particular case where K' is the complex of differential forms on У relatively to S, we thus obtain a resolution of the structural sheaf of the crystalline topos ("Poincare' lemma") . ' In Chapter VII, we prove (in a new way) the fundamental property of crystalline cohomology: If X is a closed.

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