THE SYNTOMIC TOPOLOGY ON A SCHEME

Abstract

In 1949, Andre Wei!, [25], declared his now well known conjectures concerning the number of solutions of polynomial equations over finite fields. His conjectures suggested a deep connection between the arith­ metic of algebraic varieties defined over finite fields and the topology of algebraic varieties defined over the complex numbers. Wei! explained that if one had a suitable good cohomology theory for abstract vari­ eties, analogous to the ordinary cohomology of varieties defined over C, one could deduce his conjectures from various standard properties of the cohomology theory. This observation was one of the princi­ pal motivations for the introduction of various cohomology theories into abstract algebraic geometry. In 1955, [21], Serre introduced the first cohomology theory into abstract algebraic geometry using coher­ ent sheaves on algebraic varieties with respect to the Zariski topology. This was an algeoraic analogue of the notion of coherent sheaves in an­ alytic geometry. Some years later, Grothendieck, [4], inspired by some of the Serre's ideas. He could obtain a good theory by considering the variety together }Yith all its unramified covj:)_rs. This was the beginning of his theory of etale topology, developed jointly with M. Artin, which he used to define the p-adic cohomology. The crystalline cohomology of Grothendieck, [8], and Berthelot, [1], gives another similar coho­ mological interpretation of the Wei!conjectures. In, [8], Grothendieck established a relation between etale cohomology and de Rham coho­ mology. We recall very briefly his ideas as follows: Let p be a prime number and X be a smooth projective scheme over the discrete valuation ring Zp with a generic fibre X= X Xz" Qp where QP is the algebraic closure of Qp and a special fibre Y = X0z"Qp. The• Qp-adic etale cohomology associated to the generic fibre X is denoted by Vx = H*(Xet, QP) ; ancrthe crystaline cohomology associated to the special fibre Y is denoted by Dy = H*(Xzar: n /Q"); where n /Qp is the complex of differential forms on X.