Download Rigid Cohomology by Bernard Le Stum PDF
By Bernard Le Stum
Courting again to paintings of Berthelot, inflexible cohomology seemed as a standard generalization of Monsky-Washnitzer cohomology and crystalline cohomology. it's a p-adic Weil cohomology appropriate for computing Zeta and L-functions for algebraic types on finite fields. furthermore, it really is powerful, within the experience that it supplies algorithms to compute the variety of rational issues of such kinds. this can be the 1st publication to provide an entire remedy of the idea, from complete dialogue of the entire fundamentals to descriptions of the very newest advancements. effects and proofs are integrated that aren't to be had somewhere else, neighborhood computations are defined, and plenty of labored examples are given. This available tract could be of curiosity to researchers operating in mathematics geometry, p-adic cohomology thought, and comparable cryptographic components.
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Extra resources for Rigid Cohomology
Example text
We start developing now the geometric ground for this theory. 11 Let P be a (locally closed) formal subscheme of PN V and ∩ P . Then, we have X := AN k k ]X[P = BN (0, 1+ ) ∩ PK . 6 that ]X[P = BN (0, 1+ ) ∩ PK . 12 Let A be a V-algebra of finite presentation, X := SpecA and rig V := XK . Let V[T1 , . . , TN ] → A 20 Tubes be a presentation of A, X ⊂ AN V the corresponding inclusion, Y the algebraic and P := Y . Then, we have closure of X in PN V rig rig ]Yk [Y = YK = YK ⊃ XK and ]Xk [Y = XK = BN (0, 1+ ) ∩ XK ⊂ YK .
2 Tubes of radius one 21 of X as a formal subscheme of P so that ]X[P = {x ∈ PK , |f1 (x)|, . . , |fr (x)| < 1 and ∃j ∈ {1, . . , s}, |gj (x)| = 1}. This is easily seen to be an admissible open subset. More precisely, admissible open subsets are stable under finite union and finite intersections and, we know that for each g = 1, . . , s, {x ∈ PK , |gj (x)| = 1} is a Weierstrass domain and, in particular, is an admissible open subset. We are therefore reduced to the case s = 0, which means that X is closed.
Mixed immersions) of formal embeddings Xi _ / Pi _ X /P is a covering (resp. an open covering, resp. a mixed covering) if it is so at all steps. 15 If Xi _ / Pi _ X /P is an open covering or a finite mixed covering of formal embeddings, then ]X[P = ∪i ]Xi [Pi is an admissible covering. Proof First of all, since the tube only depends on an open neighborhood, we may assume that Pi = P for each i. Now, for open coverings, our assertion is a direct consequence of the continuity of the specialization map.