# Introduction to Etale Cohomology by Günter Tamme

By Günter Tamme

Étale Cohomology is likely one of the most vital equipment in glossy Algebraic Geometry and quantity concept. It has, within the final a long time, introduced basic new insights in mathematics and algebraic geometric issues of many purposes and lots of very important effects. The publication provides a quick and simple creation into the realm of Abelian different types, Derived Functors, Grothendieck Topologies, Sheaves, common Étale Cohomology, and Étale Cohomology of Curves.

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Example text

Reg] (resp. ) @ Th e category L Con (resp . LPCon) of locally connected (resp. locally pathconnected) topological spaces [and continuous maps ]. (Hint. 1 Special categorical properties of topological constructs Completeness and cocompleteness 1. 1. 1 T h eor em. Let C be a construct. 2. 2. 35 (b) For any set X , any family ((Xi , ~i) )iEl of C- objecis indexed by some class I and any family (Ii : Xi ----+ X)iEl of maps indexed by I there exists a unique C- siruciure C. i) , I i, X , 1) , i. e.

A m onom orphism iff f : X --* Y 'is injective. I b) an epimo rphism iff f : X ---t Y is surjectiv e. P ro of a) ex) Let x , y E X such that f (x) = f(y) · x : (X , ~rI) --* (X, O defined by x (z) = x for each z E X and y : (X, ~rI) ---t ( X ,~) defined y(z) = y for each z E X are C-morphisms (cf. 2 2)) such th at f o x = I 0 y. e. x = y. Sinc e I is injective 1(X' ) = 8(x') for each x' E X' . Thus 1 = 8. (X,~) be Cmorphisms such th a t f 0 1 = f 0 8. 2. 5 Theorem. b) a) (indirect) . Suppose t hat f is not surjective.

JX;))iEI a family of semiuniform convergence spac es, (j; : X ---+ X i)iEI a family of maps, then JX = {F E F(X x X) : (j; x j;)(F) E J X, for each i E I} is th e initial SUConv-structure on X with resp ect to th e given data. e. such t hat A sati sfies the following condit ions: 1) X E A, 2) A E A implies X\A E A , 3) UnEIN An E A whenever (An)nElN is a sequence in At» A map f : (X , A) ---+ (X' , A') between measur abl e spaces is called measurable provided th at f - I[A' ] E A for each A' E A' .