By Eberhard Freitag
This e-book is anxious with some of the most very important advancements in algebraic geometry over the past a long time. In 1949 Andr? Weil formulated his recognized conjectures in regards to the numbers of ideas of diophantine equations in finite fields. He himself proved his conjectures via an algebraic thought of Abelian types within the one-variable case. In 1960 seemed the 1st bankruptcy of the "El?ments de G?ometrie Alg?braique" par A. Grothendieck (en collaboration avec J. Dieudonn?). In those "El?ments" Grothendieck developed a brand new starting place of algebraic geometry with the declared goal to return to an explanation of the Weil conjectures through a brand new algebraic cohomology thought. Deligne succeded in proving the Weil conjectures at the foundation of Grothendiecks rules. the purpose of this "Ergebnisbericht" is to enhance as self-contained as attainable and as brief as attainable Grothendiecks 1-adic cohomology concept together with Delignes monodromy idea and to give his unique evidence of the Weil conjectures.
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Extra info for Etale Cohomology and the Weil Conjecture (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge A Series of Modern Surveys in Mathematics)
Example text
If G is a group then a unit in 7l.. (G) is an element with a two-sided multi plicative inverse. The elements of the group ± G = {gig E G } v { - gig E G} are called the trivial units, and all others are called the non-trivial units of 7l.. ( G) . 6) A . Suppose that G is an abelian group such that 7l.. (G) has non-trivial units . (G) matrix A which cannot be transformed to an identity matrix by any finite sequence of the operations (I)-(V). B. The group G = 7l.. 5 is an abelian group such that 7l..
9), determines the deformation (M, u K) A (M, u K) up L 2 L, L, = K Ll L2 , reI L 2 . J It follows directly from the second definition that f* is a group homo morphism. 6) it follows directly that g* f* = (gf) * . 2) There is a covariant functor from the category offinite CW complexes and cellular maps to the category of abelian groups and group homomorphisms given by L ...... Wh(L) and (f: L I -+ L2) ...... U* : Wh(L 1) -+ Wh(L 2 ))' Moreover iff ':::!. g then f* = g * . PROOF: The reader having done his duty, we need only verify that if f ':::!.
Matrices and formal deformations Given a homotopically trivial CW pair, we have shown that it can be transformed into a pair in simplified form. So consider a simplified pair (K, L) ; K = L u Uej u Uer + 1 where the ej are trivially attached at eO . W + 1 : 81' + 1 --+ L u U ej, where 'Pi is a characteristic map for er+ 1 . (K" L ; eO) in the homotopy r + exact sequence of the triple (K, K , L). Since, however, freely homotopic r attaching maps give (7. 1 ) the same result up to simple-homotopy type, we do not wish to be bound to homotopies keeping the base point fixed.