By Philippe Gille
This booklet is the 1st accomplished, sleek creation to the speculation of valuable easy algebras over arbitrary fields. ranging from the fundamentals, it reaches such complicated effects because the Merkurjev-Suslin theorem. This theorem is either the fruits of labor initiated by way of Brauer, Noether, Hasse and Albert and the start line of present learn in motivic cohomology idea by means of Voevodsky, Suslin, Rost and others. Assuming just a reliable historical past in algebra, yet no homological algebra, the publication covers the elemental thought of crucial easy algebras, equipment of Galois descent and Galois cohomology, Severi-Brauer forms, residue maps and, eventually, Milnor K-theory and K-cohomology. The final bankruptcy rounds off the idea by way of providing the implications in confident attribute, together with the concept of Bloch-Gabber-Kato. The booklet is appropriate as a textbook for graduate scholars and as a reference for researchers operating in algebra, algebraic geometry or K-theory.
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Extra resources for Central Simple Algebras and Galois Cohomology
Our next goal is to classify all central simple k-algebras split by K by means of a single cohomology set. This should then carry a product operation, for tensor product induces a natural commutative and associative product operation on the set of isomorphism classes of central simple algebras, as shown by the following lemma. 4 If A and B are central simple k-algebras split by K , then so is A ⊗k B. 1, it is enough to verify the isomorphism of matrix algebras Mn (K ) ⊗ K Mm (K ) ∼ = Mnm (K ). 1.
We construct a norm map N K |k : K 1 (A K ) → K 1 (A) as follows. Fixing an isomorphism φ : Endk (K ) ∼ = Mn (k) gives rise to a composite map id⊗φ φ∗ : A K = A ⊗k K → A ⊗k Endk (K ) −−−−−→ A ⊗k Mn (k) ∼ = Mn (A). We then deﬁne the norm map N K |k as the composite φ∗ ∼ K (A), K 1 (A K ) → K 1 (Mn (A)) → 1 where the second map is the inverse of the isomorphism of the previous lemma. Since the conjugation action of the group GLn (k) (and even of GLn (A)) on Mn (A) induces a trivial action on K 1 (Mn (A)), we conclude that the map above is independent of the choice of φ.
Denoting by In the identity matrix, another computation shows that A 0 0 A−1 = In A In 0 In A 0 −In 0 In −A−1 In 0 In In 0 (9) and similarly for the other terms. The ﬁrst three terms on the right-hand side are upper or lower triangular matrices with 1s in the diagonal, so the lemma applies. For the fourth, notice that 0 −In In 0 = In −In In 0 In −In 0 In In In 0 In , so the lemma applies again. Using Whitehead’s lemma we may easily calculate K 1 -groups of ﬁelds. 4 For a ﬁeld k the natural map k × = GL1 (k) → GL∞ (k) ∼ K (k).
Central Simple Algebras and Galois Cohomology by Philippe Gille