189-571B: Higher Algebra II

Assignment 5. Due: Wednesday, April 4.

1. Let $A$ be a central simple algebra over $F$ and let $K$ be an $F$- subalgebra of $A$ that is a field.

(a) Show that the degree $d$ of $K$ over $F$ divides $n$, where $n^2 := \dim_{F}(A)$.

(b) Let $B$ be the centraliser of $K$ in $A$. Show that $B$ is a central simple algebra over $K$. What is its dimension over $K$?

(c) Show that the $K$-algebra $A\otimes_F K$ is isomorphic to a matrix ring with entries in $B$, and conclude that the class of $A$ and $B$ are equal in the Brauer group of $K$. (Hint: observe that $A$ can be equipped with the structure of a $B^{\rm op}$-module, and construct an explicit $K$-algebra homomorphism $A\otimes K \rightarrow {\rm End}_{B^{\rm op}}(A)$.)

2. Let $G$ be a finite group of cardinality $n$ and let $M$ be an (abelian) $G$-module. Prove that every element of $H^1(G,M)$ has order dividing $n$.

3. Keeping the same assumptions as in the previous question, show that $H^1(G,M)$ is finite if $M$ is finitely generated as a $G$-module.

4. Let $K/F$ be a Galois extension with Galois group $G$. Show that the $K$- algebra $K[\varepsilon]$, where $\varepsilon^2 = 0$, has automorphism group isomorphic to $K^\times$. Give a direct classification of the forms of $K[\varepsilon]$ over $F$ (i.e., the $F$-algebras whose tensor product over $K$ becomes isomorphic to $K[\varepsilon]$). Use this to deduce Hilbert's theorem 90, asserting that $H^1(G,K^\times)=1$.

5. A commutative $F$-algebra $A$ is called an étale algebra over $F$ if it is isomorphic to a finite product of finite seperable field extensions of $F$. If $A$ is an étale algebra of rank $n$, show that there is a Galois extension $K$ of $F$ for which $A\otimes_F K$ is isomorphic to $K^n$ as a $K$-algebra.

6. Show that the group of $K$-algebra automorphisms of $K^n$ is isomorphic to the symmetric group $S_n$ on $n$ elements. If $G$ is any group acting trivially on $S_n$, show that $H^1(G,S_n)$ is in bijection with the {\em conjugacy classes} of homomorphisms from $G$ to $S_n$, i.e., the collection of isomorphism classes of permuation representations of $G$ of degree $n$.

7. Let ${\bf EA}_n(K/F)$ be the set of $F$-isomorphism classes of étale algebras of rank $n$ which become isomorphic to $K^n$ after being tensored with $K$. It is a pointed set, in which the distinguished element is the split étale algebra $F^n$. Use the work you have done in exercises $5$ and $6$ to describe a canonical identification of pointed sets $$ \iota: {\bf EA}_n(K/F) \rightarrow H^1(G, S_n),$$ where $G= {\rm Gal}(K/F)$, as before. Describe $\iota(A)$ as carefully and concretely as you can when $A\in {\bf EA}_n(K/F)$ is represented by the algebra $F[x]/p(x)$, where $p(x)$ is a seperable polynomial of degree $n$ with coefficients in $F$.

8. Problem 6, page 164 of ``Advanced Algebra".

9. Problem 7, page 164 of ``Advanced Algebra".

10. If $G$ is a cyclic group of order $n$ having $\sigma$ as a generator, and $M$ is an abelian $G$-module, show by an elementary argument, working directly with the definitions, that $$ H^1(G,M) = M^{N=0}/(\sigma-1)M,$$ where $N = 1 + \sigma + \cdots + \sigma^{n-1}$.