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 $N = 1 + \sigma + \cdots + \sigma^{n-1}$.