Exercise 10, § 13

Let K be a commutative ring, A a K-algebra, and P an $(A$, A)-bimodule. An extension of A by P is a triple $(B, i, p)$, where B is a K-algebra, $p: B\rightarrow A$ a surjective homomorphism of K-algebras, and $i: P\rightarrow B$ an injective K-linear homomorphism with image Ker $p$, satisfying $i(p(b)x) =b i(x)$ and $i(x p(b)) =i(x)b$ for $x\in P$ and $b\in B$. An isomorphism from an extension $(B, i, p)$ to an extension $(B', i', p')$ is an isomorphism of K-algebras $u: B\rightarrow B'$ such that $p'\circ u=p$ and $u\circ i=i'$.

a) Let $(B, i, p)$ be an extension of A by P; the image of $i$ is a two-sided ideal of B with square zero. Conversely, if $p: B\rightarrow A$ is a surjective homomorphism of K-algebras whose kernel P has square zero and $i$ is the canonical injection of P into B, then the triple $(B, i, p)$ is an extension of A by P.

b) Let $i_0: P\rightarrow A\oplus P$ and $p_0: A\oplus P\rightarrow A$ be the canonical mappings. There exists a unique K-algebra structure on $A\oplus P$ such that $(A\oplus P, i_0, p_0)$ is an extension of A by P. Prove that the automorphism group of this extension can be identified with the group of K-derivations from A to P.

c) Let $(B, i, p)$ be an extension of A by P. Prove that the following conditions are equivalent:

(i) The extension $(B, i, p)$ is isomorphic to the extension $(A\oplus P, i_0, p_0)$.

(ii) There exists a homomorphism of K-algebras $s: A\rightarrow B$ such that $p\circ s=$ Id$_A$.

(iii) There exists a subalgebra S of B such that $B = S\oplus i(P)$.

If these conditions are satisfied, then we say that the extension $(B, i, p)$ is trivial. d) Let $(B, i, p)$ and $(B', i', p')$ be extensions of A by P. Let C be the subalgebra of $B\times B'$ consisting of the elements $(b, b')$ such that $p(b) =p'(b')$, let $\mathfrak{c}$ be the two-sided ideal of C consisting of the elements $(i(x),-i'(x))$ for $x\in P$, and let $B''$ be the algebra $C/\mathfrak{c}$ and $\pi : C\rightarrow B''$ the canonical homomorphism. Let $i'': P\rightarrow B''$ and $p'': B''\rightarrow A$ be the homomorphisms defined by $i''(x) =\pi (i(x),0)$ and $p''(\pi (b, b')) =$ $p(b)$ for $x\in P,b\in B,b'\in B'$. Prove that $(B'', i'', p'')$ is an extension of A by P; it is called the amalgamated sum of $(B, i, p)$ and $(B', i', p')$. Prove that we thus define a commutative group law on the set Ex$_K(A,P)$ of isomorphism classes of extensions of A by P, whose neutral element is the class of the trivial extension. Moreover, the action of K on Ex$_K(A,P)$ given by $\lambda (B, i, p) = (B, \lambda^{-1}i, p)$ for $\lambda \in K,\lambda \not= 0$ defines a K-module structure on Ex$_K(A,P)$.

e) Let $u: P\rightarrow P'$ be a homomorphism of $(A$, A)-bimodules, and let $(B, i, p)$ be an extension of A by P. Let $B'$ be the amalgamated sum $B\oplus_PP'$ (II, §1, p. 381, Exercise 5); for $b\in B, x'\in P'$, denote the class of $(b, x')$ in $B'$ by $[b, x']$. Let $i': P'\rightarrow B',p': B'\rightarrow A$, and $v: B\rightarrow B'$ be the mappings define by

Prove that there exists a unique algebra structure on $B'$ such that $(B', i', p')$ is an extension of A by $P'$ and $v$ an algebra homomorphism. We say that $(B', i', p')$ is the extension of A by $P'$ deduced from $(B, i, p)$ via $u$; every extension $(C, j, q)$ of A by $P'$ for which there exists an algebra homomorphism $w: B\rightarrow C$ satisfying $q\circ w=p$ and $w\circ i=j\circ v$ is isomorphic to $(B', i', p')$.

f ) The mapping Hom$_{(A,A)}(P,P')\times$ Ex$_K(A,P)\rightarrow$ Ex$_K(A,P')$ that sends a pair $(u,(B, i, p))$ to the class of the extension deduced from $(B, i, p)$ via $u$ is K-bilinear. g) Let $f: A'\rightarrow A$ be a K-algebra homomorphism, and let $(B, i, p)$ be an extension of A by P. Define likewise the structure of an algebra on the fibered product $B''= B\times_AA'$ and an extension $(B'', i'', p'')$ of $A'$ by P, said to be deduced from $(B, i, p)$ via $f$.

h) Suppose, moreover, that the algebra A is commutative. Prove that the classes of extensions $(B, i, p)$ of A by P such that the algebra B is commutative form a K-submodule of Ex$_K(A,P)$, which we denote by Ex$^c_K(A,P)$.