Exercise 3, § 3

Let $k$ be a field and $A$ a commutative $k$-algebra. For every integer $i \geqslant 1$ put $B_i = A \otimes_k A \otimes_k \cdots \otimes_k A$ and endow $B_i$ with the structure of an $A$-algebra obtained by making $A$ act on the first component. Put $C_1 = B_2, C_2 = B_3, C_3 = B_4 \oplus B_3$, and define $A$-linear mappings $d_2 : C_2 \to C_1$ and $d_3 : C_3 \to C_2$ by the formulas

¹ If $A$ is a local, noetherian and complete algebra over a (discrete) field $k$, this definition coincides with that given in VIII, p. 98, exerc. 30, according to Exercise 5 below.

for every choice of elements $x, y, z, \alpha, \beta$ of $A$. Thus one obtains a complex of $A$-modules

If $N$ is an $A$-module, we shall say that a $k$-bilinear mapping $f : A \times A \to N$ is a 2-cocycle if one has the identity $x f(y, z) - f(xy, z) + f(x, yz) - zf(x, y) = 0$ for $x, y$ and $z$ in $A$, and that it is symmetric if one has $f(x, y) = f(y, x)$ for every pair $(x, y) \in A \times A$.

a) The following conditions are equivalent: (i) the complex $\mathrm{Hom}_A(C_k(A), N)$ is acyclic for every $A$-module $N$; (ii) whatever the $A$-module $N$, every symmetric 2-cocycle $f : A \times A \to N$ is a 1-coboundary, that is to say, of the form $f(a, b) = ag(b) + bg(a) - g(ab)$ with $g \in \mathrm{Hom}_k(A, N)$; (iii) $A$ is a smooth algebra over $k$.

b) If $A$ is a field, the preceding conditions are also equivalent to the condition that the complex $C_k(A)$ be acyclic.