Exercise 4, § 3

On supposes the ring A commutative.

Let G be a group, K a simplicial scheme (X, p. 177, exercise 19). An operation of G in K is a homomorphism of G into the group of bijective simplicial mappings from K into itself.

a) An operation of G in K defines an operation of G on the A-module S(K, A) (loc. cit.); prove that this action makes S(K, A) a complex of $A^{(G)}$-modules.

b) Suppose that G operates freely on the underlying set of K. Prove that the $A^{(G)}$-modules $S_n(K, A)$ are free.

c) Suppose that the simplicial scheme K is conical. Prove that the $A^{(G)}$-complex S(K, A) defines a left resolution of A, endowed with the structure of $A^{(G)}$-module for which $e_g a = a$ for all $g \in G, a \in A$.

d) Let $|G|$ be the simplicial scheme $(G, \mathscr{F})$, where $\mathscr{F}$ is the set of finite subsets of G. The group G operates by left translations on the simplicial scheme $|G|$; prove that the $A^{(G)}$-complex S(|G|, A) is a free resolution of the $A^{(G)}$-module A.

e) Let $B(A^{(G)}, A)$ be the standard resolution of the $A^{(G)}$-module A; for $n \geq 0$, the $A^{(G)}$-module $B_n(A^{(G)}, A)$ is identified with $(A^{(G)})^{\otimes(n+1)}$. Prove that the a-linear homomorphism $u : S(|G|, A) \to B(A^{(G)}, A)$, defined by

is an isomorphism of $A^{(G)}$-complexes.