Exercise 2, § 3

Algebra, X, p. 178

We suppose that the ring A is commutative. Let B be an A-algebra; for $n\geq0$ , we denote by $\mathscr A^n(B)$ the tensor product over A of $(n+1)$ modules equal to B. We put $\mathscr A^n(B)=0$ for $n<0$ and $\mathscr A(B)=\bigoplus_n\mathscr A^n(B)$ .

EXERCISES

define a graded A-endomorphism d of degree +1 of $\mathscr{A}(B)$ by setting:

d(b_0 \otimes \cdots \otimes b_n) = \sum_{i=0}^{n+1} (-1)^i b_0 \otimes \cdots \otimes b_{i-1} \otimes 1_B \otimes b_i \otimes \cdots \otimes b_n \quad \text{for } b_0, \ldots, b_n \in B .

a) Prove that $d \circ d = 0$ , so that $(\mathscr{A}(B), d)$ is a complex of A-modules. If M is an A-module, we denote by $\mathscr{A}(B, M)$ the complex such that $\mathscr{A}''(B, M) = \mathscr{A}''(B) \otimes_A M$ , with differential $d \otimes 1_M$ .

b) Suppose that the algebra B is a faithfully flat A-module (X, p. 169, exercise 9). Prove that the complex $\mathscr{A}(B, M)$ defines a right resolution of the A-module M. (It is enough to verify that the complex $B \otimes_A \mathscr{A}(B, M)$ defines a resolution of $B \otimes_A M$ ; construct an A-linear homotopy between these two complexes.)

Solution

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