Exercise 13, § 2

Let $I$ be a finite set; we shall denote by $(e_i)_{i \in I}$ the canonical basis of the $\mathbf{Z}$-module $\mathbf{Z}^I$. An I-*complex* (resp. I-*precomplex*) of A-modules is called an A-module C graded of type $\mathbf{Z}^I$, endowed with a family of A-endomorphisms $(d_i)_{i \in I}$, such that $d_i$ is graded of degree $(-e_i)$, satisfying:

If Card (I) = 1, an I-complex is a complex; if Card (I) = 2, an I-complex is a bicomplex (exercise 11). If $(C, (d_i))$ and $(C', (d'_i))$ are two I-complexes (resp. I-precomplexes), a morphism from $(C, (d_i))$ into $(C', (d'_i))$ is a graded homomorphism of degree zero $u : C \to C'$ such that $d'_i \circ u = u \circ d_i$ for all $i \in I$.

a) Let $(C, (d_i))$ be an I-complex. If one endows $C$ with the total grading (of type $\mathbf{Z}$) deduced from its grading, show that $(C, \sum_{i \in I} d_i)$ is a complex of $A$-modules, called the complex associated with the I-complex $(C, (d_i))$.

Show that a morphism of I-complexes induces a morphism of the associated complexes.

b) Let $C$ be an I-precomplex, and let $R$ be a total order relation on $I$, denoted by $\leq$. For $k = (k_i)_{i \in I}$, $x \in C_k$ and $i \in I$, put

Show that $(C, (\delta_i))$ is an I-complex, denoted $C_R$.

c) Let $R_\alpha, R_\beta$ be two total order relations on $I$, denoted by $\leq_\alpha$ and $\leq_\beta$. Let $u_{\beta \alpha}$ be the endomorphism of $C$ defined by $u(x) = (-1)^{\varepsilon(k)} x$ for $x \in C_k$, with $\varepsilon(k) = \sum_{i <_\alpha j <_\beta i} k_i k_j$. Show that $u_{\beta \alpha}$ is a morphism of I-complexes from $C_{R_\alpha}$ into $C_{R_\beta}$.

d) Let $S$ be the set of total order relations on $I$; we endow $S$ with the trivial preorder relation (that is, such that $R_\alpha \leq R_\beta$ whatever $R_\alpha, R_\beta \in S$). Show that the system $(C_{R_\alpha}, u_{\alpha \beta})$ is a projective system of I-complexes, relative to $S$; its limit $\hat{C}$ is an I-complex, called the I-complex associated with the I-precomplex $C$. Show that any morphism of I-precomplexes induces a morphism of the associated I-complexes.