Exercise 18, § 2

Let $E$ be a spectral sequence (exercise 14), such that the $A$-module $E_2$ is of finite length. Let $\chi$ be the set of classes of $A$-modules of finite length; now prove that one has $\chi(E_r)=\chi(E_2)=\chi(E_\infty)$ for every $r\geq2$ (X, p. 41). If moreover $E$ converges towards an $A$-module graded $G$, the latter is of finite length and one has $\chi(G)=\chi(E_\infty)$.

☡19) A simplicial scheme is called a pair $(K,\mathscr F)$, where $K$ is a set and $\mathscr F$ a set of finite subsets of $K$, called faces of $K$, such that every subset of a face is a face and every point of $K$ belongs to at least one face. A simplicial mapping of a simplicial scheme $(K,\mathscr F)$ into a simplicial scheme $(L,\mathscr G)$ is a mapping from $K$ into $L$ which transforms every face of $K$ into a face of $L$.

a) Let $n$ be an integer $\geq0$. An $n$-simplex of the simplicial scheme $K$ is called every sequence $(\alpha_0,\ldots,\alpha_n)$ of elements of $K$ such that the $\alpha_i$ belong to one and the same face of $K$. We denote by $S_n(K,A)$ the free $A$-module generated by the $n$-simplices for $n\geq0$, and we set $S_n(K,A)=0$ for $n<0$. We define an $A$-homomorphism

by setting $d_n=0$ for $n\leq0$ and

for every $n$-simplex $(\alpha_0,\ldots,\alpha_n)$ (the sign $\widehat{\ }$ over a letter means that it is to be omitted). Now prove that $d_{n-1}\circ d_n=0$ for every $n$, so that $(S(K,A),d)$ is a complex of $A$-modules. One defines analogously an $A$-complex $C(K,A)$ by setting $C^n(K,A)=\operatorname{Hom}_A(S_n(K,\mathbf Z),A)$, $\delta^n=\operatorname{Hom}(d_{n+1},1_A)$ for every $n$. For $n\geq0$, one sometimes denotes by $H_n(K,A)$ (resp. $H^n(K,A)$) the $A$-module $H_n(S(K,A))$ (resp. $H^n(C(K,A))$).

b) Let $K,L$ be two simplicial schemes, $f:K\rightarrow L$ a simplicial application. Now prove that $f$ induces morphisms of complexes $S(f):S(K,A)\rightarrow S(L,A)$ and $C(f):C(L,A)\rightarrow C(K,A)$, whence $A$-homomorphisms $f_*:H(K,A)\rightarrow H(L,A)$ and $f^*:H(L,A)\rightarrow H(K,A)$.

c) Let $g:K\rightarrow L$ be a second simplicial application. We say that $f$ and $g$ are simplicially homotopic if, for every face $\sigma$ of $K$, the set $f(\sigma)\cup g(\sigma)$ is a face of $L$. Now prove that the morphisms $S(f)$ and $S(g)$ (resp. $C(f)$ and $C(g)$) are then homotopic (define a homotopy $h$ by putting

for every $n$-simplex $(\alpha_0,\ldots,\alpha_n)$).

d) We say that the simplicial scheme $K$ is conical if there exists a point $s$ of $K$ such that, for every face $\sigma$ of $K$, $\sigma\cup\{s\}$ is a face. Now prove that there then exist homotopisms of $S(K,A)$ and of $C(K,A)$ onto $A$.

e) Let $D(K,A)$ be the subcomplex of $S(K,A)$ generated by the simplexes $(\alpha_0,\ldots,\alpha_n)$ for which two of the $\alpha_i$ are equal, and by the elements of the form

for every $n$-simplex $(\alpha_0,\ldots,\alpha_n)$ and every permutation $\sigma$ of $\{0,\ldots,n\}$. Now prove that $D(K,A)$ is a subcomplex of $S(K,A)$, and that the quotient passing application

is a homotopism (a reciprocal homotopism is obtained by choosing a total order on $K$ and associating with the class modulo $D(K,A)$ of a simplex $(\alpha_0,\ldots,\alpha_n)$ the simplex

where $\sigma$ is the permutation such that $\alpha_{\sigma(0)}<\cdots<\alpha_{\sigma(n)}$ if all the $\alpha_i$ are different, and $0$ in the contrary case).

*20) Let $R$ be a complete discrete valuation ring (AC, VI, § 3, no 6), with residue field $k$ of characteristic $p>0$. Suppose that the field of fractions of $R$ is of characteristic $0$. Denote by $v$ the normalized valuation of $\mathbf{R}$, by $\mathfrak m$ the maximal ideal of $\mathbf{R}$ and by $\mathbf{R}^*$ the multiplicative group of the units of $\mathbf{R}$. For every integer $n\geq 0$, let $\mathbf{R}^*(n)$ be the subgroup of $\mathbf{R}^*$ formed by the elements $x$ such that $v(x-1)\geq n$.

a) Let $r$ be an integer, $r>v(p)/(p-1)$. Show that the series

converges for $x\in\mathfrak m^r$ and defines an isomorphism of the additive group $\mathfrak m^r$ onto the group $\mathbf{R}^*(r)$.

b) Suppose that $k$ is a finite field with $q$ elements. Let $n$ be an integer $\geq 1$, $\mu_n$ the subgroup of the elements $x$ of $\mathbf{R}^*$ such that $x^n=1$. Show that the groups $\mu_n$ and $\mathbf{R}^*/(\mathbf{R}^*)^n$ are finite and that one has

(Let $\mathscr C$ be the set of classes of finite $\mathbf{Z}$-modules, $\varphi:K(\mathscr C)\to\mathbf{Q}^*$ the homomorphism defined by

$\mathscr C^{(r)}$ the complex of $\mathbf{Z}$-modules such that $C_p^{(r)}=0$ for $p\neq0,1$, $C_0^{(r)}=C_1^{(r)}=\mathbf{R}^*(r)$, $d_1(x)=x^n$ for $x\in\mathbf{R}^*(r)$. Prove that $\varphi(H(\mathscr C^{(r)}))$ is independent of $r$, then use a.).)*

21) Suppose that the ring A is a commutative algebra over a commutative ring $k$; the A-module $D_k(A)$ of the $k$-derivations of A is then identified with the dual of the A-module $\Omega^1_{A/k}$, which defines on $\Omega_{A/k}$ a structure of left $\Lambda_A(D_k(A))$-module (cf. III, p. 165). Prove the formula

where $\omega\in\Omega^p_{A/k}$, $X_0,\ldots,X_p\in D_k(A)$, and where the sign $\widehat{\ }$ over a letter means that it is to be omitted. (Reduce to the case $p=1$.)

22) We retain the conventions of the preceding exercise; if M is an A-module, $\nabla$ a connection on M and if $X\in D_k(A)$, we denote by $\nabla_X$ the $k$-endomorphism $(1_M\otimes X)\circ\nabla$ of M.

(a) Let $M_1,M_2$ be two A-modules, provided with connections ${}^1\nabla,{}^2\nabla$; prove that one defines a connection $\nabla$ on $M_1\oplus M_2$ (resp. $M_1\otimes_A M_2$, resp. $\operatorname{Hom}_A(M_1,M_2)$) by putting $\nabla(m_1+m_2)={}^1\nabla(m_1)+{}^2\nabla(m_2)$ for $m_1\in M_1$, $m_2\in M_2$ (resp. $\nabla_X(m_1\otimes m_2)={}^1\nabla_X(m_1)\otimes m_2+m_1\otimes{}^2\nabla_X(m_2)$ for $X\in D_k(A)$, $m_1\in M_1$ and $m_2\in M_2$, resp. $\nabla_X(u)={}^2\nabla_X\circ u-u\circ{}^1\nabla_X$ for $X\in D_k(A)$, $u\in\operatorname{Hom}_A(M_1,M_2)$.

(b) Let M be an A-module, $\nabla:M\to M\otimes_A\Omega^1_{A/k}$ a connection on M. If $X,Y\in D_k(A)$, we denote by $R(X,Y)$ the A-endomorphism of M such that $R(X,Y)(m)=\langle R(m),X\wedge Y\rangle$ for every $m\in M$. Prove the formula: $R(X,Y)=[\nabla_X,\nabla_Y]-\nabla_{[X,Y]}$ (use Exercise 21).

(c) Suppose that the A-module M is projective, so that the curvature homomorphism R is identified with an element of $\operatorname{End}_A(M)\otimes_A\Omega^2_{A/k}$. If $\operatorname{End}_A(M)$ is provided with the connection $\nabla$ defined in (a), prove that one has $\nabla^2R=0$.