(2)

c) If F is of type $\mathscr{C}$, then so is the cokernel of u, and we have

(3)

The proposition follows from the existence of exact sequences

(4)

Let us prove the corollary by induction on $n$; the cases $n= 0$ and $n= 1$ are trivial. So let $n\geqslant 1$, and let

be an exact sequence of modules of type $\mathscr{C}$. By Proposition 1, the kernel F of $u_n$ is a module of type $\mathscr{C}$, and we have

Moreover, we have an exact sequence

and the induction hypothesis gives the relation

We then immediately deduce from (5) and (6) that $\sum^{n+1}_{i=0}(-1)^i\varphi (E_i) = 0$, and the corollary follows.

a) If the modules M and N are of type $\mathscr{C}$, then so are the modules $M\cap N$ and M + N, and we have

b) If the modules $E/M$ and $E/N$ are of type $\mathscr{C}$, then so are the modules $E/(M\cap N)$ and $E/(M + N)$, and we have

Assertions a) and b) follow from the existence of the exact sequences

and

(II, §1, No. 7, p. 207, Proposition 10).

Since $\mathscr{C}$ is hereditary, the modules $E_0/E_1$ and $E_1$ are of type $\mathscr{C}$, and we have $\varphi (E) =\varphi (E_0/E_1) +\varphi (E_1)$. Since the sequence $(E_{i+1})_{0\leqslant i\leqslant n-1}$ is a composition series of $E_1$, Proposition 3 follows by induction on $n$.

b) Let G be an abelian group, and let $\varphi :\mathscr{C}\rightarrow G$ be an additive function of modules. There exists a unique homomorphism $u: K(\mathscr{C})\rightarrow G$ such that we have $\varphi (E) =u([E]_{\mathscr{C}})$ for every module E of type $\mathscr{C}$.

Assertion a) is obvious. Let us prove b). There exists a homomorphism $u':\mathbf{Z}^{(\mathscr{C})}\rightarrow G$ that extends $\varphi$. Since $\varphi$ is additive, we have $u'(r_{\mathscr{E}}) = 0$ for every exact sequence $(\mathscr{E})$ of modules of type $\mathscr{C}$; therefore, R is contained in the kernel of $u'$. Consequently, when passing to the quotient, $u'$ defines a homomorphism $u$ from $K(\mathscr{C})$ to G, and it is clear that we have $\varphi (E) =u([E]_{\mathscr{C}})$ for every A-module of type $\mathscr{C}$. The group $K(\mathscr{C})$ is generated by the set of elements $[E]_{\mathscr{C}}$ for E running through $\mathscr{C}$; the uniqueness of $u$ follows.

Let $\mathscr{C}$ and $\mathscr{D}$ be additive sets of classes of A-modules such that $\mathscr{C}\subset \mathscr{D}$. Since the mapping $E\mapsto [E]_{\mathscr{D}}$ fromn$\mathscr{C}$ to the Grothendieck groupnK($\mathscr{D}$) is additive, there exists a homomorphism $\gamma_{\mathscr{D},\mathscr{C}}: K(\mathscr{C})\rightarrow K(\mathscr{D})$, called canonical, characterized by the formula $\gamma_{\mathscr{D},\mathscr{C}}([E]_{\mathscr{C}}) = [E]_{\mathscr{D}}$ for every module E of type $\mathscr{C}$. It is not always injective (VIII, p. 210, Exercise 13).

The stated condition is sufficient because it implies the relations

which give $[E]_{\mathscr{C}}= [F]_{\mathscr{C}}$.

Now suppose that we have $[E]_{\mathscr{C}}= [F]_{\mathscr{C}}$. By the construction of the group $K(\mathscr{C})$, there exist two finite families of exact sequences of modules of type $\mathscr{C}$

for $i\in I$ and

for $j\in J$ such that in $\mathbf{Z}^{(\mathscr{C})}$, we have

cl(E) $-$ cl(F) $=\sum_{j\in J}r_{\mathscr{H}_j}-\sum_{i\in I}r_{\mathscr{G}_i}$.

More explicitly, this relation can be written as

(7) cl(E) $+\sum_{i\in I}$ cl(G$_i$) $+\sum_{j\in J}$ cl(H$'_j$) $+\sum_{j\in J}$ cl(H$''_j$)

= cl(F) $+\sum_{i\in I}$ cl(G$'_i$) $+\sum_{i\in I}$ cl(G$''_i$) $+\sum_{j\in J}$ cl(H$_j$).

Set $G =\bigoplus_{i\in I}G_i, G'=\bigoplus_{i\in I}G'_i$, etc. By passing to the direct sums, we obtain the exact sequences

$(\mathscr{G})$ 0 // ${G'}^p$ // G $^q$ // $G''$ // $0$,

$(\mathscr{H})$ 0 // ${H'}^r$ // H $^s$ // $H''$ // 0

consisting of modules of type $\mathscr{C}$.

Next, let $M_1, . . . ,M_m,N_1, . . . ,N_n$ be A-modules of type $\mathscr{C}$. If we have $\sum^m_{i=1}$ cl(M$_i$) $=\sum^n_{j=1}$ cl(N$_j$) in the group $\mathbf{Z}^{(\mathscr{C})}$, then we have $m=n$ and there exists a permutation $\sigma \in \mathfrak{S}_m$ such that cl(M$_i$) $=$ cl(N$_{\sigma(i)}$) for every $1\leqslant i\leqslant m$ (I, §7, No. 9, p. 95, Proposition 11). Consequently, the modules $\bigoplus^m_{i=1}M_i$ and $\bigoplus^n_{j=1}N_j$ are isomorphic. In particular, from (7), we deduce the existence of an isomorphism from $E\oplus Q$ to $F\oplus P$, where we have set

We also set

The modules L, M, P, Q are of type $\mathscr{C}$, and the sequence

$(\mathscr{E})$ 0 // L $^{\lambda}$ // P $^\mu$ // M // $0$,

where we define $\lambda$ and $\mu$ by

is exact. We construct an exact sequence

in the same way; this concludes the proof.

The set $\mathscr{C}$ is a commutative monoid for the law of composition $(E,E')\mapsto$ cl(E$\oplus E'$). We sometimes denote the group of differences of the commutative monoid $\mathscr{C}$ (I, §2, No. 4, p. 20) by $K'(\mathscr{C})$ and call it the Grothendieck group of $\mathscr{C}$ for direct sums. For any module E of type $\mathscr{C}$, we denote the image of cl(E) in $K'(\mathscr{C})$ by $[E]'_{\mathscr{C}}$.

b) Let G be an abelian group, and let $\varphi :\mathscr{C}\rightarrow G$ be a weakly additive function of modules. There exists a unique group homomorphism $u: K'(\mathscr{C})\rightarrow$ G such that we have $\varphi (E) =u([E]'_{\mathscr{C}})$ for every module E of type $\mathscr{C}$.

c) Let E and F be modules of type $\mathscr{C}$. We have $[E]'_{\mathscr{C}}= [F]'_{\mathscr{C}}$ if and only if there exists a module M of type $\mathscr{C}$ such that $E\oplus M$ is isomorphic to $F\oplus M$.

Assertion a) is obvious. Assertion b) follows from (I, §2, No. 4, p. 19, Theorem 1) and assertion c) from (I, §2, No. 4, p. 18, Proposition 6).

Since the mapping $E\mapsto [E]_{\mathscr{C}}$ from $\mathscr{C}$ to $K(\mathscr{C})$ is a weakly additive function of modules, we can deduce a homomorphism $u: K'(\mathscr{C})\rightarrow K(\mathscr{C})$ from it. This homomorphism is surjective but is not always an isomorphism (VIII, p. 191, Remark 2).

Let $R'$ be the subgroup of $\mathbf{Z}^{(\mathscr{C})}$ generated by the elements of the form $r_{\mathscr{E}}$, where $\mathscr{E}$ is a split exact sequence of A-modules of type $\mathscr{C}$, that is, by the elements of the form cl(E$'\oplus E''$)$-$ cl(E$'$)$-$ cl(E$''$), where $E'$ and $E''$ are modules of type $\mathscr{C}$. The canonical mapping from $\mathscr{C}$ to the quotient group $\mathbf{Z}^{(\mathscr{C})}/R'$ extends to a group homomorphism $v: K'(\mathscr{C})\rightarrow \mathbf{Z}^{(\mathscr{C})}/R'$. This is an isomorphism. Indeed, the canonical mapping from $\mathscr{C}$ to $K'(\mathscr{C})$ extends to a group homomorphism from $\mathbf{Z}^{(\mathscr{C})}$ to $K'(\mathscr{C})$ whose kernel contains $R'$, and therefore to a homomorphism $v':\mathbf{Z}^{(\mathscr{C})}/R'\rightarrow K'(\mathscr{C})$ by passing to the quotient; it is clear that $v$ and $v'$ are inverse bijections.

An element of $K'(\mathscr{C})$ is called effective if it is of the form $[E]'_{\mathscr{C}}$ for an A-module E of type $\mathscr{C}$. The set of effective elements of $K'(\mathscr{C})$ is denoted by $K'(\mathscr{C})^+$.

0 // ${E'}^i$ // E $^p$ // $E''$ // 0

an exact sequence. We have $\ell_S(E) =\ell_S(E') +\ell_S(E'')$.

Let $\Sigma '$ and $\Sigma ''$ be Jordan–Hölder series of $i(E')$ and $E/i(E')$, respectively. There exists a Jordan–Hölder series Σ of E whose sequence of quotients can be obtained by juxtaposing the sequence of quotients of Σ and that of $\Sigma '$ (I, §4, No. 8, p. 44).

Formula (8) follows from Proposition 3 applied to a Jordan–Hölder series of E. By Lemma 1, for every element S of $\mathscr{S}$, there exists a $\mathbf{Z}$-linear mapping $\varphi_S$ from $K(\mathscr{C})$ to $\mathbf{Z}$ such that $\varphi_S([E]_{\mathscr{C}}) =\ell_S(E)$ for every module E of type $\mathscr{C}$. In particular, we have $\varphi_S([S]_{\mathscr{C}}) = 1$ and $\varphi_S([S']_{\mathscr{C}}) = 0$ for every $S'\not= S$ in $\mathscr{S}$. It follows that the elements of the form $[S]_{\mathscr{C}}$ (for $S\in \mathscr{S}$) are linearly independent over $\mathbf{Z}$; these elements generate $K(\mathscr{C})$ by formula (8).

Indeed, we have $[E]_{\mathscr{C}}=\sum_{S\in\mathscr{S}}\ell_S(E)[S]_{\mathscr{C}}$ and an analogous formula for F, and E is isomorphic to F if and only if we have $\ell_S(E) =\ell_S(F)$ for every $S\in \mathscr{S}$ (VIII, p. 72).

Let P be a system of representatives consisting of irreducible elements of A (VII, §1, No. 3, p. 3). The maximal ideals of A are the ideals $pA$ for $p\in P$; every simple A-module is therefore isomorphic to a unique module $A/pA$. Moreover (VII, §1, No. 3, p. 4, Theorem 2), the abelian group $L^*/A^*$ is free and has the family $(pA^*)_{p\in P}$ as a basis. Hence there exists an isomorphism $\varphi$ from R(A) to $L^*/A^*$ characterized by $\varphi ([A/pA]) =pA^*$ for every $p\in P$.

Let $a\not= 0$ in A. There exist an integer $r\geqslant 0$, elements $p_1, . . . , p_r$ of P, and an element $u$ of $A^*$ such that we have $a=up_1\cdots p_r$. The module $A/aA$ admits the composition series defined by

and the module $E_{i-1}/E_i= (p_1\cdots p_{i-1}A)/(p_1\cdots p_iA)$ is isomorphic to $A/p_iA$. We therefore have (VIII, p. 185, Proposition 3)

Suppose that there exists an $a\in A$ such that $a_M= 0_M$ and $a_{M'}= 1_{M'}$. Let S be a simple A-module. If cl(S) belongs to the support $\mathscr{S}_M$ of M, then the A-module S is isomorphic to one of the quotients of a Jordan–Hölder series of M and we have $a_S= 0_S$. Likewise, if cl(S) belongs to the support $\mathscr{S}_{M'}$ of M’, then we have $a_S= 1_S$. It follows that $\mathscr{S}_M$ and $\mathscr{S}_{M'}$ are disjoint.

Conversely, suppose that the sets $\mathscr{S}_M$ and $\mathscr{S}_{M'}$ are disjoint. They are finite because M and $M'$ are finite-dimensional over K. Every simple A-module S whose class belongs to $\mathscr{S}_M\cup \mathscr{S}_{M'}$ is finite-dimensional over K and a fortiori over the field End$_A(S)$. By Corollary 1 of Proposition 4 (VIII, p. 83), there exists an element $b\in A$ such that we have $b_S= 0_S$ (resp. $b_S= 1_S$) for every simple A-module S whose class belongs to $\mathscr{S}_M$ (resp. $\mathscr{S}_{M'}$). Let $(M_i)_{0\leqslant i\leqslant n}$ be a Jordan–Hölder series of M. By the above, we have $bM_i\subset M_{i+1}$ for $0\leqslant i < n$ and therefore $(b^n)_M= 0_M$. The existence of a natural number $m$ such that $((1-b)^m)_{M'}= 0_{M'}$ is proved in the same way. Set $P(X) = 1-(1-X^n)^m$ and $a= P(b)$. The polynomial P(X) is a multiple of $X^n$, so we have $a_M= 0_M$, while the polynomial $1-P(X)$ is a multiple of $(1-X)^m$, so we have $a_{M'}= 1_{M'}$. This concludes the proof.

Let L be an extension of K. If M is an A-module that is finite-dimensional over K, then $M_{(L)}$ is an $A_{(L)}$-module that is finite-dimensional over L. Moreover, for every exact sequence

of A-modules, the sequence of $A_{(L)}$-modules

deduced from it by extension of scalars is exact (II, §7, No. 7, p. 308, Proposition 14). Hence there exists a unique ring homomorphism $u: R_K(A)\rightarrow$ $R_L(A_{(L)})$ such that $u([M]) = [M_{(L)}]$ for every A-module M that is finite-dimensional over K (No. 5).

By Lemma 2, there exists an element $a\in A$ such that $a_S= 0$ and $a_T= 1$. The element $1\otimes a$ of $A_{(L)}$ acts as 0 on $S_{(L)}$ and as 1 on $T_{(L)}$. By Lemma 2, the supports of the $A_{(L)}$-modules $S_{(L)}$ and $T_{(L)}$ are disjoint.

Let us prove Theorem 1. Let $\mathscr{S}$ be the set of classes of simple A-modules that are finite-dimensional over K. The family $([S])_{S\in\mathscr{S}}$ is a basis of the $\mathbf{Z}$-module $R_K(A)$. Let $S\in \mathscr{S}$. The $A_{(L)}$-module $S_{(L)}$ is not zero, so its support is not empty. Let $S'$ be an element of this support. By Lemma 1 of VIII, p. 190, there exists a homomorphism $f_{S'}: R_L(A_{(L)})\rightarrow \mathbf{Z}$ such that $f_{S'}([E]) =\ell_{S'}(E)$ for every $A_{(L)}$-module E that is finite-dimensional over L. We have $f_{S'}([S_{(L)}])\not= 0$ by construction and $f([T_{(L)}]) = 0$ for every $T\in \mathscr{S}-\{S\}$ by Lemma 3. We have therefore proved that the elements of $R_L(A_{(L)})$ of the form $[S_{(L)}]$ for $S\in \mathscr{S}$ are linearly independent over $\mathbf{Z}$. It follows that the homomorphism $u$ is injective.

Let $S\in \mathscr{S}$, and let $S'$ be an element of the support of $S_{(L)}$. For every $\xi \in R_K$(A), the coordinate of $\xi$ of index [S] in the basis $([S])_{S\in\mathscr{S}}$ is $f_{S'}(u(\xi ))/[S_{(L)}: S']$. It follows that if $u(\xi )$ is effective, then so is $\xi$.