4.56 (1973)

Let $R$ be a commutative Noetherian ring with 1, and $\Lambda$ an $R$-algebra, which is finitely generated as $R$-module. Put
$T = \{U \in \operatorname{Mod} \Lambda \mid \exists$ an exact $\Lambda$-sequence $0 \to P \to \Lambda^{(n)} \to U \to 0$ for some $n$, with $P_{\mathfrak{m}} \cong \Lambda^{(n)}_{\mathfrak{m}}$ for every maximal ideal $\mathfrak{m}$ of $R \}$.
Denote by $G(T)$ the Grothendieck group of $T$ relative to short exact sequences.
$\qquad$ a) Describe $G(T)$, in particular, what does it mean: $[U] = [V]$ in $G(T)$?
$\qquad$ b) Conjecture: if $\dim(\max(R)) = d < \infty$, and there are two epimorphisms $\varphi : \Lambda^{(n)} \to U$, $\psi : \Lambda^{(n)} \to V$, $n > d$, and $[U] = [V]$ in $G(T)$, then $\operatorname{Ker} \varphi = \operatorname{Ker} \psi$.