The Stacks project

Lemma 15.75.8. Let $R$ be a ring. Let $\mathfrak p \subset R$ be a prime. Let $M^\bullet $ and $N^\bullet $ be bounded complexes of finite projective $R$-modules representing the same object of $D(R)$. Then there exists an $f \in R$, $f \not\in \mathfrak p$ such that there is an isomorphism (!) of complexes

\[ M^\bullet _ f \oplus P^\bullet \cong N^\bullet _ f \oplus Q^\bullet \]

where $P^\bullet $ and $Q^\bullet $ are finite direct sums of trivial complexes, i.e., complexes of the form the form $\ldots \to 0 \to R_ f \xrightarrow {1} R_ f \to 0 \to \ldots $ (placed in arbitrary degrees).

Proof. If we have an isomorphism of the type described over the localization $R_\mathfrak p$, then using that $R_\mathfrak p = \mathop{\mathrm{colim}}\nolimits R_ f$ (Algebra, Lemma 10.9.9) we can descend the isomorphism to an isomorphism over $R_ f$ for some $f$. Thus we may assume $R$ is local and $\mathfrak p$ is the maximal ideal. In this case the result follows from the uniqueness of a “minimal” complex representing a perfect object, see Lemma 15.75.6, and the fact that any complex is a direct sum of a trivial complex and a minimal one (Algebra, Lemma 10.102.2). $\square$

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