The Stacks project

Lemma 15.75.6. Let $(R, \mathfrak m, \kappa )$ be a local ring. Let $K \in D(R)$ be pseudo-coherent. Set $d_ i = \dim _\kappa H^ i(K \otimes _ R^\mathbf {L} \kappa )$. Then $d_ i < \infty $ and for some $b \in \mathbf{Z}$ we have $d_ i = 0$ for $i > b$. Then there exists a complex

\[ \ldots \to R^{\oplus d_{b - 2}} \to R^{\oplus d_{b - 1}} \to R^{\oplus d_ b} \to 0 \to \ldots \]

representing $K$ in $D(R)$. Moreover, this complex is unique up to isomorphism(!).

Proof. Observe that $K \otimes _ R^\mathbf {L} \kappa $ is pseudo-coherent as an object of $D(\kappa )$, see Lemma 15.64.12. Hence the cohomology spaces are finite dimensional and vanish above some cutoff. Every object of $D(\kappa )$ is isomorphic in $D(\kappa )$ to a complex $E^\bullet $ with zero differentials. In particular $E^ i \cong \kappa ^{\oplus d_ i}$ is finite free. Applying Lemma 15.75.5 we obtain the existence.

If we have two complexes $F^\bullet $ and $G^\bullet $ with $F^ i$ and $G^ i$ free of rank $d_ i$ representing $K$. Then we may choose a map of complexes $\beta : F^\bullet \to G^\bullet $ representing the isomorphism $F^\bullet \cong K \cong G^\bullet $, see Derived Categories, Lemma 13.19.8. The induced map of complexes $\beta \otimes 1 : F^\bullet \otimes _ R^\mathbf {L} \kappa \to G^\bullet \otimes _ R^\mathbf {L} \kappa $ must be an isomorphism of complexes as the differentials in $F^\bullet \otimes _ R^\mathbf {L} \kappa $ and $G^\bullet \otimes _ R^\mathbf {L} \kappa $ are zero. Thus $\beta ^ i : F^ i \to G^ i$ is a map of finite free $R$-modules whose reduction modulo $\mathfrak m$ is an isomorphism. Hence $\beta ^ i$ is an isomorphism and we win. $\square$

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