Lemma 15.77.4 (0G9A)—The Stacks project

Table of contents
Part 1: Preliminaries
Chapter 15: More on Algebra
Section 15.77: Recognizing perfect complexes
Lemma 15.77.4 (cite)

Lemma 15.77.4. Let $R$ be a ring. Let $K$ be a pseudo-coherent object of $D(R)$. Let $a, b \in \mathbf{Z}$. The following are equivalent

$K$ has projective-amplitude in $[a, b]$,
$K$ is perfect of tor-amplitude in $[a, b]$,
$\mathop{\mathrm{Ext}}\nolimits ^ i_ R(K, N) = 0$ for all finitely presented $R$-modules $N$ and all $i \not\in [-b, -a]$,
$H^ n(K) = 0$ for $n > b$ and $\mathop{\mathrm{Ext}}\nolimits ^ i_ R(K, N) = 0$ for all finitely presented $R$-modules $N$ and all $i > -a$, and
$H^ n(K) = 0$ for $n \not\in [a - 1, b]$ and $\mathop{\mathrm{Ext}}\nolimits ^{-a + 1}_ R(K, N) = 0$ for all finitely presented $R$-modules $N$.

Proof. From the final statement of Lemma 15.74.2 we see that (2) implies (1). If (1) holds, then $K$ can be represented by a complex of projective modules $P^ i$ with $P^ i = 0$ for $i \not\in [a, b]$. Since projective modules are flat (as summands of free modules), we see that $K$ has tor-amplitude in $[a, b]$, see Lemma 15.66.3. Thus by Lemma 15.74.2 we see that (2) holds.

In conditions (3), (4), (5) the assumed vanishing of ext groups $\mathop{\mathrm{Ext}}\nolimits ^ i_ R(K, M)$ for $M$ of finite presentation is equivalent to the vanishing for all $R$-modules $M$ by Lemma 15.65.1 and Algebra, Lemma 10.11.3. Thus the equivalence of (1), (3), (4), and (5) follows from Lemma 15.68.2. $\square$

The Stacks project

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