The Stacks project

Lemma 15.123.4. Let $R$ be a ring. Let $M$ be an object of $D(R)$. The following are equivalent

  1. $M$ is invertible in $D(R)$, see Categories, Definition 4.42.4, and

  2. for every prime ideal $\mathfrak p \subset R$ there exists an $f \in R$, $f \not\in \mathfrak p$ such that $M_ f \cong R_ f[-n]$ for some $n \in \mathbf{Z}$.

Moreover, in this case

  1. $M$ is a perfect object of $D(R)$,

  2. $M = \bigoplus H^ n(M)[-n]$ in $D(R)$,

  3. each $H^ n(M)$ is a finite projective $R$-module,

  4. we can write $R = \prod _{a \leq n \leq b} R_ n$ such that $H^ n(M)$ corresponds to an invertible $R_ n$-module.

Proof. Assume (2). Consider the object $R\mathop{\mathrm{Hom}}\nolimits _ R(M, R)$ and the composition map

\[ R\mathop{\mathrm{Hom}}\nolimits (M, R) \otimes _ R^\mathbf {L} M \to R \]

Checking locally we see that this is an isomorphism; we omit the details. Because $D(R)$ is symmetric monoidal we see that $M$ is invertible.

Assume (1). Observe that an invertible object of a monoidal category has a left dual, namely, its inverse. Thus $M$ is perfect by Lemma 15.123.3. Consider a prime ideal $\mathfrak p \subset R$ with residue field $\kappa $. Then we see that $M \otimes _ R^\mathbf {L} \kappa $ is an invertible object of $D(\kappa )$. Clearly this implies that $\dim H^ i(M \otimes _ R^\mathbf {L} \kappa )$ is nonzero exactly for one $i$ and equal to $1$ in that case. By Lemma 15.74.6 this gives (2).

In the proof above we have seen that (a) holds. Let $U_ n \subset \mathop{\mathrm{Spec}}(R)$ be the union of the opens of the form $D(f)$ such that $M_ f \cong R_ f[-n]$. Clearly, $U_ n \cap U_{n'} = \emptyset $ if $n \not= n'$. If $M$ has tor amplitude in $[a, b]$, then $U_ n = \emptyset $ if $n \not\in [a, b]$. Hence we see that we have a product decomposition $R = \prod _{a \leq n \leq b} R_ n$ as in (d) such that $U_ n$ corresponds to $\mathop{\mathrm{Spec}}(R_ n)$, see Algebra, Lemma 10.24.3. Since $D(R) = \prod _{a \leq n \leq b} D(R_ n)$ and similary for the category of modules parts (b), (c), and (d) follow immediately. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0FNT. Beware of the difference between the letter 'O' and the digit '0'.