The Stacks project

Lemma 15.79.2. Let $R$ be a Noetherian ring. If $R$ is a strong generator for $D_{perf}(R)$, then $R$ is regular of finite dimension.

Proof. Assume $D_{perf}(R) = \langle R \rangle _ n$ for some $n \geq 1$. For any finite $R$-module $M$ we can choose a complex

\[ P = ( P^{-n - 1} \xrightarrow {d^{-n - 1}} P^{-n} \xrightarrow {d^{-n}} P^{-n + 1} \xrightarrow {d^1} \ldots \xrightarrow {d^{-1}} P^0) \]

of finite free $R$-modules with $H^ i(P) = 0$ for $i = -n, \ldots , - 1$ and $M \cong \mathop{\mathrm{Coker}}(d^{-1})$. Note that $P$ is in $D_{perf}(R)$. For any $R$-module $N$ we can compute $\mathop{\mathrm{Ext}}\nolimits ^ n_ R(M, N)$ the finite free resolution $P$ of $M$, see Algebra, Section 10.71 and compare with Derived Categories, Section 13.27. In particular, the sequence above defines an element

\[ \xi \in \mathop{\mathrm{Ext}}\nolimits ^ n_ R(\mathop{\mathrm{Coker}}(d^{-1}), \mathop{\mathrm{Coker}}(d^{-n - 1})) = \mathop{\mathrm{Ext}}\nolimits ^ n_ R(M, \mathop{\mathrm{Coker}}(d^{-n - 1})) \]

and for any element $\overline{\xi }$ in $\mathop{\mathrm{Ext}}\nolimits ^ n_ R(M, N)$ there is a $R$-module map $\varphi : \mathop{\mathrm{Coker}}(d^{-n - 1}) \to N$ such that $\varphi $ maps $\xi $ to $\overline{\xi }$. For $j = 1, \ldots , n - 1$ consider the complexes

\[ K_ j = (\mathop{\mathrm{Coker}}(d^{-n - 1}) \to P^{-n + 1} \to \ldots \to P^{-j}) \]

with $\mathop{\mathrm{Coker}}(d^{-n - 1})$ in degree $-n$ and $P^ t$ in degree $t$. We also set $K_ n = \mathop{\mathrm{Coker}}(d^{-n - 1})[n]$. Then we have maps

\[ P \to K_1 \to K_2 \to \ldots \to K_ n \]

which induce vanishing maps on cohomology. By Lemma 15.79.1 since $P \in D_{perf}(R) = \langle R \rangle _ n$ we find that the composition of this maps is zero in $D(R)$. Since $\mathop{\mathrm{Hom}}\nolimits _{D(R)}(P, K_ n) = \mathop{\mathrm{Hom}}\nolimits _{K(R)}(P, K_ n)$ by Derived Categories, Lemma 13.19.8 we conclude $\xi = 0$. Hence $\mathop{\mathrm{Ext}}\nolimits ^ n_ R(M, N) = 0$ for all $R$-modules $N$, see discussion above. It follows that $M$ has projective dimension $\leq n - 1$ by Algebra, Lemma 10.109.8. Since this holds for all finite $R$-modules $M$ we conclude that $R$ has finite global dimension, see Algebra, Lemma 10.109.12. We finally conclude by Algebra, Lemma 10.110.8. $\square$


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Your email address will not be published.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 0FXI. Beware of the difference between the letter 'O' and the digit '0'.