The Stacks project

40.8 Cohen-Macaulay presentations

Given any groupoid $(U, R, s, t, c)$ with $s, t$ flat and locally of finite presentation there exists an “equivalent” groupoid $(U', R', s', t', c')$ such that $s'$ and $t'$ are Cohen-Macaulay morphisms (and locally of finite presentation). See More on Morphisms, Section 37.22 for more information on Cohen-Macaulay morphisms. Here “equivalent” can be taken to mean that the quotient stacks $[U/R]$ and $[U'/R']$ are equivalent stacks, see Groupoids in Spaces, Section 78.20 and Section 78.25.

Lemma 40.8.1. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid over $S$. Assume $s$ and $t$ are flat and locally of finite presentation. Then there exists an open $U' \subset U$ such that

  1. $t^{-1}(U') \subset R$ is the largest open subscheme of $R$ on which the morphism $s$ is Cohen-Macaulay,

  2. $s^{-1}(U') \subset R$ is the largest open subscheme of $R$ on which the morphism $t$ is Cohen-Macaulay,

  3. the morphism $t|_{s^{-1}(U')} : s^{-1}(U') \to U$ is surjective,

  4. the morphism $s|_{t^{-1}(U')} : t^{-1}(U') \to U$ is surjective, and

  5. the restriction $R' = s^{-1}(U') \cap t^{-1}(U')$ of $R$ to $U'$ defines a groupoid $(U', R', s', t', c')$ which has the property that the morphisms $s'$ and $t'$ are Cohen-Macaulay and locally of finite presentation.

Proof. Apply Lemma 40.6.1 with $g = \text{id}$ and $\mathcal{Q} =$“locally of finite presentation”, $\mathcal{R} =$“flat and locally of finite presentation”, and $\mathcal{P}=$“Cohen-Macaulay”, see Remark 40.6.3. This gives us an open $U' \subset U$ such that Let $t^{-1}(U') \subset R$ is the largest open subscheme of $R$ on which the morphism $s$ is Cohen-Macaulay. This proves (1). Let $i : R \to R$ be the inverse of the groupoid. Since $i$ is an isomorphism, and $s \circ i = t$ and $t \circ i = s$ we see that $s^{-1}(U')$ is also the largest open of $R$ on which $t$ is Cohen-Macaulay. This proves (2). By More on Morphisms, Lemma 37.22.7 the open subset $t^{-1}(U')$ is dense in every fibre of $s : R \to U$. This proves (3). Same argument for (4). Part (5) is a formal consequence of (1) and (2) and the discussion of restrictions in Groupoids, Section 39.18. $\square$


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Your email address will not be published.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 04LK. Beware of the difference between the letter 'O' and the digit '0'.