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 37.20 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 76.19 and Section 76.24.

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 37.20.7 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. $\square$


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