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
t^{-1}(U') \subset R is the largest open subscheme of R on which the morphism s is Cohen-Macaulay,
s^{-1}(U') \subset R is the largest open subscheme of R on which the morphism t is Cohen-Macaulay,
the morphism t|_{s^{-1}(U')} : s^{-1}(U') \to U is surjective,
the morphism s|_{t^{-1}(U')} : t^{-1}(U') \to U is surjective, and
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
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