Lemma 76.27.3. Let S be a scheme. Let f : X \to Y be a morphism of algebraic spaces over S. Assume the fibres of f are locally Noetherian. The following are equivalent
f is Gorenstein,
f is flat and for some surjective étale morphism V \to Y where V is a scheme, the fibres of X_ V \to V are Gorenstein algebraic spaces, and
f is flat and for any étale morphism V \to Y where V is a scheme, the fibres of X_ V \to V are Gorenstein algebraic spaces.
Given x \in |X| with image y \in |Y| the following are equivalent
f is Gorenstein at x, and
\mathcal{O}_{Y, \overline{y}} \to \mathcal{O}_{X, \overline{x}} is flat and \mathcal{O}_{X, \overline{x}}/ \mathfrak m_{\overline{y}}\mathcal{O}_{X, \overline{x}} is Gorenstein.
Proof.
Given an étale morphism V \to Y where V is a scheme choose a scheme U and a surjective étale morphism U \to X \times _ Y V. Consider the commutative diagram
\xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y }
Let u \in U with images x \in |X|, y \in |Y|, and v \in V. Then f is Gorenstein at x if and only if U \to V is Gorenstein at u (by definition). Moreover the morphism U_ v \to X_ v = (X_ V)_ v is surjective étale. Hence the scheme U_ v is Gorenstein if and only if the algebraic space X_ v is Gorenstein. Thus the equivalence of (1), (2), and (3) follows from the corresponding equivalence for morphisms of schemes, see Duality for Schemes, Lemma 48.24.4 by a formal argument.
Proof of equivalence of (a) and (b). The corresponding equivalence for flatness is Morphisms of Spaces, Lemma 67.30.8. Thus we may assume f is flat at x when proving the equivalence. Consider a diagram and x, y, u, v as above. Then \mathcal{O}_{Y, \overline{y}} \to \mathcal{O}_{X, \overline{x}} is equal to the map \mathcal{O}_{V, v}^{sh} \to \mathcal{O}_{U, u}^{sh} on strict henselizations of local rings, see Properties of Spaces, Lemma 66.22.1. Thus we have
\mathcal{O}_{X, \overline{x}}/ \mathfrak m_{\overline{y}}\mathcal{O}_{X, \overline{x}} = (\mathcal{O}_{U, u}/\mathfrak m_ v \mathcal{O}_{U, u})^{sh}
by Algebra, Lemma 10.156.4. Thus we have to show that the Noetherian local ring \mathcal{O}_{U, u}/\mathfrak m_ v \mathcal{O}_{U, u} is Gorenstein if and only if its strict henselization is. This follows immediately from Dualizing Complexes, Lemma 47.22.3 and the definition of a Gorenstein local ring as a Noetherian local ring which is a dualizing complex over itself.
\square
Comments (0)