48.25 Gorenstein morphisms

This section is one in a series. The corresponding sections for normal morphisms, regular morphisms, and Cohen-Macaulay morphisms can be found in More on Morphisms, Sections 37.19, 37.20, and 37.21.

The following lemma says that it does not make sense to define geometrically Gorenstein schemes, since these would be the same as Gorenstein schemes.

Lemma 48.25.1. Let $X$ be a locally Noetherian scheme over the field $k$. Let $k'/k$ be a finitely generated field extension. Let $x \in X$ be a point, and let $x' \in X_{k'}$ be a point lying over $x$. Then we have

$\mathcal{O}_{X, x}\text{ is Gorenstein} \Leftrightarrow \mathcal{O}_{X_{k'}, x'}\text{ is Gorenstein}$

If $X$ is locally of finite type over $k$, the same holds for any field extension $k'/k$.

Proof. In both cases the ring map $\mathcal{O}_{X, x} \to \mathcal{O}_{X_{k'}, x'}$ is a faithfully flat local homomorphism of Noetherian local rings. Thus if $\mathcal{O}_{X_{k'}, x'}$ is Gorenstein, then so is $\mathcal{O}_{X, x}$ by Dualizing Complexes, Lemma 47.21.8. To go up, we use Dualizing Complexes, Lemma 47.21.8 as well. Thus we have to show that

$\mathcal{O}_{X_{k'}, x'}/\mathfrak m_ x \mathcal{O}_{X_{k'}, x'} = \kappa (x) \otimes _ k k'$

is Gorenstein. Note that in the first case $k \to k'$ is finitely generated and in the second case $k \to \kappa (x)$ is finitely generated. Hence this follows as property (A) holds for Gorenstein, see Dualizing Complexes, Lemma 47.23.1. $\square$

The lemma above guarantees that the following is the correct definition of Gorenstein morphisms.

Definition 48.25.2. Let $f : X \to Y$ be a morphism of schemes. Assume that all the fibres $X_ y$ are locally Noetherian schemes.

1. Let $x \in X$, and $y = f(x)$. We say that $f$ is Gorenstein at $x$ if $f$ is flat at $x$, and the local ring of the scheme $X_ y$ at $x$ is Gorenstein.

2. We say $f$ is a Gorenstein morphism if $f$ is Gorenstein at every point of $X$.

Here is a translation.

Lemma 48.25.3. Let $f : X \to Y$ be a morphism of schemes. Assume all fibres of $f$ are locally Noetherian. The following are equivalent

1. $f$ is Gorenstein, and

2. $f$ is flat and its fibres are Gorenstein schemes.

Proof. This follows directly from the definitions. $\square$

Proof. Follows from Lemma 48.24.2 and the definitions. $\square$

Lemma 48.25.5. A syntomic morphism is Gorenstein. Equivalently a flat local complete intersection morphism is Gorenstein.

Proof. Recall that a syntomic morphism is flat and its fibres are local complete intersections over fields, see Morphisms, Lemma 29.30.11. Since a local complete intersection over a field is a Gorenstein scheme by Lemma 48.24.5 we conclude. The properties “syntomic” and “flat and local complete intersection morphism” are equivalent by More on Morphisms, Lemma 37.59.8. $\square$

Lemma 48.25.6. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms. Assume that the fibres $X_ y$, $Y_ z$ and $X_ z$ of $f$, $g$, and $g \circ f$ are locally Noetherian.

1. If $f$ is Gorenstein at $x$ and $g$ is Gorenstein at $f(x)$, then $g \circ f$ is Gorenstein at $x$.

2. If $f$ and $g$ are Gorenstein, then $g \circ f$ is Gorenstein.

3. If $g \circ f$ is Gorenstein at $x$ and $f$ is flat at $x$, then $f$ is Gorenstein at $x$ and $g$ is Gorenstein at $f(x)$.

4. If $f \circ g$ is Gorenstein and $f$ is flat, then $f$ is Gorenstein and $g$ is Gorenstein at every point in the image of $f$.

Proof. After translating into algebra this follows from Dualizing Complexes, Lemma 47.21.8. $\square$

Lemma 48.25.7. Let $f : X \to Y$ be a flat morphism of locally Noetherian schemes. If $X$ is Gorenstein, then $f$ is Gorenstein and $\mathcal{O}_{Y, f(x)}$ is Gorenstein for all $x \in X$.

Proof. After translating into algebra this follows from Dualizing Complexes, Lemma 47.21.8. $\square$

Lemma 48.25.8. Let $f : X \to Y$ be a morphism of schemes. Assume that all the fibres $X_ y$ are locally Noetherian schemes. Let $Y' \to Y$ be locally of finite type. Let $f' : X' \to Y'$ be the base change of $f$. Let $x' \in X'$ be a point with image $x \in X$.

1. If $f$ is Gorenstein at $x$, then $f' : X' \to Y'$ is Gorenstein at $x'$.

2. If $f$ is flat and $x$ and $f'$ is Gorenstein at $x'$, then $f$ is Gorenstein at $x$.

3. If $Y' \to Y$ is flat at $f'(x')$ and $f'$ is Gorenstein at $x'$, then $f$ is Gorenstein at $x$.

Proof. Note that the assumption on $Y' \to Y$ implies that for $y' \in Y'$ mapping to $y \in Y$ the field extension $\kappa (y')/\kappa (y)$ is finitely generated. Hence also all the fibres $X'_{y'} = (X_ y)_{\kappa (y')}$ are locally Noetherian, see Varieties, Lemma 33.11.1. Thus the lemma makes sense. Set $y' = f'(x')$ and $y = f(x)$. Hence we get the following commutative diagram of local rings

$\xymatrix{ \mathcal{O}_{X', x'} & \mathcal{O}_{X, x} \ar[l] \\ \mathcal{O}_{Y', y'} \ar[u] & \mathcal{O}_{Y, y} \ar[l] \ar[u] }$

where the upper left corner is a localization of the tensor product of the upper right and lower left corners over the lower right corner.

Assume $f$ is Gorenstein at $x$. The flatness of $\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}$ implies the flatness of $\mathcal{O}_{Y', y'} \to \mathcal{O}_{X', x'}$, see Algebra, Lemma 10.100.1. The fact that $\mathcal{O}_{X, x}/\mathfrak m_ y\mathcal{O}_{X, x}$ is Gorenstein implies that $\mathcal{O}_{X', x'}/\mathfrak m_{y'}\mathcal{O}_{X', x'}$ is Gorenstein, see Lemma 48.25.1. Hence we see that $f'$ is Gorenstein at $x'$.

Assume $f$ is flat at $x$ and $f'$ is Gorenstein at $x'$. The fact that $\mathcal{O}_{X', x'}/\mathfrak m_{y'}\mathcal{O}_{X', x'}$ is Gorenstein implies that $\mathcal{O}_{X, x}/\mathfrak m_ y\mathcal{O}_{X, x}$ is Gorenstein, see Lemma 48.25.1. Hence we see that $f$ is Gorenstein at $x$.

Assume $Y' \to Y$ is flat at $y'$ and $f'$ is Gorenstein at $x'$. The flatness of $\mathcal{O}_{Y', y'} \to \mathcal{O}_{X', x'}$ and $\mathcal{O}_{Y, y} \to \mathcal{O}_{Y', y'}$ implies the flatness of $\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}$, see Algebra, Lemma 10.100.1. The fact that $\mathcal{O}_{X', x'}/\mathfrak m_{y'}\mathcal{O}_{X', x'}$ is Gorenstein implies that $\mathcal{O}_{X, x}/\mathfrak m_ y\mathcal{O}_{X, x}$ is Gorenstein, see Lemma 48.25.1. Hence we see that $f$ is Gorenstein at $x$. $\square$

Lemma 48.25.9. Let $f : X \to Y$ be a morphism of schemes which is flat and locally of finite type. Then formation of the set $\{ x \in X \mid f\text{ is Gorenstein at }x\}$ commutes with arbitrary base change.

Proof. The assumption implies any fibre of $f$ is locally of finite type over a field and hence locally Noetherian and the same is true for any base change. Thus the statement makes sense. Looking at fibres we reduce to the following problem: let $X$ be a scheme locally of finite type over a field $k$, let $K/k$ be a field extension, and let $x_ K \in X_ K$ be a point with image $x \in X$. Problem: show that $\mathcal{O}_{X_ K, x_ K}$ is Gorenstein if and only if $\mathcal{O}_{X, x}$ is Gorenstein.

The problem can be solved using a bit of algebra as follows. Choose an affine open $\mathop{\mathrm{Spec}}(A) \subset X$ containing $x$. Say $x$ corresponds to $\mathfrak p \subset A$. With $A_ K = A \otimes _ k K$ we see that $\mathop{\mathrm{Spec}}(A_ K) \subset X_ K$ contains $x_ K$. Say $x_ K$ corresponds to $\mathfrak p_ K \subset A_ K$. Let $\omega _ A^\bullet$ be a dualizing complex for $A$. By Dualizing Complexes, Lemma 47.25.3 $\omega _ A^\bullet \otimes _ A A_ K$ is a dualizing complex for $A_ K$. Now we are done because $A_\mathfrak p \to (A_ K)_{\mathfrak p_ K}$ is a flat local homomorphism of Noetherian rings and hence $(\omega _ A^\bullet )_\mathfrak p$ is an invertible object of $D(A_\mathfrak p)$ if and only if $(\omega _ A^\bullet )_\mathfrak p \otimes _{A_\mathfrak p} (A_ K)_{\mathfrak p_ K}$ is an invertible object of $D((A_ K)_{\mathfrak p_ K})$. Some details omitted; hint: look at cohomology modules. $\square$

Lemma 48.25.10. In Situation 48.16.1 let $f : X \to Y$ be a morphism of $\textit{FTS}_ S$. Let $x \in X$. If $f$ is flat, then the following are equivalent

1. $f$ is Gorenstein at $x$,

2. $f^!\mathcal{O}_ Y$ is isomorphic to an invertible object in a neighbourhood of $x$.

In particular, the set of points where $f$ is Gorenstein is open in $X$.

Proof. Set $\omega ^\bullet = f^!\mathcal{O}_ Y$. By Lemma 48.18.4 this is a bounded complex with coherent cohomology sheaves whose derived restriction $Lh^*\omega ^\bullet$ to the fibre $X_ y$ is a dualizing complex on $X_ y$. Denote $i : x \to X_ y$ the inclusion of a point. Then the following are equivalent

1. $f$ is Gorenstein at $x$,

2. $\mathcal{O}_{X_ y, x}$ is Gorenstein,

3. $Lh^*\omega ^\bullet$ is invertible in a neighbourhood of $x$,

4. $Li^* Lh^* \omega ^\bullet$ has exactly one nonzero cohomology of dimension $1$ over $\kappa (x)$,

5. $L(h \circ i)^* \omega ^\bullet$ has exactly one nonzero cohomology of dimension $1$ over $\kappa (x)$,

6. $\omega ^\bullet$ is invertible in a neighbourhood of $x$.

The equivalence of (1) and (2) is by definition (as $f$ is flat). The equivalence of (2) and (3) follows from Lemma 48.24.4. The equivalence of (3) and (4) follows from More on Algebra, Lemma 15.77.1. The equivalence of (4) and (5) holds because $Li^* Lh^* = L(h \circ i)^*$. The equivalence of (5) and (6) holds by More on Algebra, Lemma 15.77.1. Thus the lemma is clear. $\square$

Lemma 48.25.11. Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation. Let $x \in X$ with image $s \in S$. Set $d = \dim _ x(X_ s)$. The following are equivalent

1. $f$ is Gorenstein at $x$,

2. there exists an open neighbourhood $U \subset X$ of $x$ and a locally quasi-finite morphism $U \to \mathbf{A}^ d_ S$ over $S$ which is Gorenstein at $x$,

3. there exists an open neighbourhood $U \subset X$ of $x$ and a locally quasi-finite Gorenstein morphism $U \to \mathbf{A}^ d_ S$ over $S$,

4. for any $S$-morphism $g : U \to \mathbf{A}^ d_ S$ of an open neighbourhood $U \subset X$ of $x$ we have: $g$ is quasi-finite at $x$ $\Rightarrow$ $g$ is Gorenstein at $x$.

In particular, the set of points where $f$ is Gorenstein is open in $X$.

Proof. Choose affine open $U = \mathop{\mathrm{Spec}}(A) \subset X$ with $x \in U$ and $V = \mathop{\mathrm{Spec}}(R) \subset S$ with $f(U) \subset V$. Then $R \to A$ is a flat ring map of finite presentation. Let $\mathfrak p \subset A$ be the prime ideal corresponding to $x$. After replacing $A$ by a principal localization we may assume there exists a quasi-finite map $R[x_1, \ldots , x_ d] \to A$, see Algebra, Lemma 10.125.2. Thus there exists at least one pair $(U, g)$ consisting of an open neighbourhood $U \subset X$ of $x$ and a locally1 quasi-finite morphism $g : U \to \mathbf{A}^ d_ S$.

Having said this, the lemma translates into the following algebra problem (translation omitted). Given $R \to A$ flat and of finite presentation, a prime $\mathfrak p \subset A$ and $\varphi : R[x_1, \ldots , x_ d] \to A$ quasi-finite at $\mathfrak p$ the following are equivalent

1. $\mathop{\mathrm{Spec}}(\varphi )$ is Gorenstein at $\mathfrak p$, and

2. $\mathop{\mathrm{Spec}}(A) \to \mathop{\mathrm{Spec}}(R)$ is Gorenstein at $\mathfrak p$.

3. $\mathop{\mathrm{Spec}}(A) \to \mathop{\mathrm{Spec}}(R)$ is Gorenstein in an open neighbourhood of $\mathfrak p$.

In each case $R[x_1, \ldots , x_ n] \to A$ is flat at $\mathfrak p$ hence by openness of flatness (Algebra, Theorem 10.129.4), we may assume $R[x_1, \ldots , x_ n] \to A$ is flat (replace $A$ by a suitable principal localization). By Algebra, Lemma 10.168.1 there exists $R_0 \subset R$ and $R_0[x_1, \ldots , x_ n] \to A_0$ such that $R_0$ is of finite type over $\mathbf{Z}$ and $R_0 \to A_0$ is of finite type and $R_0[x_1, \ldots , x_ n] \to A_0$ is flat. Note that the set of points where a flat finite type morphism is Gorenstein commutes with base change by Lemma 48.25.8. In this way we reduce to the case where $R$ is Noetherian.

Thus we may assume $X$ and $S$ affine and that we have a factorization of $f$ of the form

$X \xrightarrow {g} \mathbf{A}^ n_ S \xrightarrow {p} S$

with $g$ flat and quasi-finite and $S$ Noetherian. Then $X$ and $\mathbf{A}^ n_ S$ are separated over $S$ and we have

$f^!\mathcal{O}_ S = g^!p^!\mathcal{O}_ S = g^!\mathcal{O}_{\mathbf{A}^ n_ S}[n]$

by know properties of upper shriek functors (Lemmas 48.16.3 and 48.17.3). Hence the equivalence of (a), (b), and (c) by Lemma 48.25.10. $\square$

Lemma 48.25.12. The property $\mathcal{P}(f)=$“the fibres of $f$ are locally Noetherian and $f$ is Gorenstein” is local in the fppf topology on the target and local in the syntomic topology on the source.

Proof. We have $\mathcal{P}(f) = \mathcal{P}_1(f) \wedge \mathcal{P}_2(f)$ where $\mathcal{P}_1(f)=$“$f$ is flat”, and $\mathcal{P}_2(f)=$“the fibres of $f$ are locally Noetherian and Gorenstein”. We know that $\mathcal{P}_1$ is local in the fppf topology on the source and the target, see Descent, Lemmas 35.22.15 and 35.26.1. Thus we have to deal with $\mathcal{P}_2$.

Let $f : X \to Y$ be a morphism of schemes. Let $\{ \varphi _ i : Y_ i \to Y\} _{i \in I}$ be an fppf covering of $Y$. Denote $f_ i : X_ i \to Y_ i$ the base change of $f$ by $\varphi _ i$. Let $i \in I$ and let $y_ i \in Y_ i$ be a point. Set $y = \varphi _ i(y_ i)$. Note that

$X_{i, y_ i} = \mathop{\mathrm{Spec}}(\kappa (y_ i)) \times _{\mathop{\mathrm{Spec}}(\kappa (y))} X_ y.$

and that $\kappa (y_ i)/\kappa (y)$ is a finitely generated field extension. Hence if $X_ y$ is locally Noetherian, then $X_{i, y_ i}$ is locally Noetherian, see Varieties, Lemma 33.11.1. And if in addition $X_ y$ is Gorenstein, then $X_{i, y_ i}$ is Gorenstein, see Lemma 48.25.1. Thus $\mathcal{P}_2$ is fppf local on the target.

Let $\{ X_ i \to X\}$ be a syntomic covering of $X$. Let $y \in Y$. In this case $\{ X_{i, y} \to X_ y\}$ is a syntomic covering of the fibre. Hence the locality of $\mathcal{P}_2$ for the syntomic topology on the source follows from Lemma 48.24.6. $\square$

[1] If $S$ is quasi-separated, then $g$ will be quasi-finite.

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