## 47.23 Formal fibres

This section is a continuation of More on Algebra, Section 15.51. There we saw there is a (fairly) good theory of Noetherian rings $A$ whose local rings have Cohen-Macaulay formal fibres. Namely, we proved (1) it suffices to check the formal fibres of localizations at maximal ideals are Cohen-Macaulay, (2) the property is inherited by rings of finite type over $A$, (3) the fibres of $A \to A^\wedge$ are Cohen-Macaulay for any completion $A^\wedge$ of $A$, and (4) the property is inherited by henselizations of $A$. See More on Algebra, Lemma 15.51.4, Proposition 15.51.5, Lemma 15.51.6, and Lemma 15.51.7. Similarly, for Noetherian rings whose local rings have formal fibres which are geometrically reduced, geometrically normal, $(S_ n)$, and geometrically $(R_ n)$. In this section we will see that the same is true for Noetherian rings whose local rings have formal fibres which are Gorenstein or local complete intersections. This is relevant to this chapter because a Noetherian ring which has a dualizing complex is an example.

Lemma 47.23.1. Properties (A), (B), (C), (D), and (E) of More on Algebra, Section 15.51 hold for $P(k \to R) =$“$R$ is a Gorenstein ring”.

Proof. Since we already know the result holds for Cohen-Macaulay instead of Gorenstein, we may in each step assume the ring we have is Cohen-Macaulay. This is not particularly helpful for the proof, but psychologically may be useful.

Part (A). Let $K/k$ be a finitely generated field extension. Let $R$ be a Gorenstein $k$-algebra. We can find a global complete intersection $A = k[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c)$ over $k$ such that $K$ is isomorphic to the fraction field of $A$, see Algebra, Lemma 10.158.11. Then $R \to R \otimes _ k A$ is a relative global complete intersection. Hence $R \otimes _ k A$ is Gorenstein by Lemma 47.21.7. Thus $R \otimes _ k K$ is too as a localization.

Proof of (B). This is clear because a ring is Gorenstein if and only if all of its local rings are Gorenstein.

Part (C). Let $A \to B \to C$ be flat maps of Noetherian rings. Assume the fibres of $A \to B$ are Gorenstein and $B \to C$ is regular. We have to show the fibres of $A \to C$ are Gorenstein. Clearly, we may assume $A = k$ is a field. Then we may assume that $B \to C$ is a regular local homomorphism of Noetherian local rings. Then $B$ is Gorenstein and $C/\mathfrak m_ B C$ is regular, in particular Gorenstein (Lemma 47.21.3). Then $C$ is Gorenstein by Lemma 47.21.8.

Part (D). This follows from Lemma 47.21.8. Part (E) is immediate as the condition does not refer to the ground field. $\square$

Lemma 47.23.2. Let $A$ be a Noetherian local ring. If $A$ has a dualizing complex, then the formal fibres of $A$ are Gorenstein.

Proof. Let $\mathfrak p$ be a prime of $A$. The formal fibre of $A$ at $\mathfrak p$ is isomorphic to the formal fibre of $A/\mathfrak p$ at $(0)$. The quotient $A/\mathfrak p$ has a dualizing complex (Lemma 47.15.9). Thus it suffices to check the statement when $A$ is a local domain and $\mathfrak p = (0)$. Let $\omega _ A^\bullet$ be a dualizing complex for $A$. Then $\omega _ A^\bullet \otimes _ A A^\wedge$ is a dualizing complex for the completion $A^\wedge$ (Lemma 47.22.1). Then $\omega _ A^\bullet \otimes _ A K$ is a dualizing complex for the fraction field $K$ of $A$ (Lemma 47.15.6). Hence $\omega _ A^\bullet \otimes _ A K$ is isomorphic to $K[n]$ for some $n \in \mathbf{Z}$. Similarly, we conclude a dualizing complex for the formal fibre $A^\wedge \otimes _ A K$ is

$\omega _ A^\bullet \otimes _ A A^\wedge \otimes _{A^\wedge } (A^\wedge \otimes _ A K) = (\omega _ A^\bullet \otimes _ A K) \otimes _ K (A^\wedge \otimes _ A K) \cong (A^\wedge \otimes _ A K)[n]$

as desired. $\square$

Here is the verification promised in Divided Power Algebra, Remark 23.9.3.

Lemma 47.23.3. Properties (A), (B), (C), (D), and (E) of More on Algebra, Section 15.51 hold for $P(k \to R) =$“$R$ is a local complete intersection”. See Divided Power Algebra, Definition 23.8.5.

Proof. Part (A). Let $K/k$ be a finitely generated field extension. Let $R$ be a $k$-algebra which is a local complete intersection. We can find a global complete intersection $A = k[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c)$ over $k$ such that $K$ is isomorphic to the fraction field of $A$, see Algebra, Lemma 10.158.11. Then $R \to R \otimes _ k A$ is a relative global complete intersection. It follows that $R \otimes _ k A$ is a local complete intersection by Divided Power Algebra, Lemma 23.8.9.

Proof of (B). This is clear because a ring is a local complete intersection if and only if all of its local rings are complete intersections.

Part (C). Let $A \to B \to C$ be flat maps of Noetherian rings. Assume the fibres of $A \to B$ are local complete intersections and $B \to C$ is regular. We have to show the fibres of $A \to C$ are local complete intersections. Clearly, we may assume $A = k$ is a field. Then we may assume that $B \to C$ is a regular local homomorphism of Noetherian local rings. Then $B$ is a complete intersection and $C/\mathfrak m_ B C$ is regular, in particular a complete intersection (by definition). Then $C$ is a complete intersection by Divided Power Algebra, Lemma 23.8.9.

Part (D). This follows by the same arguments as in (C) from the other implication in Divided Power Algebra, Lemma 23.8.9. Part (E) is immediate as the condition does not refer to the ground field. $\square$

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