Lemma 47.21.8. Let $A \to B$ be a flat local homomorphism of Noetherian local rings. The following are equivalent

$B$ is Gorenstein, and

$A$ and $B/\mathfrak m_ A B$ are Gorenstein.

Lemma 47.21.8. Let $A \to B$ be a flat local homomorphism of Noetherian local rings. The following are equivalent

$B$ is Gorenstein, and

$A$ and $B/\mathfrak m_ A B$ are Gorenstein.

**Proof.**
Below we will use without further mention that a local Gorenstein ring has finite injective dimension as well as Lemma 47.21.5. By More on Algebra, Remark 15.62.21 we have

\[ \mathop{\mathrm{Ext}}\nolimits ^ i_ A(\kappa _ A, A) \otimes _ A B = \mathop{\mathrm{Ext}}\nolimits ^ i_ B(B/\mathfrak m_ A B, B) \]

for all $i$.

Assume (2). Using that $R\mathop{\mathrm{Hom}}\nolimits (B/\mathfrak m_ A B, -) : D(B) \to D(B/\mathfrak m_ A B)$ is a right adjoint to restriction (Lemma 47.13.1) we obtain

\[ R\mathop{\mathrm{Hom}}\nolimits _ B(\kappa _ B, B) = R\mathop{\mathrm{Hom}}\nolimits _{B/\mathfrak m_ A B}(\kappa _ B, R\mathop{\mathrm{Hom}}\nolimits (B/\mathfrak m_ A B, B)) \]

The cohomology modules of $R\mathop{\mathrm{Hom}}\nolimits (B/\mathfrak m_ A B, B)$ are the modules $\mathop{\mathrm{Ext}}\nolimits ^ i_ B(B/\mathfrak m_ A B, B) = \mathop{\mathrm{Ext}}\nolimits ^ i_ A(\kappa _ A, A) \otimes _ A B$. Since $A$ is Gorenstein, we conclude only a finite number of these are nonzero and each is isomorphic to a direct sum of copies of $B/\mathfrak m_ A B$. Hence since $B/\mathfrak m_ A B$ is Gorenstein we conclude that $R\mathop{\mathrm{Hom}}\nolimits _ B(B/\mathfrak m_ B, B)$ has only a finite number of nonzero cohomology modules. Hence $B$ is Gorenstein.

Assume (1). Since $B$ has finite injective dimension, $\mathop{\mathrm{Ext}}\nolimits ^ i_ B(B/\mathfrak m_ A B, B)$ is $0$ for $i \gg 0$. Since $A \to B$ is faithfully flat we conclude that $\mathop{\mathrm{Ext}}\nolimits ^ i_ A(\kappa _ A, A)$ is $0$ for $i \gg 0$. We conclude that $A$ is Gorenstein. This implies that $\mathop{\mathrm{Ext}}\nolimits ^ i_ A(\kappa _ A, A)$ is nonzero for exactly one $i$, namely for $i = \dim (A)$, and $\mathop{\mathrm{Ext}}\nolimits ^{\dim (A)}_ A(\kappa _ A, A) \cong \kappa _ A$ (see Lemmas 47.16.1, 47.20.2, and 47.21.2). Thus we see that $\mathop{\mathrm{Ext}}\nolimits ^ i_ B(B/\mathfrak m_ A B, B)$ is zero except for one $i$, namely $i = \dim (A)$ and $\mathop{\mathrm{Ext}}\nolimits ^{\dim (A)}_ B(B/\mathfrak m_ A B, B) \cong B/\mathfrak m_ A B$. Thus $B/\mathfrak m_ A B$ is Gorenstein by Lemma 47.16.1. $\square$

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)