The Stacks project

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 $g \circ f$ 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. $\square$

Comments (2)

Comment #7718 by Andrew on

There's a typo in (4) with .

Post a comment

Your email address will not be published.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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0C11. Beware of the difference between the letter 'O' and the digit '0'.