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 $f$.