# finite homological dimension

Hi, I found the following in the proof of a theorem: $Z \subset Y \times M$ where $M$ is a smooth projective variety over $\mathbb{C}$, $Y$ is a scheme and $Z$ is a subscheme of the product, flat over $Y$. Then $\mathcal{O}_Z$ is of finite homological dimension (is quasi-isomorphic to a bounded complex of locally free sheaves), as an $\mathcal{O}_{ Y \times M}$-module. Why? (all schemes are of finite type over $\mathbb{C}$) Thank you.

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This is proved in SGA 6; the main part is in Expos'e I, Corollaire 3.5(b) and the following Example 3.6 (2nd item). –  Jason Starr Jun 26 '12 at 12:57
Also, probably you should see Cor.4.14 / Def. 4.14.1 and Cor. 5.8.1. Flatness implies finite Tor dimension. –  Jason Starr Jun 26 '12 at 13:02
thank you, Jason :) –  emmy Jul 14 '12 at 8:49