# finite tor dimension

Hi. Can, every one, give me an example of finite surjective morphism of finite tor dimension (but not flat!) between reduced schemes or complex analytic spaces... Thank you.

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Consider a smooth surface $Y$ with a point $p\in Y$. Let $X$ be obtained by gluing two copies of $Y$ at $p$, with the obvious morphism $X \to Y$. This is surjective and finite, and has finite Tor-dimension (because $Y$ is regular, hence every morphism to $Y$ has finite Tor dimension). However, it is not flat (for example, because $X$ is not Cohen-Macaulay).

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thank you very much Angelo. In fact, all flat and surjective morphims with no Cohen-Macaulay fibers gives, by finite projection or Noether quasi-normalization, finite tor dimension morphism... –  kaddar Jun 18 '10 at 7:17