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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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  • $\begingroup$ 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... $\endgroup$
    – kaddar
    Commented Jun 18, 2010 at 7:17

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