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Suppose $X$ and $Y$ are both regular schemes and the morphism $X \to Y$ is a local complete intersection (meaning it is a composition of a regular immersion and a smooth morphism, but in general not flat, e.g. $\mathbb{A}^2$ blowup the origin). Are there any conditions which could ensure that the morphism $X \times_Y X \to X$ (second projection) is a local complete intersection again?

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Just to clarify, are you looking for conditions that are weaker than flatness? The property of being both flat and local complete intersection is stable under arbitrary base change. –  Jason Starr Mar 26 '13 at 12:30
    
Problem solved. Thanks! –  marker Mar 26 '13 at 23:48

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