# Flatness of the canonical projections

Given two smooth projective schemes $X$ and $Y$ over some algebraically closed fields $k$, one has the product $X\times Y$ with the projections $\pi_X$ and $\pi_Y$.

Now i have a coherent sheaf $M$ on $X$ and a coherent sheaf $N$ on $Y$, and i have locally free resolutions $M_{\*}\rightarrow M$ of length m and $N_{\*}\rightarrow N$ of length n for cohomological computations. But i want to work with $\pi_X^{\*}M$ and $\pi_Y^{\*}N$ on $X\times Y$. The easiest way would be if i could pullback the resolutions too, so i don't have to get new resolutions on the product. But as the pullback is just right exact in general, one needs flatness of the projections for this.

So i tried to see if the projectiosn are flat. But for example $\pi_X$ is given by the base change of $Y\rightarrow Spec(k)$ by $X\rightarrow Spec(k)$. Since $Y\rightarrow Spec(k)$ is flat, and flatness is preserved under base change one has that $\pi_X$ is flat. Could it be that easy?

## 1 Answer

Yes. If $f: X\to S$ is a flat morphism and $S'\to S$ is an arbitrary morphism, then the projection $f_{S'}: X\times_S S'\to S'$ is flat. A reference is EGA IV.2.1.4.

• Thanks for the confirmation and the reference. I often find myself thinking: This should be in EGA, but i just can't find it. – TonyS Nov 4 '10 at 15:59
• This is also in Hartshorne, III.9.2 (but really it's just the statement that if M a flat R-module, and R --> S is a ring map, then M \times S is a flat S module). – mdeland Nov 4 '10 at 16:50