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Let $X\to S$ be a flat morphism of noetherian schemes. I know that I can check smoothness on the geometric fibers to see whether $X\to S$ is smooth.

Let $T\to S$ be a surjective morphism. Under what conditions can I check smoothness of $X\to S$ by checking the smoothness of $X\times_S T\to T$?

For instance, is "finite flat surjective" allowed? Is this what we call fppf descent?

Do I need $X\to S$ to be flat for this to work?

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    Smoothness is a fpqc-local property: see Lemma 02VL in the Stacks project. So yes, you can check smoothness by taking a surjective fppf base change. – Zhen Lin Sep 02 '13 at 09:58

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Descent is an extension of the idea of local properties of morphisms (or schemes).

A property $\mathcal{P}$ of a morphism $f:X \to S$ is local if it can be checked locally i.e. if $\{U_i\}$ is an open covering of $S$ and for each $i$ the induced map $f_i:f^{-1}(U_i) \to U_i$ has property $\mathcal{P}$ then $f$ has property $\mathcal{P}$. Note that this can be rephrased as: if $\{U_i \to S \}$ is a family of open immersions s.t. $\cup U_i=S$ and for each $i$ the base change $f_i:X \times_S U_i \to U_i$ has property $\mathcal{P}$ then $f$ has property $\mathcal{P}$. We can also replace the family $\{U_i \to S \}$ with the single covering map $U=\coprod U_i \to S$, which is open and surjective.

Descent is the same notion applied to a different topology on category of schemes, that is allowing other coverings such as etale/fppf/fpqc coverings.

Now, the property "smoothness" descends through fppf coverings. So if $T \to S$ is a fppf covering map i.e. flat, locally of finite presentation and surjective and $X \times_S T \to T$ is smooth then so is $X \to S$.

Uri Brezner
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