# SGA4 Exposé IX 2.14 - Isotrivialization of a constructible sheaf on a stratification

In SGA4 Exposé IX 2.14, there is a statement that given a constructible sheaf $$F$$ on a qcqs scheme $$X$$, not only can we choose a constructible stratification $$\{X_i\to X\}$$ of $$X$$ such that the pullback of $$F$$ to each $$X_i$$ is locally constant constructible, we can in fact choose this stratification to be isotrivial, that is, such that there exists a finite étale cover $$U_i\to X_i$$ with the property that the pullback of $$F$$ to each $$U_i$$ is a constant sheaf.

The reference for this is IX.2.8.1, which tells us that a for a surjective locally finite presentation map of schemes $$Y\to X$$, there exists a constructible stratification $$\{X_i\to X\}$$ together with a finite étale map $$U_i\to X_i$$ and a finite locally free radiciel map $$V_i\to U_i$$ such that the pullback of $$Y$$ to each $$V_i$$ admits a section.

Since $$F\to X$$ is a finite presentation étale algebraic space over $$X$$, we can cover it by some étale affine $$F^\prime \to F$$, we can apply the theorem to show that the data above exist to construct sections, and since the map $$V_i\to U_i$$ is a universal homeomorphism, sections over $$V_i$$ and $$U_i$$ will be the same, so we can just consider the case where we have the data $$U_i\to X_i\to X$$.

But I'm not clear on how to show that this proves the claim. What I tried was some argument approximating the base by Noetherian schemes and finding some étale sheaf over some index that pulls back to $$F$$ on $$X$$. Moreover, using the stratification of a constructible sheaf, you can assume it's locally constant constructible and with all fibres nonempty. Then in the Noetherian case, you apply induction on the size of the fibres by finding a global section over some finite étale $$U\to X$$, then taking the subsheaf complementary to the global section over $$U$$ (which is still lcc) and applying the inductive hypothesis.

Is this the 'easy argument' that the text references, or is there something easier?

• Hmm, my guess is that to go from locally constant to constant, one should apply 2.8.1 to a cover $X'_i \to X_i$ over which $\mathscr F$ becomes constant. After refining the stratification $\{X_i \to X\}$, this gives surjective morphisms $$Y''_i \stackrel{g_i}\to Y'_i \stackrel{h_i}\to X_i$$ such that $g_i$ is finite radicial and $h_i$ finite étale and $\mathscr F|_{Y''_i}$ is constant. Then VIII.1.1 tells you that $\mathscr F|_{Y'_i}$ is already constant. Nov 6, 2020 at 22:20
• @R.vanDobbendeBruyn Ah, great. That seems to work perfectly! You should answer the question so I can accept your answer! Nov 7, 2020 at 0:57