5
$\begingroup$

Let $X$ be a topological space. A stratification of $X$ is a decomposition into a collection of subsets $X = \amalg_i \ X_i$ (called strata) such that $X_i$'s are open in its closure, and $\partial X_i = \overline{X_i} \setminus X_i$ is again a disjiont union of strata. A sheaf $\mathcal{F}$ on $X$ is constructible with respect to a stratification $\amalg \ X_i$ if $\mathcal{F}\big|_{X_i}$ are locally constant sheaves for all $i$.

I heard if the stratification satisfies some conditions, then we can calculate $H^i(X;\mathcal{F})$ by using the \v{C}ech complex of a certain open cover associated to the stratification. Does anyone know if there's reference about this? Or maybe the details are not hard?

By the way, I know there's probably a page in Stack Project about this. But I concern more about manifold so I'm looking for one with more topological setting.

Thanks.

$\endgroup$
  • $\begingroup$ I don't know such reference but the case of a locally constant sheaf already doesn't seems obvious to me. There are lot of computations in the paper " The decomposition theorem, perverse sheaves and the topology of algebraic maps", but sadly very few justifications. Maybe a toy example is worth working if you don't know it already, $X = \Bbb C^*$, $F$ is a local system with stalk $M$, i.e a representation $\rho : \Bbb Z \to GL(M)$. Then, $H^0(X, F) = M^{\rho}$, and $H^1 = \text{coker}(M \overset{\text{id} - \rho}{\to} M)$. $\endgroup$ – user171326 Jun 17 '17 at 5:44

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.