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I'm writing some notes for some students which just finished a first course in scheme theory. There I would like to talk about constructible sheaves, but I found it hard to give a compelling motivation for these objects. (And I don't like to give a definition without at least trying to explain why this is a nice thing to consider.)

Since the students mostly never saw what a stratification is, perhaps it is best to begin with local systems and then focus on the fact that the derived pushforward of a local system is not necessarily a local system, but constructible sheaves are closed under the usual six functors.

I would like to know what the community here thinks of such approach and if there are other possible motivations.

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    $\begingroup$ arxiv.org/abs/1303.3255 $\endgroup$ May 2, 2021 at 18:36
  • $\begingroup$ Name of @ViditNanda's reference: Curry - Sheaves, cosheaves and applications. $\endgroup$
    – LSpice
    May 2, 2021 at 18:52
  • $\begingroup$ Dear @ViditNanda, could you explain further why you suggest this reference? I see that constructible sheaves are discussed there in the 11th chapter. But they motivate it using a lot of stuff a student which has just finished a first course in scheme theory don't know. (However, I definitely agree that this is a beautiful thesis!) $\endgroup$
    – Gabriel
    May 2, 2021 at 19:00
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    $\begingroup$ Constructible sheaves are usually defined in either the analytic or etale topology. Have your students already seen the definition of sheaves and sheaf cohomology in one of these topologies, and the motivation for that, and you literally just need to motivate the definition of constructible? Or are you trying to motivate the whole package of technology? $\endgroup$
    – Will Sawin
    May 2, 2021 at 19:03
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    $\begingroup$ @Gabriel If I had to do it, I would start with an étale cover $f:X\to Y$, and ask them what kind of sheaf $f_*\mathbb{Z}$ or $f_*\mathbb{Z}/n$ would be. Now suppose that it was only a branched cover.... $\endgroup$ May 2, 2021 at 19:05

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Even if you're only interested in say cohomology with coefficients in the constant sheaf, working with constructible sheaves gives you extra flexibility and is more amenable to inductive proofs.

Here is a basic theorem in the topology of algebraic varieties one of whose proofs could serve as a motivation. I discussed it in some MO answer, maybe I'll link it later, and I learned this from Lazarsfeld's book "Positivity in Algebraic Geometry" (volume 1).

Theorem. Let $X\subseteq \mathbf{C}^n$ be an affine algebraic variety, i.e. a closed subset cut out by polynomial equations. Then $H^i(X, \mathbf{Z}) = 0$ for $i>\dim X$.

Here $\dim X$ is the algebraic dimension, can be defined in various ways, see any textbook on algebraic geometry. The "dimension" (in whatever sense) of $X$ as a topological space is thus $2\dim X$. The above result is really special to affine varieties: the projective space $\mathbf{C}P^n$ has dimension $n$ and nonzero cohomology in even degrees $i\leq 2n$.

One proof of the above result uses Morse theory (a careful analysis of $X$ by means of an auxiliary function $\mathbf{C}^n\to [0,\infty)$ such as $\sum |z_i|^2$), and shows a bit more: $X$ is homotopy equivalent to a CW complex with cells in dimensions $\leq n$ (the Andreotti-Frankel theorem). This is not the proof I'd like to mention.

The proof which generalizes well to other contexts, e.g. to $\ell$-adic cohomology, due to Michael Artin, proceeds by induction on $d=\dim X$. By the Noether normalization lemma, there exists a finite morphism $$ f\colon X\to \mathbf{C}^d $$ ("finite" here is equivalent to "proper with finite fibers"). How is the cohomology of $X$ related to cohomology of $\mathbf{C}^d$? You really need sheaf cohomology to answer that. In this case (because the map is finite!) we obtain isomorphisms $$ H^i(X, \mathbf{Z}) \simeq H^i(\mathbf{C}^d, f_*\mathbf{Z}). $$ The sheaf $f_*\mathbf{Z}$ on the right hand side is no longer the constant sheaf, but can be shown to be a constructible sheaf. (Over a Zariski dense open subset of $\mathbf{C}^n$, it will be a locally constant sheaf of rank $e$ where $e$ is the degree of the finite map.) Therefore the theorem will follow from a more general statement below.

Theorem. Let $F$ be a sheaf on $\mathbf{C}^d$, constructible with respect to a Zariski stratification. Then $H^i(\mathbf{C}^d, F)=0$ for $i>d$.

Now we are able to proceed by induction on $d$: we take a linear projection $$ p\colon \mathbf{C}^d \to \mathbf{C}^{d-1}. $$ If $p$ is generic, then the cohomology of $F$ fits inside a long exact sequence (a form of the Leray spectral sequence): $$ \cdots \to H^i(\mathbf{C}^{d-1}, p_* F)\to H^i(\mathbf{C}^d, F) \to H^{i-1}(\mathbf{C}^{d-1}, R^1 p_* F)\to \cdots $$ Here $R^1 p_* F$ is the first higher push-forward, and (again for $p$ generic) the sheaves $p_* F = R^0 p_* f$ and $R^1 p_* F$ will have stalks which compute the cohomology of the fibers: $$ (R^i p_* F)_y = H^i(p^{-1}(y), F). $$ (I am telling this backwards: one shows the above formula for all $i$, and since we know the theorem for $d=1$, we know that $R^i p_* F = 0$ for $i>1$, and then we get the long exact sequence.) One can show that $p_* F$ and $R^1 p_* F$ are again constructible with respect to an algebraic stratification, and we proceed by induction. (The case $d=1$ still has to be done by hand.)

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