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In algebraic geometry, an etale sheaf on a Noetherian scheme is called constructible if the scheme has a finite stratification by locally closed subschemes such that the pullback of the sheaf to each of the subschemes is lcc (locally constant constructible). This definition seems to make sense both for sheaves of abelian groups and sheaves of sets. There is a similar definition in algebraic topology and I have no doubt that analogues of my questions can be formulated in the algebraic topology world.

I don't really have such a good feeling for these constructible sheaves, and, when re-reading the definition recently, I realised that I could easily formulate simple questions about building such things, that I could not answer. Let me ask the most general question first, which might be too general to have a reasonable answer.

General question. Say I have a Noetherian scheme $X$ and I am given a stratification by finitely many locally closed subschemes $U_i$. Say I have lcc sheaves on each of the $U_i$. What additional data do I need to glue these sheaves together to get a constructible sheaf on $X$?

If I am lucky then there is some slick cocycle-related answer and we can all go home happy. But I am not a big fan of these general types of question, so here are a few completely specific special cases---the simplest non-trivial (as far as I am concerned) ones, which I am already a bit hesitant on.

Q2 Let $X$ be the affine line over the complexes. Let $P$ denote the origin and let $U$ be $X\backslash P$. Up to isomorphism, how many constructible sheaves of sets are there on $X$, whose pullback to $U$ is the constant sheaf of sets of size 1 (represented by the identity $U\to U$) and whose pullback to $P$ is the constant sheaf of sets of size 2 (represented by $P\coprod P\to P$)?

There are, I guess, two notions of isomorphism: one can imagine that the stalk above $P$ is a given set of size 2 and demand that isomorphisms must induce the identity on this set, or one can just demand that the isomorphism induces a bijection on the stalk at $P$. I've noticed that different people interpret the question in different ways so let me just ask that the isomorphism induces an arbitrary bijection, to fix ideas.

I am a little scared of sheaves of sets because the initial and the terminal objects in the category of sets are not isomorphic, and this tripped me up a bit when I was trying to answer Q2. We don't have such a problem in the category of abelian groups so perhaps that's a safer place to be:

Q3 $X$ the affine line, $P$ the origin and $U$ the rest, as before. How many isomorphism classes of constructible sheaves of abelian groups on $X$ are there, whose stalk at $P$ is cyclic of order 4 and whose pullback to $U$ is constant and cyclic of order 2?

What is the motivation for Q2 and Q3? There are several answers. The motivation is certainly not "I need to know the answers to these questions!". One possible answer is "I feel like I cannot say I understand constructible sheaves unless I can answer these basic questions". Another is "what I really want to know is the answer to the general question, but I am worried the general question is too vague so I am asking special cases of it in order to get a feeling for the general question".

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2 Answers 2

up vote 9 down vote accepted

I wonder if this may suffice for your purposes: suppose you have a scheme $X$, cut up into an open subscheme $U$ and a closed subscheme $Z$ supported on the complement of $U$ in $X$. Let $j:U\to X$ and $i:Z\to X$ be in the inclusions. Let $F$ be an \'etale sheaf of sets on $X$. One derives from $F$ a triple $$ (j^*F, i^*F, \phi: i^*F\to i^*j_*j^*F). $$ The $\phi$ piece comes from restricting the adjunction morphism $F\to j_*j^*F$ to $Z$. The formation of the triple is functorial in $F$. The resulting functor provides an equivalence of categories from the category of \'etale sheaves on $X$ to the category of triples $$ (A,B,\phi:B\to i^*j_*A), $$ where $\phi$ is any sheaf morphism.

To the extent that you can compute $j_*$ and $i^*$, you can thus describe sheaves on $X$ in terms of sheaves on $U$ and $Z$. Building up from the bottom of a stratification of $X$ by locally closed subschemes, you could thus (granting an ability to compute $i^*$ and $j_*$) describe sheaves on $X$ constructible with respect to your stratification. (Performing the computations is on par with proving that $j_*$ preserves constructibility.)

For example, in your Q2, you are extending $A = 1_U$ across $Z = P$, and you want $B = 1_Z + 1_Z$. The sheaf $i_*j^*A$ is $1_P$. Since this sheaf is terminal, there is a unique constructible $F$ on $X$ that restricts to $A$ on $U$ and $B$ on $Z$; it corresponds to the triple $(A,B,\phi)$, where $\phi:B\to i^*j_*A$ is the unique sheaf morphism.

In your Q3, since you are working with abelian sheaves, you should consider $A=(\mathbf{Z}/2\mathbf{Z})_U$ and $B=(\mathbf{Z}/4\mathbf{Z})_Z$ as abelian sheaves, and you should require $\phi$ to be a homomorphism of abelian sheaves. Since $$ i^*j_*A = (\mathbf{Z}/2\mathbf{Z})_Z, $$ there are two choices for $\phi$ in a triple $(A,B,\phi)$, corresponding to $1\mapsto 1$ and $1\mapsto 0$.

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As a kind of added reference, the procedure above as known as Artin gluing. –  Torsten Ekedahl Feb 2 '11 at 5:06
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Artin gluing is explain in SGA 4 (tome I, exp 4, I think) and Milne's book, if you'd like a reference. –  JBorger Feb 2 '11 at 6:19
    
This is lovely, and furthermore it's a bit irritating because in the past I have been well aware of the results you mention :-/ . Does it answer the entire question, in fact? How about if I have $P$ a point in $L$ a line in $X$ a plane, and I give you sheaves on $P$, $L\backslash P$ and $X\backslash L$? I figure that first I can glue to get a sheaf on $L$ and then glue again. Yeah, I think your answer gets to the heart of the question. Thanks. –  Kevin Buzzard Feb 2 '11 at 21:55

Fantastic answer! To continue with the question, how about some easy examples of how to compute the cohomology of a constructible sheaf? Or the derived push-forward or pull-back?

For example, if we take the two given examples of a constructible sheaf on $\mathbb{P}^1$ (instead of $\mathbb{A}^1$ to make things easier) with fibers $\mathbb{Z}/2$ and $\mathbb{Z}/4$, what are the cohomology groups associated to these sheaves?

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