As a student of homotopy theory or algebraic topology, I have a certain outlook as to how one ought to think of a cohomology theory. There are axioms that help us with rudimentary computations, there are some spectral sequences and then there is Brown representability. This is far away from the starting point of looking at the singular (co)chain complex or a simplicial complex and trying to compute its homology, it seems a bit more refined. Even the ring structure is a bit clearer, it comes from the fact that we are mapping into a ring object.
There are more and more instances where i feel like i would benefit from understanding a bit more of sheaf cohomology than just "it's the derived functor of the global sections functor of a sheaf." This is a tidge helpful, but it does not really help too much with computations from my point of view. It feels like resolutions of sheaves are large hard objects mostly because sheaves contain so much data.
My question is essentially the following:
-
Are there homotopy theorists out there who have over come these feelings? what advice do you have? In fact any advice that someone might have that understands the uses of sheaf theory in homotopy theory would be helpful.
-
Are there things resembling the Eilenberg-Steenrod axioms for sheaf cohomology? not directly due to their classification theorem, but things that help you to compute the Sheaf cohomology like a MVS sequence or what have you. I mostly would like help in doing computations in the way that the E-S axioms do? so things like the Grothendieck-Riemann-Roch Theorem, which i am told can be used in such a way.
-
Is there a book that goes through explicit toy computations of sheaf cohomology? Are there toy examples you would suggest for getting to be more comfortable with these things? Are there examples that live on simpler spaces than schemes? these might help a bit more than others
Some addendums : I think i need to make a few comments. I am not well versed in algebraic geometry. I find this to be a fault of mine and this is an attempt to help bridge the gap. Suggestions for references are appreciated. I really appreciate all of the excellent answers so far!
thanks for your time
EDIT MAIN QUESTION: I think what i am really asking about is the six functor formalism, but i don't really know since i don't know what that is. A friend started explaining it to me and it seemed like what i am looking for, but he said he did not feel confident in writing an answer explaining them. Hopefully someone will see this edit and give a fun answer.
Best Answer
Hi Sean,
I think I great place to inhabit, be it for calculations or for conceptual understanding, is the derived category of sheaves (of abelian groups say) $D(X)$ on your topological space $X$. Here are the basic players:
For any $X$ and integer $i$, functors $H^i: D(X) \rightarrow$ sheaves of abelian groups on X (cohomology sheaves); and
For a map $f: X \rightarrow Y$, adjoint maps $f^* :D(Y) \rightarrow D(X)$ and $f_* :D(X) \rightarrow D(Y)$ (derived pullback and pushforward).
Then, for instance, the $i^{th}$ cohomology of a sheaf is just $H^i p_*$, where $p: X \rightarrow pt$.
Lots of your favorite computational tools carry over to this setting. For instance Meyer-Vietoris: if $X = U_1 \cup U_2$ with inclusions $j_1: U_1 \rightarrow X$ and $j_2: U_2 \rightarrow X$ and $j_{12}: U_1 \cap U_2 \rightarrow X$ and $F$ is a sheaf on $X$, then there is a distinguished triangle
$${j_{12}}_* j_{12}^* F \rightarrow {j_1}_* j_1^* F \oplus {j_2}_* j_2^* F \rightarrow F \rightarrow$$
(usual M-V follows by taking F constant and applying $p_*$).
What's a reference? I learned from BBD (Faisceaux Pervers, Asterisque 100), which is great, but maybe more scheme-y than you want.
P.S.: Since you're a homotopy theorist, maybe I'll mention what I think is a great perspective on what the derived category D(X) is. Basically it's just like the ordinary category of sheaves of abelian groups, except you do everything homotopically. So instead of abelian groups you take (excuse me) HZ-module spectra, and you make your sheaves satisfy homotopical descent (this homotopical descent is really the origin of things like M-V above). To make this perspective rigorous it's helpful to use infinity-category theory, as in Lurie's book Higher Topos Theory. Using this approach, one doesn't "derive" things in the sense of starting with an abelian situation and invoking derived categories and functors; rather one makes "derived" definitions, and then all of the natural operations are automatically "derived": for instance the derived f_* and f^* can be given, in the infinity-categorical setting, by the same formulas as the ordinary f_* and f^* in the classical (abelian) setting.