Sheaf / de Rham cohomology of a stack with values in a complex of abelian sheaves

Question

I am reading Differentiable Stacks and Gerbes to understand about (hyper) cohomology groups of a stack $\mathcal{X}$ with values in a complex $\mathcal{M}$ of abelian sheaves over $\mathcal{X}$.

Recall the case of cohomology of a topological space with values in a sheaf of abelian Groups. Let $X$ be a topological space and $\mathcal{F}$ be a sheaf on $X$. We want to introduce sheaf cohomology of $X$ with values in $\mathcal{F}$. We introduce global section functor $$\Gamma(X,-):(\text{abelian sheaves on } X)\rightarrow (\text{abelian groups})$$ and then, as $\Gamma(X,-)$ is left exact functor, it produces right derived functors $R^i\Gamma(X,-)$ and $R^i\Gamma(X,\mathcal{F})$ is declared as $i$th cohomology of $X$ with valued in sheaf $\mathcal{F}$ and we denoted this by $H^i(X,\mathcal{F})$. (Correct me if I have misunderstood something till here).

Let $\mathcal{X}\rightarrow Man$ be a differentiable stack. Fixing a Grothendieck topology on the category $Man$ induces a Grothendieck topology on $\mathcal{X}$. Thus, we can talk about sheaf over $\mathcal{X}$ just like a sheaf over a topological space. Sheaf over $\mathcal{X}$ is a functor $\mathcal{F}:\mathcal{X}^{op}\rightarrow (\text{Sets})$.

We consider the category of sheaves over $\mathcal{X}$ just like category of sheaves over $X$.

To imitate the notion of definition of sheaf cohomology in topological set up, we need to make sense of notion of Global section functor. $$\Gamma(\mathcal{X},-):(\text{sheaves over } \mathcal{X})\rightarrow (\text{Sets})$$

Let $\mathcal{F}$ is a sheaf on $\mathcal{X}$. We do not have (or atleast I do not see) an object in $\mathcal{X}$ that behaves like $X$ where I can just evaluate $\mathcal{F}$ and get a set.

What one can do is, see that there is a bijection $\text{Hom}_{\text{sheaves(X)}}(\underline{0},\mathcal{F})\rightarrow \mathcal{F}(X)$ from here in Vakil's notes.

Thus, $$\Gamma(X,-):(\text{abelian sheaves on } X)\rightarrow (\text{abelian groups})$$ with $\mathcal{F}\mapsto \mathcal{F}(X)$ can be seen as $$\Gamma(X,-):(\text{abelian sheaves on } X)\rightarrow (\text{abelian groups})$$ with $\mathcal{F}\mapsto \text{Hom}_{\text{sheaves(X)}}(\underline{0},\mathcal{F})$.

Second way of seeing global sections functor can be extended to stakcs, with out much changes.

$$\Gamma(\mathcal{X},-):(\text{abelian sheaves on } \mathcal{X})\rightarrow (\text{abelian groups})$$ with $\mathcal{F}\mapsto \text{Hom}_{\text{sheaves}(\mathcal{X})}(\underline{0},\mathcal{F})$.

I am guessing this is what they mean in page $18$ https://arxiv.org/pdf/math/0605694.pdf when they say global section functor.

Just for the same reason as global section functor on $X$ is left exact , the global section functor on $\mathcal{X}$ is left exact (I did not check, is there anything non trivial happening here that does not happen in case of just a topological space? I do not think so).

As this functor $\Gamma(\mathcal{X},-)$ is left exact, we can talk about right derived functors $R^i\Gamma(\mathcal{X},-)$ which are also denoted by $$H^i(\mathcal{X},-):(\text{abelian sheaves on } \mathcal{X})\rightarrow (\text{abelian groups})$$

Then the paper says something which I do not quite understand.

Passing to derived category of complexes of abelian sheaves over $\mathcal{X}$, we get the total derived functor $$R\Gamma(\mathcal{X},-):D^+(\mathcal{X})\rightarrow D^+(\text{abelian groups}).$$ For a complex $\mathcal{M}\in D^+(\mathcal{X})$ of abelian sheaves on $\mathcal{X}$, the homology groups of the complex $R\Gamma(\mathcal{X},\mathcal{M})$ are denoted by $\mathbb{H}^i(\mathcal{X},\mathcal{M})=h^i(R\Gamma(\mathcal{X},\mathcal{M}))$ are called the hypercohomology groups of $\mathcal{X}$ with values in $\mathcal{M}$.

I understand almost nothing that is written in above block. I have not seen something similar when studying sheaf cohomology on a scheme (I did not read so much).Can someone point me to similar definition in case of sheaf cohomology on a scheme. What is the necessity to go to so called complex of abelian sheaves? What am I missing here?

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Given a stack $\mathcal{X}$ they define a sheaf $\Omega^i_{\mathcal{X}}$ for each $i$ as some type of differential forms and then say, this gives a complex $\Omega$ of sheaves of $\mathbb{R}$ vector spaces over $\mathcal{X}$ which they call as de Rham complex of $\mathcal{X}$. They say, its Hypercohomology (defined above for a complex $\mathcal{M}$) is called the de Rham cohomology of $\mathcal{X}$:

$$H^i_{DR}(\mathcal{X})=\mathbb{H}^i(\mathcal{X},\Omega_{\mathcal{X}})$$ This is done in page $19$.

I am trying to relate this notion of defining de Rham cohomology using sheaf cohomology with the same thing happening i.e., going from sheaf cohomology to de Rham cohomology in case of manifold.

Any suggestions regarding this are welcome.

Answer 1

Let me try to say something meaningful about the following:

"(1) Can someone point me to similar definition in case of sheaf cohomology on a scheme. (2) What is the necessity to go to so called complex of abelian sheaves? What am I missing here?"

Let me first say something general about (2). Take your example of the algebraic de Rham complex. As Keerthi mentions in the comments, a complex is richer than its (co)homology, but not only this, a complex is richer than an abelian sheaf all on its own. This is because there is an embedding of abelian sheaves into the category of chain complexes of abelian sheaves. Furthermore, sometimes there are invariants of a scheme which cannot be encoded in a single sheaf. An example of this is the algebraic de Rham complex. It is a special cochain complex of coherent sheaves on $X$, in that it has "multiplicative structure", meaning that there are maps

$$\Omega_{X}^{p} \otimes_{\mathscr{O}_{X}} \Omega_{X}^{q} \rightarrow \Omega_{X}^{p+q}$$

compatible with the cochain complex structure of $\Omega_{X}^{\ast}$ (this is wedging forms together).

Now, suppose that we are given a bounded below cochain complex of abelian sheaves, $\mathscr{F}^{\ast}$ (more generally, just a bounded below cochain complex of objects in an abelian category). The sheaf hypercohomology of the complex $\mathscr{F}^{\ast}$ is to $\mathscr{F}^{\ast}$, is what the sheaf cohomology of some abelian sheaf $\mathscr{G}$ is to $\mathscr{G}$. What I mean is that the sheaf hypercohomology of a complex of sheaves is obtained by taking some appropriate resolution of $\mathscr{F}^{\ast}$, applying the global sections functor to it, taking the totalization of this double complex, and then taking the cohomology of the totalized complex. This is roughly the same procedure as resolving a sheaf via an injective/acyclic resolution, applying the global sections functor, and then taking the cohomology. Let me spell this procedure out in greater detail:

1) Take a Cartan-Eilenberg resolution $I^{\ast,\ast}$ of the chain complex $\mathscr{F}^{\ast}$ too obtain a first-quadrant double complex (see Chapter 5 of Weibel's "Introduction to Homological algebra").

2) Apply $\Gamma(X,-)$ to $I^{\ast,\ast}$ to obtain a double complex in $Ab$.

3) Then compute $H^{n}\left( Tot^{\ast}(I^{\ast,\ast}(X)) \right)$, where $Tot^{n}(I^{\ast,\ast}(X))= \prod_{p+q=n} I^{p,q}(X)$. This is just $\mathbb{H}^{n}(X,\mathscr{F}^{\ast})$.

More generally, we can compute the hyper-derived functor of a right/left exact functor. Ultimately, the derived category encodes all this information, but it takes a some work to see why.