Questions on hypercohomology

Question

From Wikipedia,

https://en.wikipedia.org/wiki/Hyperhomology

the definition of hypercohomology is:

Suppose $\mathcal{A}$ is an abelian category with enough injectives, and $F: \mathcal{A} \rightarrow \mathcal{B}$ is a left exact functor from $\mathcal{A}$ to another abelian category $\mathcal{B}$. For a complex $C$ of objects of $\mathcal{A}$ that is bounded on the left, its hypercohomology $\mathbb{H}^i(C)$ is constructed as:

1. Take a quasi-isomorphism $\Phi: C \rightarrow I$, where $I$ is a complex of injective objects of $\mathcal{A}$.

2. The hypercohomology $\mathbb{H}^i(C)$ is defined as $H^i(F(I))$.

I could not find a good reference on hypercohomology, maybe because it has been replaced by the concept of derived functors, as said in the Wikipedia page. I know very few on related subjects, hence I would like to bother the community with my naive questions.

Question 1: From wikipedia, this definition is independent of the choice of $I$, i.e. up to a unique isomorphism, could anyone explain why?

Question 2: Suppose we have a map $C_1 \rightarrow C_2$ between complexes of $\mathcal{A}$, how to define a map on the hypercohomologies $\mathbb{H}^i(C_1) \rightarrow \mathbb{H}^i(C_2)$?

Question 3: Suppose we have an exact sequence of complexes of $\mathcal{A}$

$$0 \rightarrow C_1 \rightarrow C_2 \rightarrow C_3 \rightarrow 0$$

how to construct the long exact sequence of hypercohomologies?

Question 4: From wikipedia, the hypercohomology can also be defined using derived categories: the hypercohomology of $C$ is just the cohomology of $F(C)$ considered as an element of the derived category of $\mathcal{B}$, but this got me a little confused. In the derived category of $\mathcal{B}$, isn't the cohomology of the complex $F(C)$ still the cohomology obtained by taking the cohomology of this complex?

Answer 1

1) Any two injective resolutions are homotopy equivalent, much as in the case of a resolution of a single object. So their images under $F$ are homotopy equivalent and have the same cohomology.

2) One nice way is to construct a functorial injective resolution, which can be done at least in complexes over a Grothendieck abelian category. This generalizes the classical construction of the injective hull of a module.

3) This is, again, done in the same way as for ordinary derived functors. Construct resolutions sufficiently carefully so as to get an exact sequence $0\to I_1\to I_2\to I_3\to 0$ of resolutions; this is levelwise split so exactness is preserved by $F$, and we apply the snake lemma.

4) Wikipedia has an error, here. $F$ itself is not defined on the derived category; rather the total derived functor $\mathbf RF$ is; it is defined by $C\mapsto F(I(C))$, where $I(C)$ is any injective resolution. It is then immediate that the hypercohomology functors are given by composing cohomology functors with $\mathbf RF$.

By the way, Wiki's terminology here is questionable and confusing. What they are calling hypercohomology should, to my mind, be called the right hyperderived functors of $F$. It's only appropriate to call them hypercohomology when $F=\Gamma$ is a global sections functor, so that its ordinary derived functors are cohomology functors. I mention this to clarify the two kinds of cohomology arising here: this $F$-hypercohomology of a complex $C$ has virtually nothing to do with the cohomology of $C$ as a complex.