Čech-de Rham Complex as a Model of Derived Global Section

---

In the previous blog, we introduced Čech-de Rham complex to compute the anomalous charge of CFT. A (wild) intuition we learned from classical BV formalism is that, the derived object could be obtained by add some additional degree (anti-fields) to the original object, which possibly could be interpreted as some kind of resolution or derived object of the original object.

The same idea also applies to the Čech-de Rham complex we introduced in the previous blog, which introduced an additional degree (the Čech degree) to the original de Rham complex.

So, it is natural to ask the following questions:

• First, could we using the derived functor to understand the Čech-de Rham complex we introduced in the previous blog?
• Second, could we use BV formalism of CFT to capture the non-trivial topological information we discussed in the previous blog?

In this blog, we will consider the first question.

Instead of giving a full introduction of derived functor, we will only give a brief review of the concepts we will use in this blog. For a full introduction, try Methods of Homological Algebra by Gelfand and Manin.

Given an Abelian category , we can formulate its derived category by:

• Replacing "bad" objects into some "good" objects.
• Replacing "bad" morphisms into "good" morphisms.

So, the question is, what are "good" objects and "good" morphisms?

There are some examples from Gelfand and Manin's book:

• Naive tensor product is "bad" (only right-exact, not exact), we need to correctly define it by using flat resolution, i.e., given , the "good" tensor product should be defined as:

  where is a flat resolution of . By definition of flat resolution, the functor is exact on , thus the derived tensor product (which could be identified with after taking homology) is "good". Namely, given a short exact sequence , the sequence:

  is exact.

• Naive Hom functor is "bad" (only left-exact, not exact), we need to correctly define it by using injective resolution, i.e., given , the "good" Hom functor should be defined as:

  where is an injective resolution of . By definition of injective resolution, the functor is exact on , thus the derived Hom functor (which could be identified with after taking cohomology) is "good". Namely, given a short exact sequence , the sequence:

  is exact.

The lesson above is, to define some good functors, we need to replace the original objects by some "good" objects (flat or injective resolution). Which hints to identify the object with the other objects which are quasi-isomorphic to it.

However, unlike the category of chain complexes , which identifies two objects if they are chain homotopic, it is quit hard to obtain an "inverse" map of a quasi-isomorphism. Using a technique called localization of category, we can formally invert all quasi-isomorphisms in and obtain the derived category . Thus, we found a first property of derived category:

• There exists a functor:

  while is a quasi-isomorphism in , is an isomorphism in .

Such a functor is universal, i.e., given any functor which sends quasi-isomorphisms to isomorphisms, there exists a unique functor such that .

Now we back to the question of finding "good" morphism. In the (co)homology level, the exact sequence is a "good" object. If one want to find such a "good" object in the chain complex level, one would find distinguished triangle, whose definition is:

While acting some "good" functors on a distinguished triangle, the image is still a distinguished triangle. Moreover, if one take (co)homology of a distinguished triangle, one would obtain a (long) exact sequence, which return to the original discussion at (co)homology level (for a proof of this fact, check here). Thus, the morphism which preserves distinguished triangle is a "good" morphism.

For example, the derived functor preserves distinguished triangle.

Now we consider an important example of derived functor, which is the derived global section of the global section functor , which would be used in the following discussion.

The original global section functor is a typically "bad" functor, which is left-exact but not exact. To define its derived functor, we need to replace the original sheaf by its injective resolution. Note that the injective resolution is -acyclic, the derived global section could be simply defined as:

which is exact and return to the original global section functor when taking -th cohomology.

In the main text, we claim that derived functors "restore" the exactness lost by their naive counterparts. This property is not an axiom but a direct consequence of a powerful mechanism in homological algebra.

The choice of resolution (injective vs. projective/flat) is precisely tailored to the type of functor we want to correct. Here, we present the general argument for why this machinery works.

The key is that, the injective/flat(projective) resolution would restore the exactness lost by left/right-exact functors.

In this section, we will review the construction of Čech complex of sheaves of bounded below complex, following the treatment in stack project.

Consider a ringed space with a bounded blow complex of presheaves of Abelian groups .

We can compute the cohomology using Čech cocycles. Namely, we consider an open cover of and form a Čech complex

which is a double complex. The associated total complex to it is the complex with degree part:

where we denote and assign the degree of intersections be . Consider an element living in , where , the differential on the total complex is given by:

where is the differential of the complex of sheaves , and the expression means the restriction of to . Thus, the total complex could be defined as:

where and is defined as above.

Now it's time to reveal the structure behind the construction of Čech-de Rham complex. The claim is, the Čech-de Rham complex is a model of the derived global section of the complex of sheaves . Here, "model" means the cohomology of the Čech-de Rham complex is isomorphic to .

Before rushing to this claim, we firstly consider a more general situation, which would be true for any bounded below complex of sheaves of Abelian groups on a topological space (not necessarily Čech-de Rham).

The idea of the proof is to find a injective sheaves which is quasi-isomorphic to ,

then the derived section would be descended to the global section of .

Thus, the map could be simply constructed by "descending" the lowest Čech degree part of the Čech complex to the global section, which is simply the augmentation map of the Čech complex.

As a conclusion, the map we want to construct is the composition of two maps:

which is easy to expect to be functorial and compatible with the functorial map defined previously.

The remain part is to show the functoriality and compatibility of the map constructed above. Namely, we need to show:

• Functoriality: the following diagram commutes in the derived category:

• Compatibility: the following diagrams commute:

  which are the compatibility with global section and refinement of open cover respectively.

A usual way to construct an injective resolution is to use Cartan-Eilenberg resolution: , where is an injective resolution. Thus, the Čech complex:

could be used to compute the derived global section , as we discussed above. Now we consider an associated double complex:

The page of the spectral sequence associated to the double complex is the cohomology of complex . Note that is an injective resolution, this cohomology is simply the Čech complex . Thus, the page is the Čech cohomology .

Finally, if one could prove that such a spectral sequence converges to the original cohomology, the spectral sequence could be indeed used to compute such cohomology. The discussion above could be formulated by the theorem below:

Now we can apply the theorem above to the complex of sheaves

Using the Poincaré lemma, we know that, for any contractible open set , the complex has cohomology only at degree . Then, using the spectral sequence above, the map:

is indeed an isomorphism! Thus, we finally showed that, the Čech-de Rham complex we introduced in the previous blog is indeed a model of the derived global section of the complex of sheaves , i.e., .