## Extensions of sheaves and the local-global spectral sequence

GIven a topological space $X$, algebraic topologists would sometimes be interested in sheaves over this space. In most cases, these sheaves are sheaves of functions with some special property, or sheaves of modules over these rings. We could casually say thet the notion of sheaf is some mix of topology and algebra (more generally, categories), which allows to remember where are localized objects. For example, the set of continuous functions over a compact space can sometimes allow to recover an isomorphic space, but its mere normed algebra structure does not immediately say things like: “i am greater here than there”, “i am positive here and negative there”…

So a sheaf is not only determined by a set of fonctions (the commonly used term is section) defined over the whole space, but also by sections over open sets, and by the restriction maps between these various sets of sections. A sheaf has also a gluing theory : if we choose sections over open sets with equal restrictions on the intersections, they must glue to a section over the union of the open sets (if not, our object does not deserve to be a sheaf, so it is only a presheaf).

Virtually anything which can be restricted to open sets and defined locally defines a sheaf: continuous functions form a sheaf, but functions with $||f||_{\infty} = 1$ do not usually form a sheaf (this property is not local). A sheaf of abelian groups is a sheaf of things which can be (locally) added and substracted, a sheaf of rings has sections which can be multiplied, and we can also define sheaves of modules over a sheaf of rings.

A morphism of sheaves $f: F \to G$ is a map which is determined by the image of localized sections. In the case of abelian groups, the kernel sheaf is the sheaf of sections of F which are locally in the kernel of f, while the quotient sheaf (cokernel) is something looking like G where sections are identified if locally it should be so. A morphism is thus injective or surjective if locally it is so. Sheaves of abelian groups (or sheaves of modules over some sheaf of rings) form an abelian category, and it makes sense to speak of the extension group $\mathrm{Ext}^1(F,G)$ for two such sheaves F and G. Suppose we are working with sheaves of modules (especially coherent sheaves) over a sheaf of rings $\mathcal O_X$. How should we compute this group ?

Traditional methods involve finding an injective resolution of G (usually only theoretical) or a projective resolution of F. Classical projective objects are free modules, but the natural corresponding sheaves $\mathcal O_X^n$ are not projective (exercise : why ?), so free resolutions do not compute Ext groups. However, they are in a certain sense locally free, locally projective, and can be used to compute Ext sheaves. Ext sheaves $\mathcal Ext(F,G)$ are defined by gluing the local collections of isomorphism classes of extensions of F and G: a section is a gluing of local extensions. It is not clear, however, whether gluing elements of Ext groups define a global extension of F and G. But what is sure is that a global extension gives locally an isomorphism class of extension, hence a morphism $Ext^1(F,G) \to H^0(\mathcal Ext^1(F,G))$. In many cases, this morphism is not bijective (for example, when F and G are themselves locally free).

The ideas of Leray and Grothendieck provided a powerful tool to compute $Ext^1(F,G)$ anyway. It is true that $Hom(F,G) = H^0(\mathcal Hom(F,G))$. From this remark, Grothendieck extended the spectral sequences invented by Leray, to compute the derived functor of the left hand side, when we “know” how to compute $H^p$ and $\mathcal Ext^q$ (so we are able to compute a composite of derived functors). It is written

$E_2^{p,q} = H^p(\mathcal Ext^q(F,G)) \implies Ext^{p+q}(F,G)$

It is based on the idea that the computation requires two resolutions : a locally projective (or locally free) resolution $P_\bullet$ of F and a open cover making these locally free sheaves projective (in algebraic geometry, a cover by affine subschemes). The desired Ext groups now appear as the cohomology of a double complex. Spectral sequences give a way to carry the necessary computations by steps. The initial data are groups $E_0^{p,q}$ (for example p-Cech cochains of $\mathcal Hom(P_q, G)$) with morphisms $d_0: E_0^{p,q} \to E_0^{p,q+1}$ (which is used to compute Ext sheaves).

Now take the cohomology of this and get groups $E_1^{p,q}$ (Cech p-cochains of $\mathcal Ext^q(F,G)$) with the other differential $d_1: E_1^{p,q} \to E_1^{p+1,q}$ (concerning Cech cohomology). Take the cohomology of this, and get $E_2^{p,q} = H^p(\mathcal Ext^q(F,G))$. But it is not finished yet : it is possible to define morphisms $d_r: E_r^{p,q} \to E_r^{p+r,q-r+1}$ and the process should go on to infinity (but it is locally finite for simple reasons). The theory of spectral sequences says the $E_\infty^{p,q}$ describe $Ext^{p+q}(F,G)$ in the following way:

there exists a decreasing filtration $F^p Ext^{p+q}(F,G)$ such that

$E_\infty^{p,q} = F^p Ext^{p+q}(F,G) / F^{p+1} Ext^{p+q}(F,G)$

The filtration says how deep you need to go into the Cech cochains to understand the object. For example, $d_3$ vanishes for p+q = 1. We see $E_3^{0,1} = Ext^1(F,G) / F^1 Ext^1(F,G)$ is some subspace of $H^0(\mathcal Ext^1(F,G))$. This is what you see if you look globally: an global extension determines local isomorphism classes of extensions.

There is also $E_3^{1,0} = E_2^{1,0} = H^1(\mathcal Hom(F,G))$. This is what you missed in the previous paragraph : gluing data with values in $\mathcal Hom(F,G)$ are needed to determine completely $Ext^1(F,G)$. We have obtained an exact sequence :

$0 \to H^1(\mathcal Hom(F,G)) \to Ext^1(F,G) \to H^0(\mathcal Ext^1(F,G))$
$\xrightarrow{d_2} H^2(\mathcal Hom(F,G)) \to 0$

What does the last arrow means ? It means that a local choice of compatible classes of extensions does not always glue to a global class of extensions. This is the obstruction I was talking about in red. It probably has an interpretation in terms of gerbes (maybe I’ll talk about gerbes in the future): a section of $\mathcal Ext^1(F,G)$ does not define an extension, but a stack over our variety/scheme/space $X$. The objects of this stack over an open set $U$ are the extensions $0 \to G_U \to E \to F_U \to 0$ of sheaves over U, whose local extension class is the given section of $\mathcal Ext^1(F,G)$. This is a gerbe : any two such objects are locally isomorphic. Moreover, given such an object, automorphisms of E preserving F and G have the form $1 + f$ where $f$ is a section of $\mathcal Hom(F,G)$. So we have found an abelian gerbe of link $\mathcal Hom(F,G)$ naturally associated to our section of $\mathcal Ext^1(F,G)$.
Now Giraud introduced gerbes in Cohomologie non-abélienne because their equivalence classes are in natural bijection with $H^2(A)$ where $A$ is the sheaf of (here abelian) groups naturally associated: the characteristic class is zero if and only if the gerbe is neutral, i.e. has a global object. This is exactly what we needed !