I'm curious if the fifth and last chapter of Grothendieck's 1955 paper (which he states in the introduction is the origin of the paper) is describing something related to Sheaf Cohomology? Is he using the notation H1(X,G) as what we usually call H0(X,G) the global sections?

In his own (translated) words from the introduction:

'In the last chapter, we define the cohomology set H1(X,G) of X with coefficients in the sheaf of groups F, so that the expected classification theorem for fibre spaces with structure sheaf G is valid. We then proceed to a careful study of the exact cohomology sequence associated with an exact sequence of sheaves e->F->G->H->e.'

Any help would be gratefully appreciated!

PS you can find the mentioned 1955 paper of Grothendieck's General theory of fibre spaces with structure sheaf here as generously pointed to by user Jeroen.


Here, $F$ is a sheaf of not necessarily abelian groups. The category of such sheaves is not abelian (simply because the category of groups is not abelian: cokernels don't have the right properties when the image of a morphism is non-normal). So there is no general theory of sheaf cohomology for such sheaves. However, one can still define the notion of a $F$-torsor over $X$ (this is easiest to imagine when $F= \underline{G}$ is a constant sheaf, but makes sense more generally), and we'd expect $H^1(X, F)$ to parametrize these gadgets. There is no longer a natural group structure on the set of $F$-torsors, but there is at least a "zero" element, corresponding to the trivial $F$-torsor.

By explicitly understanding $F$-torsors in terms of transition functions, one can give a definition of a pointed set $H^1(X, F)$ which resembles the definition of Čech cohomology for abelian sheaves and verify that this pointed set parametrizes $F$-torsors over $X$. In the case that $F$ is actually a sheaf of abelian groups, this literally gives Čech cohomology, and thereby we see that we recover the usual $H^1(X, F)$ (e.g. by the fact that Čech cohomology computes derived functor cohomology in favorable circumstances). Of course, we can still define $H^0(X, F)$ to be the group of global sections.

Moreover, we get a partial long exact sequence: if $$1 \rightarrow F \rightarrow G \rightarrow H \rightarrow 1$$ is a short exact sequence of sheaves of groups (i.e. $F$ is a normal subgroup sheaf of $G$ with quotient sheaf $H$), then we have an exact sequence: $$ 0 \rightarrow H^0(X, F) \rightarrow H^0(X, G) \rightarrow H^0(X, H) \rightarrow H^1(X, F) \rightarrow H^1(X, G) \rightarrow H^1(X, H) $$

To make sense of this sequence, consider a global section $h \in H^0(X, H)$. We can find an open cover $X = \cup_i X_i$ and sections $g_i \in H^0(X_i, G)$ mapping to $h|_{X_i}$. Then the transition functions $f_{ij} = g_i g_j^{-1} \in H^0(X_i \cap X_j, F)$ (they're in $F$ because $g_i$ and $g_j$ both map to $h|_{X_i \cap X_j}$) define an $F$-torsor on $X$ which is independent of choices. Moreover, if $h$ comes from $g \in H^0(X, G)$, we can let $g_i = g|_{X_i}$ so we immediately see that this construction produces the trivial $F$-torsor. Thus, the non-trivial content of the exactness of the above sequence at $H^1(X, F)$ is that an $F$-torsor maps to the trivial $G$-torsor if and only if it comes from this construction. Additionally, one can show that a $G$-torsor maps to the trivial $H$-torsor if and only if it comes from an $F$-torsor. Moreover, there is a "twisting" operation of the group $H^0(X, H)$ on the set $H^1(X, F)$ extending the above construction such that two $F$-torsors are in the same orbit if and only if they map to the same $G$-torsor. Since $H^1(X, F)$ is not a group in the non-abelian case, we don't get quite as nice a description of the fibers of the map $H^1(X, G) \rightarrow H^1(X, H)$.

In general, we can't do any better. It's difficult to define an object which serves the role of $H^2(X, F)$ when $F$ is non-abelian, and other ideas (gerbes, etc.) are necessary. However, if $F$ is abelian and central in $G$, then the above exact sequence extends one step further: $$ H^1(X, F) \rightarrow H^1(X, G) \rightarrow H^1(X, H) \rightarrow H^2(X, F) $$

Since $F$ is abelian, $H^1(X, F)$ is a group, and this group acts on $H^1(X, G)$ in a manner such that the orbits coincide with the fibers of $H^1(X, G) \rightarrow H^1(X, H)$. Furthermore, we can locally lift transition data $h_{ij}$ defining an $H$-torsor to elements $g_{ij}$. Looking at $g_{ij} g_{jk} g_{ik}^{-1}$ on triple overlaps defines a Čech $2$-cocycle in $H^2(X, F)$ whose cohomology class is well-defined (due to centrality of $F$ in $G$). Then exactness at $H^1(X, H)$ says that an $H$-torsor comes from a $G$-torsor if and only if this construction yields the trivial cohomology class in $H^2(X, F)$.

Serre's book Galois Cohomology has a good summary of this non-abelian cohomology in the analogous context of group cohomology.


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