Abelian Categories, or Injective Resolutions are Diagrammatic. Note Quote.

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Jean-Pierre Serre gave a more thoroughly cohomological turn to the conjectures than Weil had. Grothendieck says

Anyway Serre explained the Weil conjectures to me in cohomological terms around 1955 – and it was only in these terms that they could possibly ‘hook’ me …I am not sure anyone but Serre and I, not even Weil if that is possible, was deeply convinced such [a cohomology] must exist.

Specifically Serre approached the problem through sheaves, a new method in topology that he and others were exploring. Grothendieck would later describe each sheaf on a space T as a “meter stick” measuring T. The cohomology of a given sheaf gives a very coarse summary of the information in it – and in the best case it highlights just the information you want. Certain sheaves on T produced the Betti numbers. If you could put such “meter sticks” on Weil’s arithmetic spaces, and prove standard topological theorems in this form, the conjectures would follow.

By the nuts and bolts definition, a sheaf F on a topological space T is an assignment of Abelian groups to open subsets of T, plus group homomorphisms among them, all meeting a certain covering condition. Precisely these nuts and bolts were unavailable for the Weil conjectures because the arithmetic spaces had no useful topology in the then-existing sense.

At the École Normale Supérieure, Henri Cartan’s seminar spent 1948-49 and 1950-51 focussing on sheaf cohomology. As one motive, there was already de Rham cohomology on differentiable manifolds, which not only described their topology but also described differential analysis on manifolds. And during the time of the seminar Cartan saw how to modify sheaf cohomology as a tool in complex analysis. Given a complex analytic variety V Cartan could define sheaves that reflected not only the topology of V but also complex analysis on V.

These were promising for the Weil conjectures since Weil cohomology would need sheaves reflecting algebra on those spaces. But understand, this differential analysis and complex analysis used sheaves and cohomology in the usual topological sense. Their innovation was to find particular new sheaves which capture analytic or algebraic information that a pure topologist might not focus on.

The greater challenge to the Séminaire Cartan was, that along with the cohomology of topological spaces, the seminar looked at the cohomology of groups. Here sheaves are replaced by G-modules. This was formally quite different from topology yet it had grown from topology and was tightly tied to it. Indeed Eilenberg and Mac Lane created category theory in large part to explain both kinds of cohomology by clarifying the links between them. The seminar aimed to find what was common to the two kinds of cohomology and they found it in a pattern of functors.

The cohomology of a topological space X assigns to each sheaf F on X a series of Abelian groups HnF and to each sheaf map f : F → F′ a series of group homomorphisms Hnf : HnF → HnF′. The definition requires that each Hn is a functor, from sheaves on X to Abelian groups. A crucial property of these functors is:

HnF = 0 for n > 0

for any fine sheaf F where a sheaf is fine if it meets a certain condition borrowed from differential geometry by way of Cartan’s complex analytic geometry.

The cohomology of a group G assigns to each G-module M a series of Abelian groups HnM and to each homomorphism f : M →M′ a series of homomorphisms HnF : HnM → HnM′. Each Hn is a functor, from G-modules to Abelian groups. These functors have the same properties as topological cohomology except that:

HnM = 0 for n > 0

for any injective module M. A G-module I is injective if: For every G-module inclusion N M and homomorphism f : N → I there is at least one g : M → I making this commute

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Cartan could treat the cohomology of several different algebraic structures: groups, Lie groups, associative algebras. These all rest on injective resolutions. But, he could not include topological spaces, the source of the whole, and still one of the main motives for pursuing the other cohomologies. Topological cohomology rested on the completely different apparatus of fine resolutions. As to the search for a Weil cohomology, this left two questions: What would Weil cohomology use in place of topological sheaves or G-modules? And what resolutions would give their cohomology? Specifically, Cartan & Eilenberg defines group cohomology (like several other constructions) as a derived functor, which in turn is defined using injective resolutions. So the cohomology of a topological space was not a derived functor in their technical sense. But a looser sense was apparently current.

Grothendieck wrote to Serre:

I have realized that by formulating the theory of derived functors for categories more general than modules, one gets the cohomology of spaces at the same time at small cost. The existence follows from a general criterion, and fine sheaves will play the role of injective modules. One gets the fundamental spectral sequences as special cases of delectable and useful general spectral sequences. But I am not yet sure if it all works as well for non-separated spaces and I recall your doubts on the existence of an exact sequence in cohomology for dimensions ≥ 2. Besides this is probably all more or less explicit in Cartan-Eilenberg’s book which I have not yet had the pleasure to see.

Here he lays out the whole paper, commonly cited as Tôhoku for the journal that published it. There are several issues. For one thing, fine resolutions do not work for all topological spaces but only for the paracompact – that is, Hausdorff spaces where every open cover has a locally finite refinement. The Séminaire Cartan called these separated spaces. The limitation was no problem for differential geometry. All differential manifolds are paracompact. Nor was it a problem for most of analysis. But it was discouraging from the viewpoint of the Weil conjectures since non-trivial algebraic varieties are never Hausdorff.

Serre replied using the same loose sense of derived functor:

The fact that sheaf cohomology is a special case of derived func- tors (at least for the paracompact case) is not in Cartan-Sammy. Cartan was aware of it and told [David] Buchsbaum to work on it, but he seems not to have done it. The interest of it would be to show just which properties of fine sheaves we need to use; and so one might be able to figure out whether or not there are enough fine sheaves in the non-separated case (I think the answer is no but I am not at all sure!).

So Grothendieck began rewriting Cartan-Eilenberg before he had seen it. Among other things he preempted the question of resolutions for Weil cohomology. Before anyone knew what “sheaves” it would use, Grothendieck knew it would use injective resolutions. He did this by asking not what sheaves “are” but how they relate to one another. As he later put it, he set out to:

consider the set13 of all sheaves on a given topological space or, if you like, the prodigious arsenal of all the “meter sticks” that measure it. We consider this “set” or “arsenal” as equipped with its most evident structure, the way it appears so to speak “right in front of your nose”; that is what we call the structure of a “category”…From here on, this kind of “measuring superstructure” called the “category of sheaves” will be taken as “incarnating” what is most essential to that space.

The Séminaire Cartan had shown this structure in front of your nose suffices for much of cohomology. Definitions and proofs can be given in terms of commutative diagrams and exact sequences without asking, most of the time, what these are diagrams of.  Grothendieck went farther than any other, insisting that the “formal analogy” between sheaf cohomology and group cohomology should become “a common framework including these theories and others”. To start with, injectives have a nice categorical sense: An object I in any category is injective if, for every monic N → M and arrow f : N → I there is at least one g : M → I such that

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Fine sheaves are not so diagrammatic.

Grothendieck saw that Reinhold Baer’s original proof that modules have injective resolutions was largely diagrammatic itself. So Grothendieck gave diagrammatic axioms for the basic properties used in cohomology, and called any category that satisfies them an Abelian category. He gave further diagrammatic axioms tailored to Baer’s proof: Every category satisfying these axioms has injective resolutions. Such a category is called an AB5 category, and sometimes around the 1960s a Grothendieck category though that term has been used in several senses.

So sheaves on any topological space have injective resolutions and thus have derived functor cohomology in the strict sense. For paracompact spaces this agrees with cohomology from fine, flabby, or soft resolutions. So you can still use those, if you want them, and you will. But Grothendieck treats paracompactness as a “restrictive condition”, well removed from the basic theory, and he specifically mentions the Weil conjectures.

Beyond that, Grothendieck’s approach works for topology the same way it does for all cohomology. And, much further, the axioms apply to many categories other than categories of sheaves on topological spaces or categories of modules. They go far beyond topological and group cohomology, in principle, though in fact there were few if any known examples outside that framework when they were given.

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