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Where can I find a proof of the de Rham-Weil theorem?

Does anyone know?

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A superb exposition is contained in the first sections of Bott's ''Lectures on characteristic classes and foliations''. –  Johannes Ebert Mar 17 '12 at 11:05
Thank you Johannes –  Louis A Mar 19 '12 at 15:19

3 Answers 3

up vote 6 down vote accepted

If "the de Rham-Weil Theorem" means that you can compute cohomology using acyclic resolutions rather than injective ones, this is a standard result you can find in just about any book on homological algebra. The earliest reference I know is Grothendieck's Tohoku paper, Section 2.4.

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Are you saying that neither de Rham nor Weil had anything to do with it? Just curious... –  Igor Rivin Mar 16 '12 at 19:18
It is a more interesting question, and one that can be answered given Weil's collected papers, whether Weil published anything about it. I recall that he did some road-testing of the sheaf idea around maybe 1947, and this was probably unpublished material/Bourbaki early drafting. –  Charles Matthews Mar 16 '12 at 20:01
There was Mathematics before Grothendieck. Type "Sheaf theory history" on Google, and you will get a paper by Haynes Miller, "Leray in Offlag XVIIA: The origins of sheaf theory..." with all information you will ever need! Roger Godement –  Roger Godement Mar 17 '12 at 12:16
Roger: Thank you! I will read this. –  Steven Landsburg Mar 17 '12 at 15:27
Found it, thanks –  Louis A Mar 19 '12 at 15:21

Of course Weil did it (although I'm not able to give you the ref right away), and I even lectured on it in Paris 40 years ago. His method is very simple : (1) You first prove that your compact manifold X can be endowed with a Riemann structure (obvious locally, global result by using a smooth partition of unity). (2) By the general theory of Riemann spaces, there are plenty of "convex" open sets U in X, i.e. such that any two points of U can be joined by one and only geodesic arc in U. Such an open set is clearly homotopic to a point, so that every closed diff form on it is exact; the intersection of any two such "convex" sets is also convex. (3) That being said, choose a finite covering of X by convex open sets U(i) - no math available - and let omega (idem) be a form of degree p on X. There are forms omega(i) of degree p-1 in the U(i) such that omega = d[omega(i)] in U(i). Since U(i) inter U(j) = U(i,j) is convex, there are forms omega(i,j) of degree p-2 in the U(i,j) such that omega(j) - omega(i) = d[omega(i,j)] in U(i,j). By standard de Rham, there are forms omega(i,j,k) in the U(i,j,k) = U(i) inter U(j) inter U(k) such that omega(j,k) - omega(i,k) + omega(i,j) = d[omega(i,j,k)] in U(i,j,k), and so on. In this way, you eventually get forms of degree O whose alternate sums are closed, hence CONSTANTS, in the p to p intersections of the U(i). If you consider the abstract simplicial complex defined by the covering (U(i)), i.e. its so-called "nerve" (nerf in French), you thus associate to omega a cocyle of degree p (with real coefficients) of this complex. Etc, etc. You can do it all by your little self - matter of patience.

Before Grothendieck's Tohoku, there was a Cartan Seminar on sheaves theory (ca. 1948-49) in which everybody, at any rate in France, learned the theory, including Groth and myself. I even wrote a book on the subject (Théorie des Faisceaux, Paris, Hermann, 1957), which was still on sale (and found customers) two or three years ago, and possibly still is.

Don't infer from my answer I'm still doing mathematics. Roger Godement, Paris.

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But it's wonderful to see you participating on MathOverflow! –  Deane Yang Mar 16 '12 at 23:03
I am also very happy to see you here. Your book is one my favourites. –  Donu Arapura Mar 16 '12 at 23:31
I think that the "Séminaire Cartan" to which Roger Godement refers is this one: numdam.org/numdam-bin/feuilleter?id=SHC_1950-1951__3_ –  alvarezpaiva Mar 17 '12 at 13:27
If only, send my greetings to Grothendieck, is he still alive? –  Louis A Mar 19 '12 at 15:20

Andrew Fanoe, a former student of mine, wrote his senior thesis on various proofs of the DeRham theorem. In particular he discusses in detail Weil's very elegant proof of DeRham's theorem. Andrew's presentation uses a more modern language, but all the ideas are in Weil's paper. In any case, here is the link to Andrew's thesis. It's worth having a look at it. It is well written and may have references you might find useful.

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Thank you Liviu –  Louis A Mar 19 '12 at 15:21

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