Where can I find a proof of the de Rham-Weil theorem?
Does anyone know?
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2$\begingroup$ A superb exposition is contained in the first sections of Bott's ''Lectures on characteristic classes and foliations''. $\endgroup$– Johannes EbertMar 17, 2012 at 11:05
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3 Answers
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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.
answered Mar 16, 2012 at 18:11
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$\begingroup$ Are you saying that neither de Rham nor Weil had anything to do with it? Just curious... $\endgroup$ Mar 16, 2012 at 19:18
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$\begingroup$ 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 I recall that he did some road-testing of the sheaf idea around maybe 1947, and this was probably unpublished material/Bourbaki early drafting. $\endgroup$ Mar 16, 2012 at 20:01
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18$\begingroup$ 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 $\endgroup$ Mar 17, 2012 at 12:16
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$\begingroup$ Roger: Thank you! I will read this. $\endgroup$ Mar 17, 2012 at 15:27
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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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31$\begingroup$ But it's wonderful to see you participating on MathOverflow! $\endgroup$ Mar 16, 2012 at 23:03
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9$\begingroup$ I am also very happy to see you here. Your book is one my favourites. $\endgroup$ Mar 16, 2012 at 23:31
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2$\begingroup$ I think that the "Séminaire Cartan" to which Roger Godement refers is this one: numdam.org/numdam-bin/feuilleter?id=SHC_1950-1951__3_ $\endgroup$ Mar 17, 2012 at 13:27
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1$\begingroup$ If only, send my greetings to Grothendieck, is he still alive? $\endgroup$– Louis AMar 19, 2012 at 15:20
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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.
answered Mar 16, 2012 at 22:28