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I am very far removed from being an expert on derived categories. Every few months, however, I read a different introductory text with the hope that eventually I will have some basic grasp on this concept.

This has the added benefit that it puts various aspects of the technicalities of homological algebra, algebraic geometry and algebraic topology into a uniform context. For example, Verdier duality generalizes both Poincare duality and Serre duality, and it is easy to see how the proof of the DeRham Theorem for real manifolds is analogous to that of the DeRham Theorem for affine varieties.

In all of my readings so far, however, only cohomology (as opposed to homology) was considered. I therefore wonder how homology is treated via derived categories. In order to ask a more precise question:

Question

Is there a simple proof of Stokes' Theorem via derived categories? What is the set up of this proof, and in particular how is homology and its relationship with cohomology treated? Are there references about this that you can recommend?

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The question is: what is a reference for a proof of Stokes' Theorem using derived categories? That sounds pretty well defined for me. If people prefer, they can give a proof of Stokes' Theorem using derived categories rather than to give a reference. Is that really not well-defined? –  James D. Taylor Jun 23 '13 at 19:48
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It seems to me that a sensible version of this question would formulate some analogue of "Stokes' theorem" that perhaps one needs in one's research and ask whether it is true. At the very least, I would like to see a lot more about what one would like. –  Eleanor Von Hohlandsbourg Jun 23 '13 at 19:48
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This strikes me as a well formulated question with a lot of potential to generate informative answers from knowledgeable people --- just the sort of question I'm glad to see on MathOverflow. –  Steven Landsburg Jun 23 '13 at 19:55
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If D denotes the Verdier duality functor, then one has the direct image functor with compact support $f_{!}=Df_{∗}D$ which has a right adjoint $f^{!}=Df^{∗}D$. When $f$ is the map from your favourite space (or manifold) $X$ to the point,$f_*f^*(C)$ is the cohomology of $X$ (with coefficients in $C$),$f_!f^*(C)$ is cohomology with compact support,$f_!f^!(C)$ is homology, and $f_*f^!(C)$ is Borel-Moore homology. If you interpret these objects and the natural pairings between them through the Riemann-Hilbert correspondence which relates $D$-modules and ordinary sheaves, you will get your answer. –  Denis-Charles Cisinski Jun 23 '13 at 20:30
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This kind of things are known in the context of complex geometry. The historical references are, I guess: M. Kashiwara, The Riemann-Hilbert Problem for Holonomic Systems, Publ. RIMS, Kyoto Univ., 20 (1984), 319-365. Z. Mebkhout, Une équivalence de catégories et une autre équivalence de catégories, Comp. Math. 51 (1984) pp. 55-69. For complex algebraic varieties, a standard reference is the book of Borel et al. on algebraic D-modules. –  Denis-Charles Cisinski Jun 23 '13 at 21:01
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1 Answer

up vote 6 down vote accepted

There is an interpretation of homology using derived categories, and you can find a brief treatment around remark 3.3.10 in Kashiwara-Schapira "Sheaves on Manifolds". If $M$ is an $n$-manifold and $a: M \to \ast$ is the tautological map to a point, then the homology of $M$ with coefficients in an abelian group $F$ is the homology of the complex $Ra_! a^! F$. If we write $or_M = H^{-n}(a^!F)$ for the orientation sheaf, then the counit of the adjunction induces an "integration" map $Ra_! a^! F \to F$ that induces a map $H^n_c(M,or_M) \to F$. When $F$ is a real vector space and $M$ is smooth, you can relate this to differential forms, because the orientation sheaf is quasi-isomorphic to a suitable version of the de Rham complex. Softness of the de Rham sheaves implies you get an isomorphism $H^n_c(M, or_M) \to \frac{\Gamma_c(M, \Omega^n \otimes or_M)}{d\Gamma_c(M,\Omega^{n-1} \otimes or_M)}$. Stokes's theorem on $M$ implies a comparison between the "integration" map and "honest integration" of forms (or densities), in particular, the fact that $d\Gamma_c(M,\Omega^{n-1} \otimes or_M)$ is annihilated by honest integration.

I don't think Stokes's theorem is a statement whose proof is made easier with derived categories. Instead, I would rather say that it implies the fact that honest integration actually yields a morphism in a derived category.

Here's the setup: If we are given a smooth manifold $M$ and a $k$-form $\omega$, we can obtain a real number from any smooth map $f$ from the standard $k$-simplex $\Delta$ to $M$, by integrating $f^\ast\omega$ along $\Delta$. This produces a map of graded real vector spaces $\int_M: \Omega^\ast(M) \to \operatorname{Hom}(\operatorname{Sing}_\ast(M),\mathbf{R})$.

Here's Stokes's theorem: $\int_M$ is in fact a map of cochain complexes.

If you want to prove the theorem efficiently, you can use naturality of pullback to reduce to a simpler statement about forms on $\Delta$ itself. There will always be a step where you have to use properties of integrals. It is common to invoke Fubini's theorem (e.g., the short proof in Guillemin-Pollack), but you can do a lower-tech proof using the fact that polynomial forms uniformly approximate smooth forms on the simplex.

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"I don't think Stokes's theorem is a statement whose proof is made easier with derived categories." You made my day :-) –  alvarezpaiva Jun 24 '13 at 12:09
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