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This is not a very precise question, but I hope it will get some good answers.

As someone with a background in smooth manifold theory, I have experienced algebraic geometry as a beautiful but foreign territory. The strangeness has a lot to do with the lack of an Inverse Function Theorem. One day, after many years of thinking I knew what the word "etale" was all about, it dawned on me that what the etale site really is is a place where people like me don't have to feel so homesick: Grothendieck provided a way for the statement "infinitesimally invertible implies locally invertible" to be true in algebraic geometry, by the astounding device of changing the meaning of the word "locally".

Bott might have called this an instance of "the old French trick of turning a theorem into a definition". He referred in that way to Schwarz's derivative of a distribution (theorem: integration by parts) and Serre's definition of fibration (theorem: homotopy lifting in fiber bundles). (Yes, I know Grothendieck is not French.)

My question is, if we list some other facts from calculus or analysis that are everyday tools in smooth manifold theory or analytic geometry, do some of them also become available in algebraic geometry when the right topology is chosen? I suspect that the word "crystalline" will come in here somewhere. For example:

Existence and uniqueness of solutions of ODEs (with dependence on initial data).

Sard's Theorem (Is there some topology that is good to invoke when proving "moving lemmas"?)

Various forms of the Fundamental Theorem of Calculus, such as: Stokes's Theorem, Poincare Lemma, or just existence of antiderivatives of one-variable functions.

Added: In characteristic zero algebraic geometry, of course de Rham cohomology has the familiar property that $X\times \mathbb A^1$ looks like $X$, and (therefore) that $\mathbb A^n$ looks like a point. But is the de Rham complex a resolution of the constant sheaf in any sense? I mean, this is not true in the etale topology, even though in some sense all smooth $n$-dimensional things are etale locally the same, right?

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    $\begingroup$ Beautiful question, touches on a question I have been brewing for weeks. $\endgroup$ – B. Bischof Aug 6 '10 at 3:29
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    $\begingroup$ I think the "Poincare Lemma" in algebraic geometry is done via the "French trick": We simply define algebraic de Rham cohomology to be the hypercohomology of the de Rham complex $(\Omega_X^\bullet,d)$. $\endgroup$ – Kevin H. Lin Aug 6 '10 at 4:24
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    $\begingroup$ I mean, I think the algebraic dR complex is not a resolution of the constant sheaf, since for instance $\log z = \int \frac{1}{z} dz$ is transcendental. So, right--the Poincare lemma does not hold. But we can view the Poincare lemma as saying "the dR complex is a resolution of the constant sheaf, so we can use it to compute cohomology of the constant sheaf". So defining algebraic dR cohomology via hypercohomology of the dR complex is kind of like saying "pretend the Poincare lemma is true--pretend that the algebraic dR complex is a resolution of the constant sheaf". Theorem ~> Definition. $\endgroup$ – Kevin H. Lin Aug 6 '10 at 5:05
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    $\begingroup$ I really hesitate to try to answer any of this. All I'll say is, in birational geometry, the analog of Sard's theorem and moving lemmas works well in the vanilla Zariski topology. The notion of "general" and "very general" seems adequate so far. For uncountable fields, the complement of a countable union of hypersurfaces is dense is an algebraic Baire category theorem. Of course, over countable fields like $\overline{\mathbb{Q}}$ the matter is very different. But then some of the things that follow from arguments about very general points are expected to fail in that setup, eg cycles. $\endgroup$ – Eugene Eisenstein Aug 6 '10 at 5:48
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    $\begingroup$ Although this is a very broad question, I hope some people will give it a try... I am more and more convinced that the importation of concepts from analysis (smooth on the one hand and complex analytic on the other) into algebraic geometry is one of its most subtle aspect. $\endgroup$ – Simon Pepin Lehalleur Aug 6 '10 at 21:27
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This is by no means a comprehensive answer, but I'll risk some remarks. Briefly, my impression is that topology often tells one what to expect, but does not always tell how to prove it. In case it matters, this is an impression of someone whose first and true love is geometric topology, but who is interested in algebraic geometry as well.

There are some topological notions that have analogs in algebraic geometry. The best known is perhaps the \'etale cohomology. It has some properties very similar to the "topological" cohomology, i.e. the cohomology of constant or more generally, constructible sheaves. There is the Mayer-Vietoris sequence (for a Zariski open cover); furthermore one can define \'etale constructible sheaves, which gives the relative cohomology of a couple (a variety, a closed subvariety). One can define the constructible derived category, and there are the "six operations": the direct and inverse image, the direct and inverse image with compact support, RHom and the derived tensor product. Moreover, there is the Verdier duality (and hence, the Poincar\'e duality as well). There is the cohomology class of a cycle and so one can define the Chern classes of a vector bundle.

There are ways to compare the \'etale cohomology and the topological cohomology. For example, let $k$ be an algebraically closed field of finite characteristic. Then we can apply the Witt vector procedure http://eom.springer.de/W/w098100.htm to it to get a complete discrete valuation ring with residue field $k$ and fraction field of characteristic 0. Then, if we have a smooth scheme over $R$, we can apply the procedure explained in SGA 4 1/2, p.54-56 to construct a morhism from the cohomology of the fiber over the maximal ideal of $R$ to the (\'etale) cohomology of the fiber over the algebraic closure of the fraction field. (And see pp. 52-53 there for an analogy with the cohomology of the preimage of a disk under a holomorphic mapping and the preimage of the origin.) Then one can use M. Artin's comparison theorem to construct an isomorphism with the usual "topological" cohomology of the constant sheaf. The resulting maps are not isomorphisms in general but they are functorial with respect to maps of smooth varieties over $R$.

Perhaps, the \'etale cohomology smooth complete varieties is a bit too close to the cohomology of complex algebraic varieties. For example, the \'etale cohomology of the projective line over an algebraically closed field with coefficients in a finite abelian group $A$ of order prime to the characteristic of the field is $A$ in degrees 0 and 2 and 0 elsewhere, just as in the complex case. But in the complex case this is ultimately a consequence of the fact that $\mathbf{C}$ is 2-dimensional over $\mathbf{R}$. So why do fields of positive characteristic know about it? To me this is a bit mysterious.

Here is a somewhat less trivial example. Morse theory gives a CW complex homotopy equivalent to a given manifold once we have a strict Morse function on the manifold. As indicated in the paper http://arxiv.org/abs/math/0301140 by D. Arapura, the algebraic analog of a cell is probably an affine variety $X$ and a constructible sheaf on it whose cohomology vanishes in degrees other than $\dim X$. Given a quasiprojective $X$ we can construct a cell decomposition (of sorts). First we replace $X$ with an affine $Y\to X$ such that the fiber over any closed point is an affine space. This is the Jouanolou trick and a proof of its existence is sketched e.g. here The Jouanolou trick. Then we can take any constructible sheaf $F$ on $X$ and pull it back to $Y$. Then we use Beilinson's lemma to choose a closed subvariety $Y'\subset Y$ such that $H^*(Y,Y',F)=0$ except maybe in degree $\dim Y$ (the existence of such a $Y'$ can be proven using the usual Morse theory if one is working over $\mathbf{C}$). Then we apply the same procedure to $Y'$ and so on. We get a filtration of $Y$ whose Leray spectral sequence will be concentrated in the 0-row. This is an analog of the cellular complex.

Since this is already way too long, let me briefly mention the differences between the algebraic and the topological cases, the way I understand them. First, there are some tools in topology that have no analog in algebraic geometry. For example, everything involving partitions of unity is a no-no. In fact I don't know any example of the use of fine sheaves in algebraic geometry. So while there is an analog of Sard's theorem, some of its consequences fail miserably. For example, there are smooth complete complex varieties that can't be embedded in any projective space. (These examples, due to Hironaka, are described e.g. in Hartshorne, Appendix B.) On the other hand, in finite characteristic there is the Frobenius automorphism which acts on everything. For complex algebraic varieties there is one of the consequences, the weight filtration, but there is no Frobenius so the proof of its existence is a bit roundabout.

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    $\begingroup$ Thanks, and sorry for the slow response. The stuff about analogues of cell structures sounds very interesting. $\endgroup$ – Tom Goodwillie Aug 13 '10 at 19:26
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I just wanted to explain a bit further what I believe Tom is saying about the Poincare lemma since there seemed to be some confusion about it. The statement of the algebraic Poincare lemma is that for a smooth scheme of finite type over $\mathbb{C}$, one can consider the formal completion at every point x of the algebraic de Rham complex $(\Omega^*,d)$. This complex should be quasi-isomorphic to $\mathbb{C}$ concentrated in degree zero. I think something like this should hold over any field of characteristic zero, but I've never seen it, so let me stick just to things that I know.

This statement is broken horribly in characteristic p, basically because of a strange phenomenon. One always has a lot of cohomology even formally locally because $d(a^p)=0$. In other words, any p^th power will be a cohomology class! This is basically it, however, because of a theorem of Cartier, which over an affine scheme over $F_p$ for simplicity says that there is an isomorphism from $\Omega^*$(no derivative)` $\mapsto H*(\Omega^*,d)$ which extends the Frobenius map. This could be thought of as a sort of Poincare lemma in characteristic p if you like.

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    $\begingroup$ There is a short lecture series (2007 Arizona winter school) by Kedlaya where he explains this char. p phenomenon carefully and makes the necessary corrections to get to Monsky-Washnitzer cohomology, with applications to the computation of zeta functions over finite fields : video : google.fr/search?tbs=vid%3A1&q=kedlaya+p-adic+cohomology lecture notes (although do not cover exactly the same ground) : swc.math.arizona.edu/aws/07/KedlayaNotes11Mar.pdf $\endgroup$ – Simon Pepin Lehalleur Aug 9 '10 at 6:53
  • $\begingroup$ Great! Thanks for the clarification, Dan. $\endgroup$ – Kevin H. Lin Aug 9 '10 at 8:56
  • $\begingroup$ It's not clear to me what I was looking for. Poincare Lemma means various things, some of which are true in characteristic zero alg geometry. I had forgotten about the Cartier isomorphism. I used to ponder that way back in the early days of cyclic homology. $\endgroup$ – Tom Goodwillie Aug 13 '10 at 19:28
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I must mention the holomorphic morse inequaties due to demailly which has lots of applications in algebraic and complex analytic geometry . It's an analog of the usual morse inequalitites.http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.296.2455&rep=rep1&type=pdf

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    $\begingroup$ To this example we may add Laumon's proof of Weil's conjecture on weights in $\ell$-adic cohomology (Transformation de Fourier constantes d'equation fonctionnelles et conjecture de Weil), which in a way takes Witten's original idea [which inspired Demailly's work] closer to Grothendieck style algebraic geometry. This is probably a separate answer in itself, though. Deligne's algebro-geometric counterpart to the Fourier transform plays a crucial role in the proof, and is worthy of mentioning by itself. $\endgroup$ – Vesselin Dimitrov Oct 3 '13 at 16:09
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Let me try to give a few answers to the first question, on ODEs.

The first answer is that you can solve ODEs in characteristic zero in the ring of formal power series. You also have the appropriate existence and uniqueness theorems for PDEs and systems of PDEs. Passing to a complete local ring is a completely valid means of localization in algebraic geometry (and also number theory, where one almost always takes a local ring to be complete). However, it has some weaknesses. In particular, when you localize at a closed point in this way, you lose track of all the other closed points.

A second approach would be to try to use a topology finer than the etale topology, like the smooth topology. (You could guess that the smooth topology works because it's known how to write the complete local ring as a limit of smooth morphisms) To pull back your ODE to some smooth covering, you need to lift whatever vector field you defined it to the covering. So we can just define a topoology on $X$ with a fixed vector field $v$ where an open set is a smooth map $f: Y \to X$ and a vector field $w$ on $Y$ such that $df(w(y)) = v(f(y))$. In this topoology, the existence of local solutions is trivial. Simply take $Y$ to be the space of pairs of a point $x \in X$ and a function on $X$ modulo the $n$th power of the maximal ideal at $x$, $n$ the order of the ODE (i.e. an $n-1$-jet on $X$). To define a connection on this bundle, and hence a lift of the vector field, observe that we have exactly enough information to extend our function to a power series that solves the ODE, and use the natural connection on power series. By construction, we have a function that satisfies this differential equation, which sends a pair $(x,f)$ to $f(x)$. This again requires characteristic zero to work properly.

A third approach would be the closest to the "old French trick", and I think requires category theory. We can simply define an open set on $X$ to be a usual open set $Y$, or perhaps an etale open set $Y$, plus a fiber functor of the Tannakian category of vector bundles with a flat connection on $Y$. We can define a point of this open set to be a point $y\in Y$ plus an isomorphism between this fiber functor and the functor taking a vector bundle with flat connection to the sections at the point $y$. We define a local section of the ODE to be a vector in the evaluation of the fiber functor at the corresponding vector bundle with flat connection (which is the vector bundle of functions mod the appropriate power of the maximal ideal, as in the previous paragraph). We can dutifully check that a local section, as we have defined it, can actually be evaluated at each point, as we have defined it (by applying the isomorphism to the chosen vector).

Of course this construction is all a bit silly, because in most cases you just want to work with the Tannakian category of vector bundles with flat connection. The previous constructions I described are not, as far as I know, practically used in algebraic geometry, but this Tannakian category is - most often in a slightly extended form as the category of $D$-modules.

A final approach (which I will only sketch) is to say perhaps the etale topology was all you needed all along. To do this, you take whatever differential equation you care about, find the corresponding vector bundle with a flat connection, then apply the Riemann-Hilbert correspondence to obtain a corresponding representation of the fundamental group of $X$. You then choose a map from the defining ring of that representation to a ring of $\ell$-adic integers for some prime $\ell$, and then extend the representation, using the fact that $GL_n (\mathbb Z_\ell)$ is profinite, to a representation of the profinite completion of the fundamental group of $X$, which is the etale fundamental group of $X$. Finally you observe that modulo any finite power of $\ell$, this representation can be trivialized etale-locally. Almost anything you'd want to do with ODEs in a cohomological direction can be transferred, eventually, into the framework of lisse $\ell$-adic sheaves by this strategy. This theory, too, is used in practical algebraic geometry.

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