TITLE: Analytic tools in algebraic geometry 
QUESTION [150 upvotes]: 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?

REPLY [4 votes]: 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.