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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