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?
Best Answer
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.