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This is a follow-up (refinement?) of this question.

In learning some algebraic topology, I've learned to think of an affine scheme as spec $R$. (I've been told that this is a legitimate use of terminology because it provides an embedding of $Rings^{op}$ into a larger category.) One construction we can do, for example, is create the "tangent scheme", which is obtained by localizing and completing. The examples I've been looking at are $\hat{\mathbb{G}}_a$, $\hat{\mathbb{G}}_m$, $(\mathbb{Z}[[t]],F)$ (FGLs (1-dimensional, commutative) more generally), $T_1C^k(\hat{\mathbb{G}}_a,\hat{\mathbb{G}}_m)\cong C^k(\hat{\mathbb{G}}_a,\hat{\mathbb{G}}_a)$ ("commutative $k$-variate FGLs satisfying the 2-cocycle condition"), etc. We then have exponential maps, which e.g. in the last case is $exp:C^k(\hat{\mathbb{G}}_a,\hat{\mathbb{G}}_a) \rightarrow C^k(\hat{\mathbb{G}}_a,\hat{\mathbb{G}}_m)$ given by $g\mapsto 1+g$.

Just to test the waters, here is my question (although please feel free to push it further or in a different direction). Presumably these exponential maps are not always injective. Once we apply this picture to a particular ring (or perhaps even before?), can we translate differential-geometric ideas like conjugate points, geodesics, "$exp$ is a local isomorphism", etc. into algebro-geometric language? When we can, which theorems for manifolds carry over to schemes and which must we discard?

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    Your question is of such a braodness, that I wonder how one can answer this but pointing you to a textbook about schemes? (In any case, what you are calling "differential-geometric ideas" involve Riemannian structures, and the schemes you are handling don't have anything similar to that...)2010-11-06
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    Great! What's a good book? I'd welcome just a small sampling of analogies, too, if you have any favorites. Also, what other differential-geometric structures were you thinking of? Riemannian was just the first one that came to mind.2010-11-06
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    Oops, I totally misread that. Yeah, I wouldn't expect intuition for smooth geometry to be immediately interpretable in an algebraic setting. But I've heard that there's still a decent notion of curvature on a scheme, for instance?2010-11-07
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    @Aaron: Dear Aaron, There is no notion of curvature for a scheme. In the case of varieties over the complex numbers, there is a deep relationship between certain algebro-geometric properties (Kodaira dimension, structure of the canonical bundle, ...) and differential geometric properties of the underlying complex analytic manifold (existence of metrics with various curvature properties); the most famous is probably the theory of Calabi--Yau varieties. But this story is not a part of scheme theory; rather, it is a part of complex algebraic and analytic geometry.2010-12-26
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    Interesting. This is exactly the sort of thing I was hoping to hear about. Is this going to be in something like Griffiths & Harris, or is it beyond the scope of that book?2010-12-27
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    Hey, check it out: http://mathoverflow.net/questions/19308/is-there-an-analogue-of-curvature-in-algebraic-geometry2011-01-07
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    And yes, you can find some of the things that Matt E is talking about in Griffiths-Harris, though that book probably pre-dates the term "Calabi-Yau". For a more modern but still introductory reference, try Huybrechts's "Complex Geometry".2011-01-07

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