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Hi,

this is a bit vague, but feel free in your answers.

If I have a scheme $X$ and a closed point $x$ on it, then how is the first infinitesimal neighborhood of the point defined, and what is the philosophy behind it? What is it useful for, when does it appear?

In particular, I would be interested in the case when $X$ is an abelian variety and $x$ is the zero point.

Furthermore, how is the formal completion of an abelian variety $X$ in the origin defined, and what do you use it for?

Regards!

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Have you looked in Hartshorne's section on formal schemes? –  Mariano Suárez-Alvarez Oct 17 '11 at 7:26
    
Definitions can be found in Hartshorne, Chap II Section 9. Philosophically, the first infinitesimal neighborhood at $x$ is a thickening, corresponding to a point and a tangent direction. It has "more" local functions than at just the point by itself. Passing to the formal completion is the algebraic equivalent of looking at holomorphic functions with power series expansions at $x$. –  Parsa Oct 17 '11 at 8:42
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Let $X$ be a scheme and $x$ a closed point. Let us call $\mathcal{I_x}$ the sheaf of ideals in $\mathcal{O}_X$ that defines $x$ Notice that the canonical embedding $x \hookrightarrow X$ is given by the sheaf map $\mathcal{O}_X \to \mathcal{O}_X/\mathcal{I_x}$. Now consider the ideal $\mathcal{I}_x^2$ and the closed subscheme defined by it. It is supported in $ x $ but it has as structure sheaf an Artin algebra with generators of degree two that essentially displays the infinitesimal information of $X$ around $x$, in particular, it contains the tangent space. In the case of an abelian variety this subscheme should determine the Lie algebra.

If $X$ is finite type over a field, say, then if $X$ is smooth then all the infinitesimal information is contained in this first order infinitesimal neighborhood. If $X$ is singular at $x$ you need the full "tangent cone" and to recover it from local data you need to pass to the limit of $\operatorname{Spec}(\mathcal{O}_X/\mathcal{I}_x^n)$ over $n \geq 1$ to get a formal scheme as suggested by Mariano.

If $X$ is an abelian variety, the completion of $X$ along the neutral point gives a formal group with topological dimension 0 but with "hidden dimension" that of $X$. To get more information about this, search for formal groups, there are several resources online an in print, especially formal groups associated to abelian varieties.

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