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!



First order infinitesimal neighborhood of a point
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.


