Formal functions and formal embeddings
Heisuke Hironaka, H. Matsumura · Journal of the Mathematical Society of Japan · 1968
This work contains two theorems, among others, which determine the field of formal-rational functions, $\hat{K}$ , along a closed algebraic set $X$ in a projective space $P$ and in an abelian variety $A$ , respectively.For obvious reasons, we assume that $X$ is connected and has a positive dimension.In the case of am- bient variety $p$ , the answer is that $\hat{K}$ is exactly the field of rational functions on $P$ .If $A$ is the ambient variety, $\hat{K}$ coincides with the field of rational func- tions on a certain abelian scheme $A^{*}$ over a certain complete local ring $R$ , which is derived from the given pair (X, $A$ ).For instance, if $X$ generates $A$ , then $R$ is nothing but the base field of $A$ and $A^{*}$ is the maximal one, say Al (X, $A$ ), among those \'etale and proper (hence abelian) extensions of $A$ which are dominated by the albanese variety of $X$ .In the general case, the origin of $A$ being chosen in $X$ with no loss of generality, let $A^{\prime}$ be the abelian sub- variety of $A$ which is generated by $X$ , and $A^{\prime\prime}=A/A^{\prime}$ .Then $R$ is the completion of the local ring of $A^{\prime\prime}$ at the origin, and $A^{*}$ is the unique etale extension of $A\times A^{g}Spec(R)$ that induces the covering Al (X, $A^{\prime}$ ) of the closed fibre $A^{\prime}$ . (There exists a non-canonical isomorphism of $A^{*}$ with the product Al (X, $A^{\prime}$ ) $\times Spec(R).)$In [1], a general problem was posed about the existence of a certain uni- versal scheme associated with any given formal scheme.The results of this paper readily imply an affirmative and explicit solution to the problem when the given formal scheme is the completion of $P$ (resp.$A$ ) along $X$ as above.Namely, in this case, $P$ itself (resp.$A^{*}$ described above) is the universal solu- tion, $i$ .$e.$ , the most dominant (in the sense of an arbitrary small neighborhood of the image of $X$ ) scheme of finite type over the ring of formal-regular func- tions, which in this case, is $k$ (resp.$R$ ).Meanwhile, Hartshorne gave an affir- mative answer to the same problem, when $X$ is a smooth (or more generally,