Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Extensions of valuation rings containing $\bf Q$ as limits of smooth
  algebras   We give a necessary and sufficient condition for an extension of valuation
rings containing $\bf Q$ to be a filtered direct limit of smooth algebras.

24
The categories of this paper are: dg-ga
12
The categories of this paper are: nucl-th
35
The categories of this paper are: mtrl-th
16
The categories of this paper are: stat
22
The categories of this paper are: chao-dyn
14
The categories of this paper are: math-ph
19
The categories of this paper are: eess
7
The categories of this paper are: hep-th
36
The categories of this paper are: cmp-lg
28
The categories of this paper are: alg-geom
33
The categories of this paper are: plasm-ph
31
The categories of this paper are: ao-sci
21
The categories of this paper are: adap-org
0
The categories of this paper are: hep-ph
10
The categories of this paper are: q-bio
17
The categories of this paper are: q-fin