Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Another proof of the almost purity theorem for perfectoid valuation
  rings   The almost purity theorem is central to the geometry of perfectoid spaces and
has numerous applications in algebra and geometry. This result is known to have
several different proofs in the case that the base ring is a perfectoid
valuation ring. We give a new proof by exploiting the behavior of Faltings'
normalized length under the Frobenius map.

22
The categories of this paper are: chao-dyn
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The categories of this paper are: solv-int
18
The categories of this paper are: econ
15
The categories of this paper are: nucl-ex
29
The categories of this paper are: funct-an
14
The categories of this paper are: math-ph
37
The categories of this paper are: supr-con
13
The categories of this paper are: quant-ph
9
The categories of this paper are: nlin
12
The categories of this paper are: nucl-th
27
The categories of this paper are: comp-gas
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The categories of this paper are: dg-ga
5
The categories of this paper are: gr-qc
21
The categories of this paper are: adap-org
4
The categories of this paper are: cond-mat
16
The categories of this paper are: stat