Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Detecting flatness over smooth bases   Given an essentially finite type morphism of schemes f: X --> Y and a
positive integer d, let f^{d}: X^{d} --> Y denote the natural map from the
d-fold fiber product, X^{d}, of X over Y and \pi_i: X^{d} --> X the i'th
canonical projection. When Y smooth over a field and F is a coherent sheaf on
X, it is proved that F is flat over Y if (and only if) f^{d} maps the
associated points of the tensor product sheaf \otimes_{i=1}^d \pi_i^*(F) to
generic points of Y, for some d greater than or equal to dim Y. The equivalent
statement in commutative algebra is an analog---but not a consequence---of a
classical criterion of Auslander and Lichtenbaum for the freeness of finitely
generated modules over regular local rings.

37
The categories of this paper are: supr-con
18
The categories of this paper are: econ
14
The categories of this paper are: math-ph
0
The categories of this paper are: hep-ph
30
The categories of this paper are: q-alg
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The categories of this paper are: chem-ph
28
The categories of this paper are: alg-geom
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The categories of this paper are: mtrl-th
26
The categories of this paper are: bayes-an
10
The categories of this paper are: q-bio
5
The categories of this paper are: gr-qc
25
The categories of this paper are: patt-sol
1
The categories of this paper are: cs
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The categories of this paper are: atom-ph
3
The categories of this paper are: physics
19
The categories of this paper are: eess