History of tag 0CKS
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assigned tag 0CKS
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2016-06-17 |
4d763d7
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Tags: Added new tags
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created statement with label lemma-max-weakly-etale-subalgebra in more-algebra.tex
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2016-06-16 |
7d0dc16 |
Maximal weakly etale subalgebras
Thanks to Ofer Gabber who writes
For a field k the class of ind-étale (equivalently weakly étale) k-algebras
is closed under taking subalgebras, quotients and tensor products,
and every k-algebra A has a largest ind-étale subalgebra A_indet .
If k' is a separable algebraic extension of k and A' the base change of A
to k' ,then A'_indet in the sense of k-algebras and in the sense of
k' algebras coincide; if furthermore k' is Galois over k then the Galois
group acts semi-linearly on A'_indet, with fixed subalgebra identified
with A_indet; hence for finite Galois extensions we get by Galois descent
that A_indet tensor (over k)k' maps isomorphically to A'_indet ,hence this
also holds for all finite separable extensions .[ In fact the formation
of the ind-étale part commutes with arbitrary ground field extension and
with tensor product, but I will not need below this more general
information.]
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