Extension of Valuations #
In this file, we define the typeclass for valuation extensions and prove basic facts about the
extension of valuations. Let A
be an R
algebra, equipped with valuations vA
and vR
respectively. Here, the extension of a valuation means that the pullback of valuation vA
to R
is equivalent to the valuation vR
on R
. We only require equivalence, not equality, of
valuations here.
Note that we do not require the ring map from R
to A
to be injective. This holds automatically
when R
is a division ring and A
is nontrivial.
A motivation for choosing the more flexible Valuation.Equiv
rather than strict equality here is
to allow for possible normalization. As an example, consider a finite extension K
of ℚ_[p]
,
which is a discretely valued field. We may choose the valuation on K
to be either:
the valuation where the uniformizer is mapped to one (more precisely,
-1
inℤₘ₀
) orthe valuation where
p
is mapped to one.
For the algebraic closure of ℚ_[p]
, if we choose the valuation of p
to be one, then the
restriction of this valuation to K
equals the second valuation, but is only equivalent to the
first valuation. The flexibility of equivalence here allows us to develop theory for both cases
without first determining the normalizations once and for all.
Main Definition #
Valuation.HasExtension vR vA
: The valuationvA
onA
is an extension of the valuationvR
onR
.
References #
- [Bourbaki, Nicolas. Commutative algebra] Chapter VI §3, Valuations.
- https://en.wikipedia.org/wiki/Valuation_(algebra)#Extension_of_valuations
Tags #
Valuation, Extension of Valuations
The class Valuation.HasExtension vR vA
states that the valuation vA
on A
is an extension of
the valuation vR
on R
. More precisely, vR
is equivalent to the comap of the valuation vA
.
- val_isEquiv_comap : vR.IsEquiv (comap (algebraMap R A) vA)
The valuation
vR
onR
is equivalent to the comap of the valuationvA
onA
Instances
Alias of Valuation.HasExtension
.
The class Valuation.HasExtension vR vA
states that the valuation vA
on A
is an extension of
the valuation vR
on R
. More precisely, vR
is equivalent to the comap of the valuation vA
.
Equations
Instances For
When K
is a field, if the preimage of the valuation integers of A
equals to the valuation
integers of K
, then the valuation on A
is an extension of the valuation on K
.
Equations
- One or more equations did not get rendered due to their size.