English
The trace map in this AKLB setup is defined as the specialization of intTraceAux to the FractionField construction, yielding a linear form B → A extending the usual notion of trace.
Русский
След в рамках данной установки AKLB определяется какSpecialization intTraceAux на FractionField, образующий линейную форму B → A,Extensions обычного следа.
LaTeX
$$$[IsIntegrallyClosed A]\\;:\\;\\text{intTrace } : B \\to_A A \\text{ is defined by }(\\text{IntTraceAux})$$$
Lean4
/-- The trace of a finite extension of integrally closed domains `B/A` is the restriction of
the trace on `Frac(B)/Frac(A)` onto `B/A`. See `Algebra.algebraMap_intTrace`. -/
noncomputable def intTrace : B →ₗ[A] A :=
haveI : IsIntegralClosure B A (FractionRing B) := IsIntegralClosure.of_isIntegrallyClosed _ _ _
haveI : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B) :=
IsIntegralClosure.isLocalization _ (FractionRing A) _ _
haveI : FiniteDimensional (FractionRing A) (FractionRing B) := .of_isLocalization A B A⁰
Algebra.intTraceAux A (FractionRing A) (FractionRing B) B
