The R-algebra structure on families of R-algebras #
The R-algebra structure on Π i : I, A i
when each A i
is an R-algebra.
Main definitions #
Equations
- Pi.algebra ι A = Algebra.mk (Pi.ringHom fun (i : ι) => algebraMap R (A i)) ⋯ ⋯
A family of algebra homomorphisms g i : B →ₐ[R] A i
defines a ring homomorphism
Pi.algHom g : B →ₐ[R] Π i, A i
given by Pi.algHom g x i = g i x
.
Equations
- Pi.algHom R A g = { toRingHom := Pi.ringHom fun (i : ι) => (g i).toRingHom, commutes' := ⋯ }
Instances For
Function.eval
as an AlgHom
. The name matches Pi.evalRingHom
, Pi.evalMonoidHom
,
etc.
Equations
- Pi.evalAlgHom R A i = { toFun := fun (f : (i : ι) → A i) => f i, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯, commutes' := ⋯ }
Instances For
Pi.algHom
commutes with composition.
Equations
- Pi.instAlgebraForall A S = Algebra.mk (Pi.ringHom fun (x : ι) => (algebraMap (S x) (A x)).comp (Pi.evalRingHom S x)) ⋯ ⋯
Function.const
as an AlgHom
. The name matches Pi.constRingHom
, Pi.constMonoidHom
,
etc.
Equations
- Pi.constAlgHom R A B = { toFun := Function.const A, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯, commutes' := ⋯ }
Instances For
When R
is commutative and permits an algebraMap
, Pi.constRingHom
is equal to that
map.
A special case of Pi.algebra
for non-dependent types. Lean struggles to elaborate
definitions elsewhere in the library without this.
Equations
- Function.algebra ι A = Pi.algebra ι fun (a : ι) => A
R
-algebra homomorphism between the function spaces ι → A
and ι → B
, induced by an
R
-algebra homomorphism f
between A
and B
.
Equations
Instances For
A family of algebra equivalences ∀ i, (A₁ i ≃ₐ A₂ i)
generates a
multiplicative equivalence between Π i, A₁ i
and Π i, A₂ i
.
This is the AlgEquiv
version of Equiv.piCongrRight
, and the dependent version of
AlgEquiv.arrowCongr
.
Equations
- One or more equations did not get rendered due to their size.