TrivialModule.lean`

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5 declarations, 0 contain sorry, 0 declared as axioms

The trivial module of a bialgebra

If A is a bialgebra over a commutative ring R, the counit A →ₐ[R] R makes R into an A-module, the trivial module. We package it as the object Bialgebra.trivialModuleCat R A of ModuleCat A.

Over a field k, restricting scalars along algebraMap k A recovers the k-module structure of k itself (Bialgebra.trivialModuleCat_smul_base), so the trivial module is one-dimensional; Bialgebra.trivialModuleCatLinearEquiv and the FiniteDimensional instance record this. That k-module structure is ModuleCat.moduleOfAlgebraModule, a scoped instance, so a file stating FiniteDimensional k (trivialModuleCat k A) must open scoped ModuleCat.Algebra.

Imports (4)

Imported by 2 files

def Bialgebra.trivialModuleCat The trivial module of the bialgebra A over R: the ring R on which A acts through the counit A →ₐ[R] R, as an object of ModuleCat A line 47 source
Source documentation
The trivial module of the bialgebra A over R: the ring R on which A acts through the counit A →ₐ[R] R, as an object of ModuleCat A.

standard axioms: Quot.sound, propext what this means
Statement uses: CommRing, Ring, Bialgebra, ModuleCat
Proof uses: ModuleCat.of, Module.compHom, AlgHom.toRingHom, Bialgebra.counitAlgHom
11 instances, coercions and auxiliary constants left out.
```
def trivialModuleCat : ModuleCat.{u} A :=
…
```
lemma Bialgebra.trivialModuleCat_smul The bialgebra A acts on its trivial module through the counit line 55 source
Source documentation
The bialgebra A acts on its trivial module through the counit. The carrier of trivialModuleCat R A is R by definition, so the product on the right is the product of R.

standard axioms: Quot.sound, propext what this means
Statement uses: CommRing, Ring, Bialgebra, Bialgebra.trivialModuleCat, Eq, HSMul.hSMul, HMul.hMul, AlgHom, Bialgebra.counitAlgHom
Proof uses: rfl
44 instances, coercions and auxiliary constants left out.
```
lemma trivialModuleCat_smul (a : A) (r : trivialModuleCat R A) :
    a • r = HMul.hMul (α := R) (β := R) (counitAlgHom R A a) r :=
…
```
Mentions: Bialgebra.trivialModuleCat
lemma Bialgebra.trivialModuleCat_smul_base Documented declaration: trivialModuleCat_smul_base line 68 source
Source documentation
The k-module structure on the trivial module of a bialgebra over a field k, obtained by restricting scalars along algebraMap k B, is the module structure of k itself: the counit is a k-algebra map, so it sends algebraMap k B c to c.

standard axioms: Quot.sound, propext what this means
Statement uses: Field, Ring, Bialgebra, Bialgebra.trivialModuleCat, Eq, HSMul.hSMul, ModuleCat.Algebra.instModuleCarrier, HMul.hMul
Proof uses: Eq.mpr, RingHom, Algebra.algebraMap, id, congrArg, rfl, AlgHom, Bialgebra.counitAlgHom, Bialgebra.trivialModuleCat_smul, AlgHom.commutes, Eq.refl
58 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
lemma trivialModuleCat_smul_base (c : k) (r : trivialModuleCat k B) :
    c • r = HMul.hMul (α := k) (β := k) c r := by
…
```
Mentions: c, Bialgebra.trivialModuleCat
def Bialgebra.trivialModuleCatLinearEquiv The trivial module of a bialgebra over a field k is k itself as a k-module; in particular it is one-dimensional line 77 source
Source documentation
The trivial module of a bialgebra over a field k is k itself as a k-module; in particular it is one-dimensional.

standard axioms: Quot.sound, propext what this means
Statement uses: Field, Ring, Bialgebra, LinearEquiv, RingHom.id, Bialgebra.trivialModuleCat, ModuleCat.Algebra.instModuleCarrier
Proof uses: LinearEquiv.mk, LinearMap.mk, AddHom.mk, Bialgebra.trivialModuleCat_smul_base
34 instances, coercions and auxiliary constants left out.
```
def trivialModuleCatLinearEquiv : trivialModuleCat k B ≃ₗ[k] k where
…
```
Mentions: Bialgebra.trivialModuleCat
instance Bialgebra.(instance at line 85) line 85 source