Cohomology.lean`

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8 declarations, 0 contain sorry, 0 declared as axioms

Cohomology of a bialgebra

Let A be a bialgebra over a field k, and let k be the trivial A-module through the counit (Bialgebra.trivialModuleCat k A). We define

Bialgebra.cohomologyRing k A: the cohomology ring H^*(A, k) = ⨁ n, Ext^n_A(k, k), a graded k-algebra under the Yoneda product;
Bialgebra.cohomology k A M: the cohomology H^*(A, M) = ⨁ n, Ext^n_A(k, M) with coefficients in an A-module M, a graded module over H^*(A, k).

Here Ext is taken in ModuleCat A, the category of all A-modules, and not in the category of finite-dimensional ones. When A is finite-dimensional over a field the two give the same groups on finite-dimensional modules: such a module admits a resolution by finitely generated free A-modules, which are again finite-dimensional and are projective in both categories, and Ext is computed from that single resolution either way. So cohomologyRing k A is the cohomology of the finite tensor category of finite-dimensional A-modules, which is what the literature means by H^*(A, k).

Two k-structures on ModuleCat A are involved, and only one of them is scoped. ModuleCat.linearOverField, giving Linear k (ModuleCat A), is a global instance, so the k-algebra structure on cohomologyRing k A registered below needs no open scoped. The k-module structure on an individual object, ModuleCat.moduleOfAlgebraModule, is a scoped instance; in both, k acts through algebraMap k A. A file that mentions FiniteDimensional k M for M : ModuleCat A therefore does have to open scoped ModuleCat.Algebra.

Imports (3)

Imported by 1 file

FormalConjecturesForMathlib.lean

abbrev Bialgebra.cohomologyRing The cohomology ring H^*(A, k) = ⨁ n, Ext^n_A(k, k) of the bialgebra A over k, where k is the trivial A-module through the counit and Ext is taken in ModuleCat A line 63 source
Source documentation
The cohomology ring H^*(A, k) = ⨁ n, Ext^n_A(k, k) of the bialgebra A over k, where k is the trivial A-module through the counit and Ext is taken in ModuleCat A. It is a graded k-algebra under the Yoneda product.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Ring, Bialgebra
Proof uses: DirectSum, Nat, CategoryTheory.Abelian.Ext, ModuleCat, Bialgebra.trivialModuleCat
12 instances, coercions and auxiliary constants left out.
```
abbrev cohomologyRing : Type u :=
…
```
abbrev Bialgebra.cohomology The cohomology H^*(A, M) = ⨁ n, Ext^n_A(k, M) of the bialgebra A over k with coefficients in the A-module M line 69 source
Source documentation
The cohomology H^*(A, M) = ⨁ n, Ext^n_A(k, M) of the bialgebra A over k with coefficients in the A-module M. It is a graded module over cohomologyRing k A under the Yoneda product.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Ring, Bialgebra, ModuleCat
Proof uses: DirectSum, Nat, CategoryTheory.Abelian.Ext, Bialgebra.trivialModuleCat
12 instances, coercions and auxiliary constants left out.
```
abbrev cohomology (M : ModuleCat.{u} A) : Type u :=
…
```
instance Bialgebra.(instance at line 72) line 72 source
instance Bialgebra.(instance at line 75) line 75 source
lemma Bialgebra.cohomologyRing_of_mul_of The product on cohomologyRing k A is the Yoneda composition of extensions line 81 source
Source documentation
The product on cohomologyRing k A is the Yoneda composition of extensions.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Ring, Bialgebra, Nat, CategoryTheory.Abelian.Ext, ModuleCat, Int, Bialgebra.trivialModuleCat, Eq, DirectSum, HMul.hMul, AddMonoidHom, DirectSum.of, HAdd.hAdd, CategoryTheory.Abelian.Ext.comp, rfl
Proof uses: DirectSum.of_mul_of
44 instances, coercions and auxiliary constants left out.
```
lemma cohomologyRing_of_mul_of {a b : ℕ}
    (x : Ext (trivialModuleCat k A) (trivialModuleCat k A) a)
    (y : Ext (trivialModuleCat k A) (trivialModuleCat k A) b) :
    (DirectSum.of _ a x : cohomologyRing k A) * DirectSum.of _ b y =
      DirectSum.of _ (a + b) (x.comp y rfl) :=
…
```
Mentions: Bialgebra.trivialModuleCat, Bialgebra.cohomologyRing
lemma Bialgebra.cohomologyRing_algebraMap_apply A scalar c : k acts on cohomologyRing k A as c times the identity in degree 0 line 90 source
Source documentation
A scalar c : k acts on cohomologyRing k A as c times the identity in degree 0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Ring, Bialgebra, Eq, Bialgebra.cohomologyRing, RingHom, Nat, CategoryTheory.Abelian.Ext, ModuleCat, Bialgebra.trivialModuleCat, Int, Algebra.algebraMap, AddMonoidHom, OfNat.ofNat, DirectSum, DirectSum.of, CategoryTheory.Abelian.Ext.mk₀, HSMul.hSMul, Quiver.Hom, CategoryTheory.CategoryStruct.id
Proof uses: rfl
63 instances, coercions and auxiliary constants left out.
```
lemma cohomologyRing_algebraMap_apply (c : k) :
    algebraMap k (cohomologyRing k A) c =
      DirectSum.of _ 0 (Ext.mk₀ (c • 𝟙 (trivialModuleCat k A))) :=
…
```
Mentions: c, Bialgebra.cohomologyRing, Bialgebra.trivialModuleCat
lemma Bialgebra.nontrivial_cohomologyRing The cohomology ring is not the zero ring: its degree-zero part contains the identity of the trivial module line 97 source
Source documentation
The cohomology ring is not the zero ring: its degree-zero part contains the identity of the trivial module. In particular a finiteness statement about cohomologyRing k A is not vacuous.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Ring, Bialgebra, Nontrivial, Bialgebra.cohomologyRing
Proof uses: Ne, Quiver.Hom, ModuleCat, Bialgebra.trivialModuleCat, CategoryTheory.CategoryStruct.id, OfNat.ofNat, Eq, congrArg, ModuleCat.Hom, LinearMap, RingHom.id, ModuleCat.Hom.hom, one_ne_zero, CategoryTheory.Abelian.Ext, Int, Nat, CategoryTheory.Abelian.Ext.mk₀, Function.Bijective.injective, CategoryTheory.Abelian.Ext.mk₀_bijective, Eq.mpr and 16 more
65 instances, coercions and auxiliary constants left out.

Tactic trace: 13 steps
Open to load the checked before-and-after goal states.
```
lemma nontrivial_cohomologyRing : Nontrivial (cohomologyRing k A) := by
…
```
Mentions: Bialgebra.cohomologyRing
lemma Bialgebra.cohomology_of_smul_of The action of cohomologyRing k A on cohomology k A M is the Yoneda composition of extensions line 115 source
Source documentation
The action of cohomologyRing k A on cohomology k A M is the Yoneda composition of extensions.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Ring, Bialgebra, ModuleCat, Nat, CategoryTheory.Abelian.Ext, Int, Bialgebra.trivialModuleCat, Eq, DirectSum, HSMul.hSMul, AddMonoidHom, DirectSum.of, HAdd.hAdd, CategoryTheory.Abelian.Ext.comp, rfl
Proof uses: DirectSum.Gmodule.of_smul_of
44 instances, coercions and auxiliary constants left out.
```
lemma cohomology_of_smul_of {M : ModuleCat.{u} A} {a b : ℕ}
    (x : Ext (trivialModuleCat k A) (trivialModuleCat k A) a)
    (y : Ext (trivialModuleCat k A) M b) :
    (DirectSum.of _ a x : cohomologyRing k A) • (DirectSum.of _ b y : cohomology k A M) =
      DirectSum.of _ (a + b) (x.comp y rfl) :=
…
```
Mentions: Bialgebra.trivialModuleCat, Bialgebra.cohomologyRing, Bialgebra.cohomology