IntersectionMultiplicity.lean`

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4 declarations, 0 contain sorry, 0 declared as axioms

Serre's intersection multiplicity

For a commutative ring R and R-modules M, N, this file defines Serre's intersection multiplicity $$\chi(M, N) = \sum_{i \ge 0} (-1)^i \ell_R(\operatorname{Tor}_i^R(M, N)),$$ using CategoryTheory.Tor on ModuleCat R.

The definition is meaningful when R is a regular local ring, M and N are finitely generated and M ⊗[R] N has finite length: then Tor_i vanishes for i > dim R, so the sum is finite, and every Tor_i has finite length, so ENat.toNat truncates nothing.

Main declarations

Module.intersectionMultiplicity R M N: the alternating sum above.
Module.intersectionMultiplicity_of_projective: if N is projective, χ(M, N) = ℓ(M ⊗ N).
Module.intersectionMultiplicity_eq_zero_of_subsingleton: if M = 0, then χ(M, N) = 0.
Module.intersectionMultiplicity_self_self: over a field k, χ(k, k) = 1.

Imports (6)

Imported by 1 file

FormalConjecturesForMathlib.lean

def Module.intersectionMultiplicity The intersection multiplicity of two modules M and N over a commutative ring R line 62 source
Source documentation
The intersection multiplicity of two modules M and N over a commutative ring R, $$\chi(M, N) = \sum_{i \ge 0} (-1)^i \ell_R(\operatorname{Tor}_i^R(M, N)).$$ Here Tor is CategoryTheory.Tor on ModuleCat R, the sum is a finsum over ℕ, and the length of each Tor module is converted to a natural number by ENat.toNat.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, AddCommGroup, Module, Int
Proof uses: finsum, Nat, HMul.hMul, HPow.hPow, Neg.neg, OfNat.ofNat, ENat.toNat, Module.length, CategoryTheory.Functor.obj, ModuleCat, CategoryTheory.Functor, CategoryTheory.Tor, ModuleCat.of
27 instances, coercions and auxiliary constants left out.
```
noncomputable def intersectionMultiplicity : ℤ :=
…
```
theorem Module.intersectionMultiplicity_of_projective If N is projective, only Tor₀(M, N) = M ⊗ N contributes to χ(M, N) line 70 source
Source documentation
If N is projective, only Tor₀(M, N) = M ⊗ N contributes to χ(M, N).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, AddCommGroup, Module, Module.Projective, Eq, Int, Module.intersectionMultiplicity, ENat.toNat, Module.length, TensorProduct
Proof uses: id, Eq.mpr, finsum, Nat, HMul.hMul, HPow.hPow, Neg.neg, OfNat.ofNat, CategoryTheory.Functor.obj, ModuleCat, CategoryTheory.Functor, CategoryTheory.Tor, ModuleCat.of, congrArg, finsum_eq_single, Ne, Nat.succ, Exists, Nat.exists_eq_succ_of_ne_zero, Subsingleton and 26 more
71 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
```
theorem intersectionMultiplicity_of_projective [Module.Projective R N] :
    intersectionMultiplicity R M N = (Module.length R (M ⊗[R] N)).toNat := by
…
```
Mentions: Module.intersectionMultiplicity
theorem Module.intersectionMultiplicity_eq_zero_of_subsingleton If M is the zero module, then χ(M, N) = 0: the functor M ⊗ - is zero, so all its left derived functors vanish line 89 source
Source documentation
If M is the zero module, then χ(M, N) = 0: the functor M ⊗ - is zero, so all its left derived functors vanish.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, AddCommGroup, Module, Subsingleton, Eq, Int, Module.intersectionMultiplicity, OfNat.ofNat
Proof uses: CategoryTheory.Limits.IsZero, ModuleCat, ModuleCat.of, ModuleCat.isZero_of_subsingleton, Nat, CategoryTheory.Functor.obj, CategoryTheory.Functor, CategoryTheory.Tor, CategoryTheory.Limits.IsZero.of_iso, HomologicalComplex, ComplexShape.down, HomologicalComplex.homologyFunctor, CategoryTheory.Functor.mapHomologicalComplex, CategoryTheory.MonoidalCategory.tensoringLeft, CategoryTheory.ProjectiveResolution.complex, CategoryTheory.projectiveResolution, CategoryTheory.ShortComplex.isZero_homology_of_isZero_X₂, HomologicalComplex.sc, CategoryTheory.Functor.map_isZero, CategoryTheory.MonoidalCategory.tensoringRight and 22 more
68 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
```
theorem intersectionMultiplicity_eq_zero_of_subsingleton [Subsingleton M] :
    intersectionMultiplicity R M N = 0 := by
…
```
Mentions: Module.intersectionMultiplicity
theorem Module.intersectionMultiplicity_self_self Normalisation: over a field k, χ(k, k) = 1 line 103 source
Source documentation
Normalisation: over a field k, χ(k, k) = 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Eq, Int, Module.intersectionMultiplicity, OfNat.ofNat
Proof uses: Eq.mpr, ENat.toNat, Module.length, TensorProduct, id, congrArg, Module.intersectionMultiplicity_of_projective, ENat, LinearEquiv.length_eq, TensorProduct.lid, Module.length_eq_one, Eq.refl
33 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem intersectionMultiplicity_self_self (k : Type u) [Field k] :
    intersectionMultiplicity k k k = 1 := by
…
```
Mentions: Module.intersectionMultiplicity