Reducing a module modulo an element of the ring

Given a commutative ring R and an element r : R, the association M ↦ M ⧸ rM extends to a functor on the category of R-modules. This functor is isomorphic to the functor of tensoring by R ⧸ (r) on either side, but can be more convenient to work with since we can work with quotient types instead of fiddling with simple tensors.

Tags

module, commutative algebra

@[reducible, inline]

abbrev QuotSMulTop {R : Type u_2} [CommRing R] (r : R) (M : Type u_1) [AddCommGroup M] [Module R M] : Type u_1

An abbreviation for M⧸rM that keeps us from having to write (⊤ : Submodule R M) over and over to satisfy the typechecker.

Equations

• QuotSMulTop r M = (M ⧸ r • ⊤)

noncomputable def QuotSMulTop.equivQuotTensor {R : Type u_2} [CommRing R] (r : R) (M : Type u_1) [AddCommGroup M] [Module R M] : QuotSMulTop r M ≃ₗ[R] TensorProduct R (R ⧸ Ideal.span {r}) M

Reducing a module modulo r is the same as left tensoring with R/(r).

Equations

• QuotSMulTop.equivQuotTensor r M = ((r • ⊤).quotEquivOfEq (Ideal.span {r} • ⊤) ⋯).trans (quotTensorEquivQuotSMul M (Ideal.span {r})).symm

noncomputable def QuotSMulTop.equivTensorQuot {R : Type u_2} [CommRing R] (r : R) (M : Type u_1) [AddCommGroup M] [Module R M] : QuotSMulTop r M ≃ₗ[R] TensorProduct R M (R ⧸ Ideal.span {r})

Reducing a module modulo r is the same as right tensoring with R/(r).

Equations

• QuotSMulTop.equivTensorQuot r M = ((r • ⊤).quotEquivOfEq (Ideal.span {r} • ⊤) ⋯).trans (tensorQuotEquivQuotSMul M (Ideal.span {r})).symm

def QuotSMulTop.map {R : Type u_2} [CommRing R] (r : R) {M : Type u_1} {M' : Type u_3} [AddCommGroup M] [Module R M] [AddCommGroup M'] [Module R M'] : (M →ₗ[R] M') →ₗ[R] QuotSMulTop r M →ₗ[R] QuotSMulTop r M'

The action of the functor QuotSMulTop r on morphisms.

Equations

• QuotSMulTop.map r = (r • ⊤).mapQLinear (r • ⊤) ∘ₗ LinearMap.codRestrict ((r • ⊤).compatibleMaps (r • ⊤)) LinearMap.id ⋯

@[simp]

theorem QuotSMulTop.map_apply_mk {R : Type u_2} [CommRing R] (r : R) {M : Type u_1} {M' : Type u_3} [AddCommGroup M] [Module R M] [AddCommGroup M'] [Module R M'] (f : M →ₗ[R] M') (x : M) : ((QuotSMulTop.map r) f) (Submodule.Quotient.mk x) = Submodule.Quotient.mk (f x)

theorem QuotSMulTop.map_comp_mkQ {R : Type u_2} [CommRing R] (r : R) {M : Type u_1} {M' : Type u_3} [AddCommGroup M] [Module R M] [AddCommGroup M'] [Module R M'] (f : M →ₗ[R] M') : (QuotSMulTop.map r) f ∘ₗ (r • ⊤).mkQ = (r • ⊤).mkQ ∘ₗ f

@[simp]

theorem QuotSMulTop.map_id {R : Type u_2} [CommRing R] (r : R) (M : Type u_1) [AddCommGroup M] [Module R M] : (QuotSMulTop.map r) LinearMap.id = LinearMap.id

@[simp]

theorem QuotSMulTop.map_comp {R : Type u_2} [CommRing R] (r : R) {M : Type u_1} {M' : Type u_3} {M'' : Type u_4} [AddCommGroup M] [Module R M] [AddCommGroup M'] [Module R M'] [AddCommGroup M''] [Module R M''] (g : M' →ₗ[R] M'') (f : M →ₗ[R] M') : (QuotSMulTop.map r) (g ∘ₗ f) = (QuotSMulTop.map r) g ∘ₗ (QuotSMulTop.map r) f

theorem QuotSMulTop.equivQuotTensor_naturality_mk {R : Type u_2} [CommRing R] (r : R) {M : Type u_1} {M' : Type u_3} [AddCommGroup M] [Module R M] [AddCommGroup M'] [Module R M'] (f : M →ₗ[R] M') (x : M) : (QuotSMulTop.equivQuotTensor r M') (((QuotSMulTop.map r) f) (Submodule.Quotient.mk x)) = (LinearMap.lTensor (R ⧸ Ideal.span {r}) f) ((QuotSMulTop.equivQuotTensor r M) (Submodule.Quotient.mk x))

theorem QuotSMulTop.equivQuotTensor_naturality {R : Type u_2} [CommRing R] (r : R) {M : Type u_1} {M' : Type u_3} [AddCommGroup M] [Module R M] [AddCommGroup M'] [Module R M'] (f : M →ₗ[R] M') : ↑(QuotSMulTop.equivQuotTensor r M') ∘ₗ (QuotSMulTop.map r) f = LinearMap.lTensor (R ⧸ Ideal.span {r}) f ∘ₗ ↑(QuotSMulTop.equivQuotTensor r M)

theorem QuotSMulTop.equivTensorQuot_naturality_mk {R : Type u_2} [CommRing R] (r : R) {M : Type u_1} {M' : Type u_3} [AddCommGroup M] [Module R M] [AddCommGroup M'] [Module R M'] (f : M →ₗ[R] M') (x : M) : (QuotSMulTop.equivTensorQuot r M') (((QuotSMulTop.map r) f) (Submodule.Quotient.mk x)) = (LinearMap.rTensor (R ⧸ Ideal.span {r}) f) ((QuotSMulTop.equivTensorQuot r M) (Submodule.Quotient.mk x))

theorem QuotSMulTop.equivTensorQuot_naturality {R : Type u_2} [CommRing R] (r : R) {M : Type u_1} {M' : Type u_3} [AddCommGroup M] [Module R M] [AddCommGroup M'] [Module R M'] (f : M →ₗ[R] M') : ↑(QuotSMulTop.equivTensorQuot r M') ∘ₗ (QuotSMulTop.map r) f = LinearMap.rTensor (R ⧸ Ideal.span {r}) f ∘ₗ ↑(QuotSMulTop.equivTensorQuot r M)

theorem QuotSMulTop.map_surjective {R : Type u_2} [CommRing R] (r : R) {M : Type u_1} {M' : Type u_3} [AddCommGroup M] [Module R M] [AddCommGroup M'] [Module R M'] {f : M →ₗ[R] M'} (hf : Function.Surjective ⇑f) : Function.Surjective ⇑((QuotSMulTop.map r) f)

theorem QuotSMulTop.map_exact {R : Type u_2} [CommRing R] (r : R) {M : Type u_1} {M' : Type u_3} {M'' : Type u_4} [AddCommGroup M] [Module R M] [AddCommGroup M'] [Module R M'] [AddCommGroup M''] [Module R M''] {f : M →ₗ[R] M'} {g : M' →ₗ[R] M''} (hfg : Function.Exact ⇑f ⇑g) (hg : Function.Surjective ⇑g) : Function.Exact ⇑((QuotSMulTop.map r) f) ⇑((QuotSMulTop.map r) g)

noncomputable def QuotSMulTop.tensorQuotSMulTopEquivQuotSMulTop {R : Type u_2} [CommRing R] (r : R) (M : Type u_1) (M' : Type u_3) [AddCommGroup M] [Module R M] [AddCommGroup M'] [Module R M'] : TensorProduct R M (QuotSMulTop r M') ≃ₗ[R] QuotSMulTop r (TensorProduct R M M')

Tensoring on the left and applying QuotSMulTop · r commute.

Equations

• One or more equations did not get rendered due to their size.

noncomputable def QuotSMulTop.quotSMulTopTensorEquivQuotSMulTop {R : Type u_2} [CommRing R] (r : R) (M : Type u_1) (M' : Type u_3) [AddCommGroup M] [Module R M] [AddCommGroup M'] [Module R M'] : TensorProduct R (QuotSMulTop r M') M ≃ₗ[R] QuotSMulTop r (TensorProduct R M' M)

Tensoring on the right and applying QuotSMulTop · r commute.

Equations

• One or more equations did not get rendered due to their size.