# mathlibdocumentation

ring_theory.derivation

# Derivations #

This file defines derivation. A derivation D from the R-algebra A to the A-module M is an R-linear map that satisfy the Leibniz rule D (a * b) = a * D b + D a * b.

## Main results #

• derivation: The type of R-derivations from A to M. This has an A-module structure.
• derivation.llcomp: We may compose linear maps and derivations to obtain a derivation, and the composition is bilinear.
• derivation.lie_algebra: The R-derivations from A to A form an lie algebra over R.
• derivation_to_square_zero_equiv_lift: The R-derivations from A into a square-zero ideal I of B corresponds to the lifts A →ₐ[R] B of the map A →ₐ[R] B ⧸ I.
• kaehler_differential: The module of kaehler differentials. For an R-algebra S, we provide the notation Ω[S⁄R] for kaehler_differential R S. Note that the slash is \textfractionsolidus.
• kaehler_differential.D: The derivation into the module of kaehler differentials.
• kaehler_differential.span_range_derivation: The image of D spans Ω[S⁄R] as an S-module.
• kaehler_differential.linear_map_equiv_derivation: The isomorphism Hom_R(Ω[S⁄R], M) ≃ₗ[S] Der_R(S, M).

## Future project #

Generalize this into bimodules.

structure derivation (R : Type u_1) (A : Type u_2) [ A] (M : Type u_3) [ M] [ M] :
Type (max u_2 u_3)

D : derivation R A M is an R-linear map from A to M that satisfies the leibniz equality. We also require that D 1 = 0. See derivation.mk' for a constructor that deduces this assumption from the Leibniz rule when M is cancellative.

TODO: update this when bimodules are defined.

Instances for derivation
@[protected, instance]
def derivation.add_monoid_hom_class {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] :
Equations
@[protected, instance]
def derivation.has_coe_to_fun {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] :
has_coe_to_fun A M) (λ (_x : A M), A → M)

Helper instance for when there's too many metavariables to apply fun_like.has_coe_to_fun directly.

Equations
theorem derivation.to_fun_eq_coe {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] (D : A M) :
@[protected, instance]
def derivation.has_coe_to_linear_map {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] :
has_coe A M) (A →ₗ[R] M)
Equations
@[simp]
theorem derivation.to_linear_map_eq_coe {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] (D : A M) :
@[simp]
theorem derivation.mk_coe {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] (f : A →ₗ[R] M) (h₁ : f 1 = 0) (h₂ : ∀ (a b : A), f (a * b) = a f b + b f a) :
@[simp, norm_cast]
theorem derivation.coe_fn_coe {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] (f : A M) :
theorem derivation.coe_injective {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] :
@[ext]
theorem derivation.ext {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] {D1 D2 : A M} (H : ∀ (a : A), D1 a = D2 a) :
D1 = D2
theorem derivation.congr_fun {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] {D1 D2 : A M} (h : D1 = D2) (a : A) :
D1 a = D2 a
@[protected]
theorem derivation.map_add {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] (D : A M) (a b : A) :
D (a + b) = D a + D b
@[protected]
theorem derivation.map_zero {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] (D : A M) :
D 0 = 0
@[simp]
theorem derivation.map_smul {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] (D : A M) (r : R) (a : A) :
D (r a) = r D a
@[simp]
theorem derivation.leibniz {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] (D : A M) (a b : A) :
D (a * b) = a D b + b D a
theorem derivation.map_sum {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] (D : A M) {ι : Type u_4} (s : finset ι) (f : ι → A) :
D (s.sum (λ (i : ι), f i)) = s.sum (λ (i : ι), D (f i))
@[simp]
theorem derivation.map_smul_of_tower {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] {S : Type u_4} [ A] [ M] [ R] (D : A M) (r : S) (a : A) :
D (r a) = r D a
@[simp]
theorem derivation.map_one_eq_zero {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] (D : A M) :
D 1 = 0
@[simp]
theorem derivation.map_algebra_map {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] (D : A M) (r : R) :
D ( A) r) = 0
@[simp]
theorem derivation.map_coe_nat {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] (D : A M) (n : ) :
D n = 0
@[simp]
theorem derivation.leibniz_pow {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] (D : A M) (a : A) (n : ) :
D (a ^ n) = n a ^ (n - 1) D a
theorem derivation.eq_on_adjoin {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] {D1 D2 : A M} {s : set A} (h : D2 s) :
D2 s)
theorem derivation.ext_of_adjoin_eq_top {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] {D1 D2 : A M} (s : set A) (hs : = ) (h : D2 s) :
D1 = D2

If adjoin of a set is the whole algebra, then any two derivations equal on this set are equal on the whole algebra.

@[protected, instance]
def derivation.has_zero {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] :
Equations
@[simp]
theorem derivation.coe_zero {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] :
0 = 0
@[simp]
theorem derivation.coe_zero_linear_map {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] :
0 = 0
theorem derivation.zero_apply {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] (a : A) :
0 a = 0
@[protected, instance]
def derivation.has_add {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] :
has_add A M)
Equations
@[simp]
theorem derivation.coe_add {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] (D1 D2 : A M) :
(D1 + D2) = D1 + D2
@[simp]
theorem derivation.coe_add_linear_map {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] (D1 D2 : A M) :
(D1 + D2) = D1 + D2
theorem derivation.add_apply {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] {D1 D2 : A M} (a : A) :
(D1 + D2) a = D1 a + D2 a
@[protected, instance]
def derivation.inhabited {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] :
Equations
@[protected, instance]
def derivation.has_smul {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] {S : Type u_4} [monoid S] [ M] [ M] [ M] :
A M)
Equations
@[simp]
theorem derivation.coe_smul {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] {S : Type u_4} [monoid S] [ M] [ M] [ M] (r : S) (D : A M) :
(r D) = r D
@[simp]
theorem derivation.coe_smul_linear_map {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] {S : Type u_4} [monoid S] [ M] [ M] [ M] (r : S) (D : A M) :
(r D) = r D
theorem derivation.smul_apply {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] (a : A) {S : Type u_4} [monoid S] [ M] [ M] [ M] (r : S) (D : A M) :
(r D) a = r D a
@[protected, instance]
def derivation.add_comm_monoid {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] :
Equations
def derivation.coe_fn_add_monoid_hom {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] :
A M →+ A → M

coe_fn as an add_monoid_hom.

Equations
@[protected, instance]
def derivation.distrib_mul_action {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] {S : Type u_4} [monoid S] [ M] [ M] [ M] :
A M)
Equations
@[protected, instance]
def derivation.is_central_scalar {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] {S : Type u_4} [monoid S] [ M] [ M] [ M] [ M] :
A M)
@[protected, instance]
def derivation.module {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] {S : Type u_4} [semiring S] [ M] [ M] [ M] :
A M)
Equations
@[protected, instance]
def derivation.is_scalar_tower {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] [ M] :
A M)
def linear_map.comp_der {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] {N : Type u_4} [ N] [ N] [ M] [ N] (f : M →ₗ[A] N) :
A M →ₗ[R] A N

We can push forward derivations using linear maps, i.e., the composition of a derivation with a linear map is a derivation. Furthermore, this operation is linear on the spaces of derivations.

Equations
@[simp]
theorem derivation.coe_to_linear_map_comp {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] (D : A M) {N : Type u_4} [ N] [ N] [ M] [ N] (f : M →ₗ[A] N) :
@[simp]
theorem derivation.coe_comp {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] (D : A M) {N : Type u_4} [ N] [ N] [ M] [ N] (f : M →ₗ[A] N) :
def derivation.llcomp {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] {N : Type u_4} [ N] [ N] [ M] [ N] :
(M →ₗ[A] N) →ₗ[A] A M →ₗ[R] A N

The composition of a derivation with a linear map as a bilinear map

Equations
def derivation.mk' {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] (D : A →ₗ[R] M) (h : ∀ (a b : A), D (a * b) = a D b + b D a) :
A M

Define derivation R A M from a linear map when M is cancellative by verifying the Leibniz rule.

Equations
@[simp]
theorem derivation.coe_mk' {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] (D : A →ₗ[R] M) (h : ∀ (a b : A), D (a * b) = a D b + b D a) :
h) = D
@[simp]
theorem derivation.coe_mk'_linear_map {R : Type u_1} {A : Type u_2} [ A] {M : Type u_3} [ M] [ M] (D : A →ₗ[R] M) (h : ∀ (a b : A), D (a * b) = a D b + b D a) :
h) = D
@[protected]
theorem derivation.map_neg {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A] [ A] {M : Type u_3} [ M] [ M] (D : A M) (a : A) :
D (-a) = -D a
@[protected]
theorem derivation.map_sub {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A] [ A] {M : Type u_3} [ M] [ M] (D : A M) (a b : A) :
D (a - b) = D a - D b
@[simp]
theorem derivation.map_coe_int {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A] [ A] {M : Type u_3} [ M] [ M] (D : A M) (n : ) :
D n = 0
theorem derivation.leibniz_of_mul_eq_one {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A] [ A] {M : Type u_3} [ M] [ M] (D : A M) {a b : A} (h : a * b = 1) :
D a = -a ^ 2 D b
theorem derivation.leibniz_inv_of {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A] [ A] {M : Type u_3} [ M] [ M] (D : A M) (a : A) [invertible a] :
D ( a) = - a ^ 2 D a
theorem derivation.leibniz_inv {R : Type u_1} [comm_ring R] {M : Type u_3} [ M] {K : Type u_2} [field K] [ M] [ K] (D : K M) (a : K) :
@[protected, instance]
def derivation.has_neg {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A] [ A] {M : Type u_3} [ M] [ M] :
has_neg A M)
Equations
@[simp]
theorem derivation.coe_neg {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A] [ A] {M : Type u_3} [ M] [ M] (D : A M) :
@[simp]
theorem derivation.coe_neg_linear_map {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A] [ A] {M : Type u_3} [ M] [ M] (D : A M) :
theorem derivation.neg_apply {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A] [ A] {M : Type u_3} [ M] [ M] (D : A M) (a : A) :
(-D) a = -D a
@[protected, instance]
def derivation.has_sub {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A] [ A] {M : Type u_3} [ M] [ M] :
has_sub A M)
Equations
@[simp]
theorem derivation.coe_sub {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A] [ A] {M : Type u_3} [ M] [ M] (D1 D2 : A M) :
(D1 - D2) = D1 - D2
@[simp]
theorem derivation.coe_sub_linear_map {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A] [ A] {M : Type u_3} [ M] [ M] (D1 D2 : A M) :
(D1 - D2) = D1 - D2
theorem derivation.sub_apply {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A] [ A] {M : Type u_3} [ M] [ M] {D1 D2 : A M} (a : A) :
(D1 - D2) a = D1 a - D2 a
@[protected, instance]
def derivation.add_comm_group {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A] [ A] {M : Type u_3} [ M] [ M] :
Equations

# Lie structures #

@[protected, instance]
def derivation.has_bracket {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A] [ A] :
has_bracket A A) A A)

The commutator of derivations is again a derivation.

Equations
@[simp]
theorem derivation.commutator_coe_linear_map {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A] [ A] {D1 D2 : A A} :
theorem derivation.commutator_apply {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A] [ A] {D1 D2 : A A} (a : A) :
D1,D2 a = D1 (D2 a) - D2 (D1 a)
@[protected, instance]
def derivation.lie_ring {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A] [ A] :
Equations
@[protected, instance]
def derivation.lie_algebra {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A] [ A] :
A A)
Equations
def diff_to_ideal_of_quotient_comp_eq {R A : Type u} {B : Type w} [comm_ring B] [ A] [ B] (I : ideal B) (f₁ f₂ : A →ₐ[R] B) (e : I).comp f₁ = I).comp f₂) :

If f₁ f₂ : A →ₐ[R] B are two lifts of the same A →ₐ[R] B ⧸ I, we may define a map f₁ - f₂ : A →ₗ[R] I.

Equations
• e = _
@[simp]
theorem diff_to_ideal_of_quotient_comp_eq_apply {R A : Type u} {B : Type w} [comm_ring B] [ A] [ B] (I : ideal B) (f₁ f₂ : A →ₐ[R] B) (e : I).comp f₁ = I).comp f₂) (x : A) :
( e) x) = f₁ x - f₂ x
def derivation_to_square_zero_of_lift {R A : Type u} {B : Type w} [comm_ring B] [ A] [ B] (I : ideal B) (hI : I ^ 2 = ) [ B] [ B] (f : A →ₐ[R] B) (e : I).comp f = (B I)) :
A I

Given a tower of algebras R → A → B, and a square-zero I : ideal B, each lift A →ₐ[R] B of the canonical map A →ₐ[R] B ⧸ I corresponds to a R-derivation from A to I.

Equations
theorem derivation_to_square_zero_of_lift_apply {R A : Type u} {B : Type w} [comm_ring B] [ A] [ B] (I : ideal B) (hI : I ^ 2 = ) [ B] [ B] (f : A →ₐ[R] B) (e : I).comp f = (B I)) (x : A) :
( e) x) = f x - B) x
def lift_of_derivation_to_square_zero {R A : Type u} {B : Type w} [comm_ring B] [ A] [ B] (I : ideal B) (hI : I ^ 2 = ) [ B] [ B] (f : A I) :

Given a tower of algebras R → A → B, and a square-zero I : ideal B, each R-derivation from A to I corresponds to a lift A →ₐ[R] B of the canonical map A →ₐ[R] B ⧸ I.

Equations
theorem lift_of_derivation_to_square_zero_apply {R A : Type u} {B : Type w} [comm_ring B] [ A] [ B] (I : ideal B) (hI : I ^ 2 = ) [ B] [ B] (f : A I) (x : A) :
x = (f x) + B) x
@[simp]
theorem lift_of_derivation_to_square_zero_mk_apply {R A : Type u} {B : Type w} [comm_ring B] [ A] [ B] (I : ideal B) (hI : I ^ 2 = ) [ B] [ B] (d : A I) (x : A) :
( x) = (B I)) x
def derivation_to_square_zero_equiv_lift {R A : Type u} {B : Type w} [comm_ring B] [ A] [ B] (I : ideal B) (hI : I ^ 2 = ) [ B] [ B] :
A I {f // I).comp f = (B I)}

Given a tower of algebras R → A → B, and a square-zero I : ideal B, there is a 1-1 correspondance between R-derivations from A to I and lifts A →ₐ[R] B of the canonical map A →ₐ[R] B ⧸ I.

Equations
@[simp]
theorem derivation_to_square_zero_equiv_lift_symm_apply {R A : Type u} {B : Type w} [comm_ring B] [ A] [ B] (I : ideal B) (hI : I ^ 2 = ) [ B] [ B] (f : {f // I).comp f = (B I)}) :
f = _
@[simp]
theorem derivation_to_square_zero_equiv_lift_apply_coe {R A : Type u} {B : Type w} [comm_ring B] [ A] [ B] (I : ideal B) (hI : I ^ 2 = ) [ B] [ B] (d : A I) :
d) =
@[reducible]
def kaehler_differential.ideal (R : Type u_1) (S : Type u_2) [comm_ring R] [comm_ring S] [ S] :
ideal S S)

The kernel of the multiplication map S ⊗[R] S →ₐ[R] S.

theorem kaehler_differential.one_smul_sub_smul_one_mem_ideal (R : Type u_1) {S : Type u_2} [comm_ring R] [comm_ring S] [ S] (a : S) :
1 ⊗ₜ[R] a - a ⊗ₜ[R] 1
def derivation.tensor_product_to {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] [ S] {M : Type u_3} [ M] [ M] [ M] (D : S M) :
S →ₗ[S] M

For a R-derivation S → M, this is the map S ⊗[R] S →ₗ[S] M sending s ⊗ₜ t ↦ s • D t.

Equations
theorem derivation.tensor_product_to_tmul {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] [ S] {M : Type u_3} [ M] [ M] [ M] (D : S M) (s t : S) :
theorem derivation.tensor_product_to_mul {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] [ S] {M : Type u_3} [ M] [ M] [ M] (D : S M) (x y : S) :
theorem kaehler_differential.submodule_span_range_eq_ideal (R : Type u_1) (S : Type u_2) [comm_ring R] [comm_ring S] [ S] :
(set.range (λ (s : S), 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1)) =

The kernel of S ⊗[R] S →ₐ[R] S is generated by 1 ⊗ s - s ⊗ 1 as a S-module.

theorem kaehler_differential.span_range_eq_ideal (R : Type u_1) (S : Type u_2) [comm_ring R] [comm_ring S] [ S] :
ideal.span (set.range (λ (s : S), 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1)) =
@[protected, instance]
def kaehler_differential.module_2 (R : Type u_1) (S : Type u_2) [comm_ring R] [comm_ring S] [ S] :
Ω[ S R ]
def kaehler_differential (R : Type u_1) (S : Type u_2) [comm_ring R] [comm_ring S] [ S] :
Type u_2

The module of Kähler differentials (Kahler differentials, Kaehler differentials). This is implemented as I / I ^ 2 with I the kernel of the multiplication map S ⊗[R] S →ₐ[R] S. To view elements as a linear combination of the form s • D s', use kaehler_differential.tensor_product_to_surjective and derivation.tensor_product_to_tmul.

We also provide the notation Ω[S⁄R] for kaehler_differential R S. Note that the slash is \textfractionsolidus.

Equations
Instances for kaehler_differential
@[protected, instance]
def kaehler_differential.module_3 (R : Type u_1) (S : Type u_2) [comm_ring R] [comm_ring S] [ S] :
module S S) Ω[ S R ]
@[protected, instance]
def kaehler_differential.module (R : Type u_1) (S : Type u_2) [comm_ring R] [comm_ring S] [ S] :
Ω[ S R ]
@[protected, instance]
def kaehler_differential.add_comm_group (R : Type u_1) (S : Type u_2) [comm_ring R] [comm_ring S] [ S] :
@[protected, instance]
def kaehler_differential.nonempty (R : Type u_1) (S : Type u_2) [comm_ring R] [comm_ring S] [ S] :
@[protected, instance]
def kaehler_differential.is_scalar_tower (R : Type u_1) (S : Type u_2) [comm_ring R] [comm_ring S] [ S] :
S S) Ω[ S R ]
@[protected, instance]
def kaehler_differential.is_scalar_tower' (R : Type u_1) (S : Type u_2) [comm_ring R] [comm_ring S] [ S] :
Ω[ S R ]
def kaehler_differential.D_linear_map (R : Type u_1) (S : Type u_2) [comm_ring R] [comm_ring S] [ S] :
S →ₗ[R] Ω[ S R ]

(Implementation) The underlying linear map of the derivation into Ω[S⁄R].

Equations
theorem kaehler_differential.D_linear_map_apply (R : Type u_1) (S : Type u_2) [comm_ring R] [comm_ring S] [ S] (s : S) :
= 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1, _⟩
def kaehler_differential.D (R : Type u_1) (S : Type u_2) [comm_ring R] [comm_ring S] [ S] :
S Ω[ S R ]

The universal derivation into Ω[S⁄R].

Equations
theorem kaehler_differential.D_apply (R : Type u_1) (S : Type u_2) [comm_ring R] [comm_ring S] [ S] (s : S) :
s = 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1, _⟩
theorem kaehler_differential.span_range_derivation (R : Type u_1) (S : Type u_2) [comm_ring R] [comm_ring S] [ S] :
def derivation.lift_kaehler_differential {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] [ S] {M : Type u_3} [ M] [ M] [ M] (D : S M) :
Ω[ S R ] →ₗ[S] M

The linear map from Ω[S⁄R], associated with a derivation.

Equations
theorem derivation.lift_kaehler_differential_apply {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] [ S] {M : Type u_3} [ M] [ M] [ M] (D : S M) (x : ) :
theorem derivation.lift_kaehler_differential_comp {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] [ S] {M : Type u_3} [ M] [ M] [ M] (D : S M) :
@[ext]
theorem derivation.lift_kaehler_differential_unique {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] [ S] {M : Type u_3} [ M] [ M] [ M] (f f' : Ω[ S R ] →ₗ[S] M) (hf : (f.comp_der) = (f'.comp_der) ) :
f = f'
theorem derivation.lift_kaehler_differential_D (R : Type u_1) (S : Type u_2) [comm_ring R] [comm_ring S] [ S] :
theorem kaehler_differential.D_tensor_product_to {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] [ S] (x : ) :
= x
theorem kaehler_differential.tensor_product_to_surjective (R : Type u_1) (S : Type u_2) [comm_ring R] [comm_ring S] [ S] :
def kaehler_differential.linear_map_equiv_derivation (R : Type u_1) (S : Type u_2) [comm_ring R] [comm_ring S] [ S] {M : Type u_3} [ M] [ M] [ M] :
(Ω[ S R ] →ₗ[S] M) ≃ₗ[S] S M

The S-linear maps from Ω[S⁄R] to M are (S-linearly) equivalent to R-derivations from S to M.

Equations