mathlib documentation

algebra.group_action_hom

Equivariant homomorphisms #

Main definitions #

Notations #

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@[nolint]
structure mul_action_hom (M' : Type u_1) (X : Type u_2) [has_scalar M' X] (Y : Type u_3) [has_scalar M' Y] :
Type (max u_2 u_3)

Equivariant functions.

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@[protected, instance]
def mul_action_hom.has_coe_to_fun (M' : Type u_1) (X : Type u_2) [has_scalar M' X] (Y : Type u_3) [has_scalar M' Y] :
has_coe_to_fun (X →[M'] Y)
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@[simp]
theorem mul_action_hom.map_smul {M' : Type u_1} {X : Type u_2} [has_scalar M' X] {Y : Type u_3} [has_scalar M' Y] (f : X →[M'] Y) (m : M') (x : X) :
⇑f (m • x) = m • ⇑f x
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@[ext]
theorem mul_action_hom.ext {M' : Type u_1} {X : Type u_2} [has_scalar M' X] {Y : Type u_3} [has_scalar M' Y] {f g : X →[M'] Y} :
(∀ (x : X), ⇑f x = ⇑g x) → f = g
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theorem mul_action_hom.ext_iff {M' : Type u_1} {X : Type u_2} [has_scalar M' X] {Y : Type u_3} [has_scalar M' Y] {f g : X →[M'] Y} :
f = g ↔ ∀ (x : X), ⇑f x = ⇑g x
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@[protected]
def mul_action_hom.id (M' : Type u_1) {X : Type u_2} [has_scalar M' X] :
X →[M'] X

The identity map as an equivariant map.

Equations
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@[simp]
theorem mul_action_hom.id_apply (M' : Type u_1) {X : Type u_2} [has_scalar M' X] (x : X) :
⇑(mul_action_hom.id M') x = x
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def mul_action_hom.comp {M' : Type u_1} {X : Type u_2} [has_scalar M' X] {Y : Type u_3} [has_scalar M' Y] {Z : Type u_4} [has_scalar M' Z] (g : Y →[M'] Z) (f : X →[M'] Y) :
X →[M'] Z

Composition of two equivariant maps.

Equations
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@[simp]
theorem mul_action_hom.comp_apply {M' : Type u_1} {X : Type u_2} [has_scalar M' X] {Y : Type u_3} [has_scalar M' Y] {Z : Type u_4} [has_scalar M' Z] (g : Y →[M'] Z) (f : X →[M'] Y) (x : X) :
⇑(g.comp f) x = ⇑g (⇑f x)
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@[simp]
theorem mul_action_hom.id_comp {M' : Type u_1} {X : Type u_2} [has_scalar M' X] {Y : Type u_3} [has_scalar M' Y] (f : X →[M'] Y) :
(mul_action_hom.id M').comp f = f
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@[simp]
theorem mul_action_hom.comp_id {M' : Type u_1} {X : Type u_2} [has_scalar M' X] {Y : Type u_3} [has_scalar M' Y] (f : X →[M'] Y) :
f.comp (mul_action_hom.id M') = f
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def mul_action_hom.to_quotient {G : Type u_16} [group G] (H : subgroup G) :
G →[G] quotient_group.quotient H

The canonical map to the left cosets.

Equations
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@[simp]
theorem mul_action_hom.to_quotient_apply {G : Type u_16} [group G] (H : subgroup G) (g : G) :
⇑(mul_action_hom.to_quotient H) g = ↑g
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def distrib_mul_action_hom.to_mul_action_hom {M : Type u_5} [monoid M] {A : Type u_6} [add_monoid A] [distrib_mul_action M A] {B : Type u_8} [add_monoid B] [distrib_mul_action M B] (self : A →+[M] B) :
A →[M] B

Reinterpret an equivariant additive monoid homomorphism as an equivariant function.

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def distrib_mul_action_hom.to_add_monoid_hom {M : Type u_5} [monoid M] {A : Type u_6} [add_monoid A] [distrib_mul_action M A] {B : Type u_8} [add_monoid B] [distrib_mul_action M B] (self : A →+[M] B) :
A →+ B

Reinterpret an equivariant additive monoid homomorphism as an additive monoid homomorphism.

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@[nolint]
structure distrib_mul_action_hom (M : Type u_5) [monoid M] (A : Type u_6) [add_monoid A] [distrib_mul_action M A] (B : Type u_8) [add_monoid B] [distrib_mul_action M B] :
Type (max u_6 u_8)

Equivariant additive monoid homomorphisms.

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@[protected, instance]
def distrib_mul_action_hom.has_coe (M : Type u_5) [monoid M] (A : Type u_6) [add_monoid A] [distrib_mul_action M A] (B : Type u_8) [add_monoid B] [distrib_mul_action M B] :
has_coe (A →+[M] B) (A →+ B)
Equations
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@[protected, instance]
def distrib_mul_action_hom.has_coe' (M : Type u_5) [monoid M] (A : Type u_6) [add_monoid A] [distrib_mul_action M A] (B : Type u_8) [add_monoid B] [distrib_mul_action M B] :
has_coe (A →+[M] B) (A →[M] B)
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@[protected, instance]
def distrib_mul_action_hom.has_coe_to_fun (M : Type u_5) [monoid M] (A : Type u_6) [add_monoid A] [distrib_mul_action M A] (B : Type u_8) [add_monoid B] [distrib_mul_action M B] :
has_coe_to_fun (A →+[M] B)
Equations
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@[norm_cast]
theorem distrib_mul_action_hom.coe_fn_coe {M : Type u_5} [monoid M] {A : Type u_6} [add_monoid A] [distrib_mul_action M A] {B : Type u_8} [add_monoid B] [distrib_mul_action M B] (f : A →+[M] B) :
⇑↑f = ⇑f
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@[norm_cast]
theorem distrib_mul_action_hom.coe_fn_coe' {M : Type u_5} [monoid M] {A : Type u_6} [add_monoid A] [distrib_mul_action M A] {B : Type u_8} [add_monoid B] [distrib_mul_action M B] (f : A →+[M] B) :
⇑↑f = ⇑f
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@[ext]
theorem distrib_mul_action_hom.ext {M : Type u_5} [monoid M] {A : Type u_6} [add_monoid A] [distrib_mul_action M A] {B : Type u_8} [add_monoid B] [distrib_mul_action M B] {f g : A →+[M] B} :
(∀ (x : A), ⇑f x = ⇑g x) → f = g
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theorem distrib_mul_action_hom.ext_iff {M : Type u_5} [monoid M] {A : Type u_6} [add_monoid A] [distrib_mul_action M A] {B : Type u_8} [add_monoid B] [distrib_mul_action M B] {f g : A →+[M] B} :
f = g ↔ ∀ (x : A), ⇑f x = ⇑g x
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@[simp]
theorem distrib_mul_action_hom.map_zero {M : Type u_5} [monoid M] {A : Type u_6} [add_monoid A] [distrib_mul_action M A] {B : Type u_8} [add_monoid B] [distrib_mul_action M B] (f : A →+[M] B) :
⇑f 0 = 0
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@[simp]
theorem distrib_mul_action_hom.map_add {M : Type u_5} [monoid M] {A : Type u_6} [add_monoid A] [distrib_mul_action M A] {B : Type u_8} [add_monoid B] [distrib_mul_action M B] (f : A →+[M] B) (x y : A) :
⇑f (x + y) = ⇑f x + ⇑f y
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@[simp]
theorem distrib_mul_action_hom.map_neg {M : Type u_5} [monoid M] (A' : Type u_7) [add_group A'] [distrib_mul_action M A'] (B' : Type u_9) [add_group B'] [distrib_mul_action M B'] (f : A' →+[M] B') (x : A') :
⇑f (-x) = -⇑f x
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@[simp]
theorem distrib_mul_action_hom.map_sub {M : Type u_5} [monoid M] (A' : Type u_7) [add_group A'] [distrib_mul_action M A'] (B' : Type u_9) [add_group B'] [distrib_mul_action M B'] (f : A' →+[M] B') (x y : A') :
⇑f (x - y) = ⇑f x - ⇑f y
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@[simp]
theorem distrib_mul_action_hom.map_smul {M : Type u_5} [monoid M] {A : Type u_6} [add_monoid A] [distrib_mul_action M A] {B : Type u_8} [add_monoid B] [distrib_mul_action M B] (f : A →+[M] B) (m : M) (x : A) :
⇑f (m • x) = m • ⇑f x
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@[protected]
def distrib_mul_action_hom.id (M : Type u_5) [monoid M] {A : Type u_6} [add_monoid A] [distrib_mul_action M A] :
A →+[M] A

The identity map as an equivariant additive monoid homomorphism.

Equations
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@[simp]
theorem distrib_mul_action_hom.id_apply (M : Type u_5) [monoid M] {A : Type u_6} [add_monoid A] [distrib_mul_action M A] (x : A) :
⇑(distrib_mul_action_hom.id M) x = x
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def distrib_mul_action_hom.comp {M : Type u_5} [monoid M] {A : Type u_6} [add_monoid A] [distrib_mul_action M A] {B : Type u_8} [add_monoid B] [distrib_mul_action M B] {C : Type u_10} [add_monoid C] [distrib_mul_action M C] (g : B →+[M] C) (f : A →+[M] B) :
A →+[M] C

Composition of two equivariant additive monoid homomorphisms.

Equations
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@[simp]
theorem distrib_mul_action_hom.comp_apply {M : Type u_5} [monoid M] {A : Type u_6} [add_monoid A] [distrib_mul_action M A] {B : Type u_8} [add_monoid B] [distrib_mul_action M B] {C : Type u_10} [add_monoid C] [distrib_mul_action M C] (g : B →+[M] C) (f : A →+[M] B) (x : A) :
⇑(g.comp f) x = ⇑g (⇑f x)
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@[simp]
theorem distrib_mul_action_hom.id_comp {M : Type u_5} [monoid M] {A : Type u_6} [add_monoid A] [distrib_mul_action M A] {B : Type u_8} [add_monoid B] [distrib_mul_action M B] (f : A →+[M] B) :
(distrib_mul_action_hom.id M).comp f = f
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@[simp]
theorem distrib_mul_action_hom.comp_id {M : Type u_5} [monoid M] {A : Type u_6} [add_monoid A] [distrib_mul_action M A] {B : Type u_8} [add_monoid B] [distrib_mul_action M B] (f : A →+[M] B) :
f.comp (distrib_mul_action_hom.id M) = f
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def mul_semiring_action_hom.to_distrib_mul_action_hom {M : Type u_5} [monoid M] {R : Type u_11} [semiring R] [mul_semiring_action M R] {S : Type u_13} [semiring S] [mul_semiring_action M S] (self : R →+*[M] S) :
R →+[M] S

Reinterpret an equivariant ring homomorphism as an equivariant additive monoid homomorphism.

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def mul_semiring_action_hom.to_ring_hom {M : Type u_5} [monoid M] {R : Type u_11} [semiring R] [mul_semiring_action M R] {S : Type u_13} [semiring S] [mul_semiring_action M S] (self : R →+*[M] S) :
R →+* S

Reinterpret an equivariant ring homomorphism as a ring homomorphism.

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@[nolint]
structure mul_semiring_action_hom (M : Type u_5) [monoid M] (R : Type u_11) [semiring R] [mul_semiring_action M R] (S : Type u_13) [semiring S] [mul_semiring_action M S] :
Type (max u_11 u_13)

Equivariant ring homomorphisms.

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@[protected, instance]
def mul_semiring_action_hom.has_coe (M : Type u_5) [monoid M] (R : Type u_11) [semiring R] [mul_semiring_action M R] (S : Type u_13) [semiring S] [mul_semiring_action M S] :
has_coe (R →+*[M] S) (R →+* S)
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@[protected, instance]
def mul_semiring_action_hom.has_coe' (M : Type u_5) [monoid M] (R : Type u_11) [semiring R] [mul_semiring_action M R] (S : Type u_13) [semiring S] [mul_semiring_action M S] :
has_coe (R →+*[M] S) (R →+[M] S)
Equations
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@[protected, instance]
def mul_semiring_action_hom.has_coe_to_fun (M : Type u_5) [monoid M] (R : Type u_11) [semiring R] [mul_semiring_action M R] (S : Type u_13) [semiring S] [mul_semiring_action M S] :
has_coe_to_fun (R →+*[M] S)
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@[norm_cast]
theorem mul_semiring_action_hom.coe_fn_coe {M : Type u_5} [monoid M] {R : Type u_11} [semiring R] [mul_semiring_action M R] {S : Type u_13} [semiring S] [mul_semiring_action M S] (f : R →+*[M] S) :
⇑↑f = ⇑f
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@[norm_cast]
theorem mul_semiring_action_hom.coe_fn_coe' {M : Type u_5} [monoid M] {R : Type u_11} [semiring R] [mul_semiring_action M R] {S : Type u_13} [semiring S] [mul_semiring_action M S] (f : R →+*[M] S) :
⇑↑f = ⇑f
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@[ext]
theorem mul_semiring_action_hom.ext {M : Type u_5} [monoid M] {R : Type u_11} [semiring R] [mul_semiring_action M R] {S : Type u_13} [semiring S] [mul_semiring_action M S] {f g : R →+*[M] S} :
(∀ (x : R), ⇑f x = ⇑g x) → f = g
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theorem mul_semiring_action_hom.ext_iff {M : Type u_5} [monoid M] {R : Type u_11} [semiring R] [mul_semiring_action M R] {S : Type u_13} [semiring S] [mul_semiring_action M S] {f g : R →+*[M] S} :
f = g ↔ ∀ (x : R), ⇑f x = ⇑g x
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@[simp]
theorem mul_semiring_action_hom.map_zero {M : Type u_5} [monoid M] {R : Type u_11} [semiring R] [mul_semiring_action M R] {S : Type u_13} [semiring S] [mul_semiring_action M S] (f : R →+*[M] S) :
⇑f 0 = 0
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@[simp]
theorem mul_semiring_action_hom.map_add {M : Type u_5} [monoid M] {R : Type u_11} [semiring R] [mul_semiring_action M R] {S : Type u_13} [semiring S] [mul_semiring_action M S] (f : R →+*[M] S) (x y : R) :
⇑f (x + y) = ⇑f x + ⇑f y
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@[simp]
theorem mul_semiring_action_hom.map_neg {M : Type u_5} [monoid M] (R' : Type u_12) [ring R'] [mul_semiring_action M R'] (S' : Type u_14) [ring S'] [mul_semiring_action M S'] (f : R' →+*[M] S') (x : R') :
⇑f (-x) = -⇑f x
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@[simp]
theorem mul_semiring_action_hom.map_sub {M : Type u_5} [monoid M] (R' : Type u_12) [ring R'] [mul_semiring_action M R'] (S' : Type u_14) [ring S'] [mul_semiring_action M S'] (f : R' →+*[M] S') (x y : R') :
⇑f (x - y) = ⇑f x - ⇑f y
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@[simp]
theorem mul_semiring_action_hom.map_one {M : Type u_5} [monoid M] {R : Type u_11} [semiring R] [mul_semiring_action M R] {S : Type u_13} [semiring S] [mul_semiring_action M S] (f : R →+*[M] S) :
⇑f 1 = 1
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@[simp]
theorem mul_semiring_action_hom.map_mul {M : Type u_5} [monoid M] {R : Type u_11} [semiring R] [mul_semiring_action M R] {S : Type u_13} [semiring S] [mul_semiring_action M S] (f : R →+*[M] S) (x y : R) :
⇑f (x * y) = (⇑f x) * ⇑f y
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@[simp]
theorem mul_semiring_action_hom.map_smul {M : Type u_5} [monoid M] {R : Type u_11} [semiring R] [mul_semiring_action M R] {S : Type u_13} [semiring S] [mul_semiring_action M S] (f : R →+*[M] S) (m : M) (x : R) :
⇑f (m • x) = m • ⇑f x
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@[protected]
def mul_semiring_action_hom.id (M : Type u_5) [monoid M] {R : Type u_11} [semiring R] [mul_semiring_action M R] :
R →+*[M] R

The identity map as an equivariant ring homomorphism.

Equations
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@[simp]
theorem mul_semiring_action_hom.id_apply (M : Type u_5) [monoid M] {R : Type u_11} [semiring R] [mul_semiring_action M R] (x : R) :
⇑(mul_semiring_action_hom.id M) x = x
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def mul_semiring_action_hom.comp {M : Type u_5} [monoid M] {R : Type u_11} [semiring R] [mul_semiring_action M R] {S : Type u_13} [semiring S] [mul_semiring_action M S] {T : Type u_15} [semiring T] [mul_semiring_action M T] (g : S →+*[M] T) (f : R →+*[M] S) :
R →+*[M] T

Composition of two equivariant additive monoid homomorphisms.

Equations
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@[simp]
theorem mul_semiring_action_hom.comp_apply {M : Type u_5} [monoid M] {R : Type u_11} [semiring R] [mul_semiring_action M R] {S : Type u_13} [semiring S] [mul_semiring_action M S] {T : Type u_15} [semiring T] [mul_semiring_action M T] (g : S →+*[M] T) (f : R →+*[M] S) (x : R) :
⇑(g.comp f) x = ⇑g (⇑f x)
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@[simp]
theorem mul_semiring_action_hom.id_comp {M : Type u_5} [monoid M] {R : Type u_11} [semiring R] [mul_semiring_action M R] {S : Type u_13} [semiring S] [mul_semiring_action M S] (f : R →+*[M] S) :
(mul_semiring_action_hom.id M).comp f = f
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@[simp]
theorem mul_semiring_action_hom.comp_id {M : Type u_5} [monoid M] {R : Type u_11} [semiring R] [mul_semiring_action M R] {S : Type u_13} [semiring S] [mul_semiring_action M S] (f : R →+*[M] S) :
f.comp (mul_semiring_action_hom.id M) = f
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def is_invariant_subring.subtype_hom (M : Type u_5) [monoid M] {R' : Type u_12} [ring R'] [mul_semiring_action M R'] (U : set R') [is_subring U] [is_invariant_subring M U] :
↥U →+*[M] R'

The canonical inclusion from an invariant subring.

Equations
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@[simp]
theorem is_invariant_subring.coe_subtype_hom (M : Type u_5) [monoid M] {R' : Type u_12} [ring R'] [mul_semiring_action M R'] (U : set R') [is_subring U] [is_invariant_subring M U] :
⇑(is_invariant_subring.subtype_hom M U) = coe
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@[simp]
theorem is_invariant_subring.coe_subtype_hom' (M : Type u_5) [monoid M] {R' : Type u_12} [ring R'] [mul_semiring_action M R'] (U : set R') [is_subring U] [is_invariant_subring M U] :
↑(is_invariant_subring.subtype_hom M U) = is_subring.subtype U