Library HoTT.Algebra.Rings.Module

Require Import WildCat.
Require Import Spaces.Nat.Core.
(* Some of the material in abstract_algebra and canonical names could be selectively exported to the user, as is done in Groups/Group.v. *)
Require Import Classes.interfaces.canonical_names.
Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup.
Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct.
Require Import Rings.Ring.

Declare Scope module_scope.

Left modules over a ring.

TODO: Right modules? We can treat right modules as left modules over the opposite ring.

Definition

An abelian group M is a left R-module when equipped with the following data:

Class IsLeftModule (R : Ring) (M : AbGroup) := {

A function lact (left-action) that takes an element r : R and an element m : M and returns an element lact r m : M.

lact : R → M → M;

Actions distribute on the left over addition in the abelian group. That is lact r (m + n) = lact r m + lact r n.

lact_left_dist :: LeftHeteroDistribute lact (+) (+);

Actions distribute on the right over addition in the ring. That is lact (r + s) m = lact r m + lact s m.

lact_right_dist :: RightHeteroDistribute lact (+) (+);

Actions are associative. That is lact (r × s) m = lact r (lact s m).

lact_assoc :: HeteroAssociative lact lact lact (.*.);

Actions preserve the multiplicative identity. That is lact 1 m = m.

lact_unit :: LeftIdentity lact 1;
}.

TODO: notation for action.

A left R-module is an abelian group equipped with a left R-module structure.

Record LeftModule (R : Ring) := {
  lm_carrier :> AbGroup;
  lm_lact :: IsLeftModule R lm_carrier;
}.

Section ModuleAxioms.
  Context {R : Ring} {M : LeftModule R} (r s : R) (m n : M).

Here we state the module axioms in a readable form for direct use.

  Definition lm_dist_l : lact r (m + n) = lact r m + lact r n := lact_left_dist r m n.
  Definition lm_dist_r : lact (r + s) m = lact r m + lact s m := lact_right_dist r s m.
  Definition lm_assoc : lact r (lact s m) = lact (r × s) m := lact_assoc r s m.
  Definition lm_unit : lact 1 m = m := lact_unit m.

End ModuleAxioms.

Facts about left modules

Section ModuleFacts.
Context {R : Ring} {M : LeftModule R} (r : R) (m : M).

Here are some quick facts that hold in modules.

The action of zero is zero.

  Definition lm_zero_l : lact 0 m = 0.
  Proof.
    apply (grp_cancelL1 (z := lact 0 m)).
    lhs_V nrapply lm_dist_r.
    f_ap.
    apply rng_plus_zero_r.
  Defined.

The action on zero is zero.

  Definition lm_zero_r : lact r (0 : M) = 0.
  Proof.
    apply (grp_cancelL1 (z := lact r 0)).
    lhs_V nrapply lm_dist_l.
    f_ap.
    apply grp_unit_l.
  Defined.

The action of -1 is the additive inverse.

  Definition lm_minus_one : lact (-1) m = -m.
  Proof.
     apply grp_moveL_1V.
     lhs nrapply (ap (_ +) (lm_unit m )^).
     lhs_V nrapply lm_dist_r.
     rhs_V nrapply lm_zero_l.
     f_ap.
     apply grp_inv_l.
  Defined.

End ModuleFacts.

Left Submodules

A subgroup of a left R-module is a left submodule if it is closed under the action of R.

Class IsLeftSubmodule {R : Ring} {M : LeftModule R} (N : M → Type) := {
ils_issubgroup :: IsSubgroup N;
is_left_submodule : ∀ r m , N m → N (lact r m);
}.

A left submodule is a subgroup of the abelian group closed under the action of R.

Record LeftSubmodule {R : Ring} (M : LeftModule R) := {
  lsm_carrier :> M → Type;
  lsm_submodule :: IsLeftSubmodule lsm_carrier;
}.

Definition subgroup_leftsubmodule {R : Ring} {M : LeftModule R}
  : LeftSubmodule M → Subgroup M
  := fun N ⇒ Build_Subgroup M N _.
Coercion subgroup_leftsubmodule : LeftSubmodule >-> Subgroup.

Submodules inherit the R-module structure of their parent.

Global Instance isleftmodule_leftsubmodule {R : Ring}
  {M : LeftModule R} (N : LeftSubmodule M)
  : IsLeftModule R N.
Proof.
  snrapply Build_IsLeftModule.
  - intros r [n n_in_N].
    ∃ (lact r n).
    by apply lsm_submodule.
  - intros r [n] [m]; apply path_sigma_hprop.
    apply lact_left_dist.
  - intros r s [n]; apply path_sigma_hprop.
    apply lact_right_dist.
  - intros r s [n]; apply path_sigma_hprop.
    apply lact_assoc.
  - intros [n]; apply path_sigma_hprop.
    apply lact_unit.
Defined.

Any left submodule of a left R-module is a left R-module.

Definition leftmodule_leftsubmodule {R : Ring}
  {M : LeftModule R} (N : LeftSubmodule M)
  : LeftModule R
  := Build_LeftModule R N _.
Coercion leftmodule_leftsubmodule : LeftSubmodule >-> LeftModule.

The submodule criterion. This is a convenient way to build submodules.

Definition Build_IsLeftSubmodule' {R : Ring} {M : LeftModule R}
  (H : M → Type) `{∀ x , IsHProp (H x )}
  (z : H zero)
  (c : ∀ r n m , H n → H m → H (n + lact r m))
  : IsLeftSubmodule H.
Proof.
  snrapply Build_IsLeftSubmodule.
  - snrapply Build_IsSubgroup'.
    + exact _.
    + exact z.
    + intros x y hx hy.
      change (sg_op ?x ?y) with (x + y).
      pose proof (p := c (-1) x y hx hy).
      rewrite lm_minus_one in p.
      exact p.
  - intros r m hm.
    rewrite <- (grp_unit_l).
    by apply c.
Defined.

Definition Build_LeftSubmodule' {R : Ring} {M : LeftModule R}
  (H : M → Type) `{∀ x , IsHProp (H x )}
  (z : H zero)
  (c : ∀ r n m , H n → H m → H (n + lact r m))
  : LeftSubmodule M.
Proof.
  pose (p := Build_IsLeftSubmodule' H z c).
  snrapply Build_LeftSubmodule.
  1: snrapply (Build_Subgroup _ H).
  2: exact p.
  rapply ils_issubgroup.
Defined.

Left R-module homomorphisms

A left module homomorphism is a group homomorphism that commutes with the action of R.

Record LeftModuleHomomorphism {R : Ring} (M N : LeftModule R) := {
  lm_homo_map :> GroupHomomorphism M N;
  lm_homo_lact : ∀ r m , lm_homo_map (lact r m) = lact r (lm_homo_map m);
}.

Definition lm_homo_id {R : Ring} (M : LeftModule R) : LeftModuleHomomorphism M M.
Proof.
  snrapply Build_LeftModuleHomomorphism.
  - exact grp_homo_id.
  - reflexivity.
Defined.

Definition lm_homo_compose {R : Ring} {M N L : LeftModule R}
  : LeftModuleHomomorphism N L → LeftModuleHomomorphism M N
  → LeftModuleHomomorphism M L.
Proof.
  intros f g.
  snrapply Build_LeftModuleHomomorphism.
  - exact (grp_homo_compose f g).
  - intros r m.
    rhs_V nrapply lm_homo_lact.
    apply (ap f).
    apply lm_homo_lact.
Defined.

Smart constructor for building left module homomorphisms from a map.

Definition Build_LeftModuleHomomorphism' {R : Ring} {M N : LeftModule R}
  (f : M → N) (p : ∀ r x y , f (lact r x + y) = lact r (f x) + f y)
  : LeftModuleHomomorphism M N.
Proof.
  snrapply Build_LeftModuleHomomorphism.
  - snrapply Build_GroupHomomorphism.
    + exact f.
    + intros x y.
      rewrite <- (lm_unit (f x)).
      set (lact 1 (f x)).
      rewrite <- (lm_unit x).
      apply p.
  - intros r m.
    simpl.
    rewrite <- (grp_unit_r (lact r m)).
    rewrite p.
    rhs_V nrapply grp_unit_r.
    apply grp_cancelL.
    specialize (p 1 0 0).
    rewrite 2 lm_unit in p.
    apply (grp_cancelL1 (z := f 0)).
    lhs_V nrapply p.
    apply ap.
    apply grp_unit_l.
Defined.

Record LeftModuleIsomorphism {R : Ring} (M N : LeftModule R) := {
  lm_iso_map :> LeftModuleHomomorphism M N;
  isequiv_lm_iso_map :: IsEquiv lm_iso_map;
}.

Definition Build_LeftModuleIsomorphism' {R : Ring} (M N : LeftModule R)
  (f : GroupIsomorphism M N) (p : ∀ r x , f (lact r x) = lact r (f x))
  : LeftModuleIsomorphism M N.
Proof.
  snrapply Build_LeftModuleIsomorphism.
  - snrapply Build_LeftModuleHomomorphism.
    + exact f.
    + exact p.
  - exact _.
Defined.

Definition lm_iso_inverse {R : Ring} {M N : LeftModule R}
  : LeftModuleIsomorphism M N → LeftModuleIsomorphism N M.
Proof.
  intros f.
  snrapply Build_LeftModuleIsomorphism.
  - snrapply Build_LeftModuleHomomorphism'.
    + exact f^-1.
    + intros r m n.
      apply moveR_equiv_V.
      rhs nrapply grp_homo_op.
      symmetry.
      f_ap.
      2: apply eisretr.
      lhs nrapply lm_homo_lact.
      apply ap.
      apply eisretr.
  - exact _.
Defined.

Category of left R-modules

TODO: define as a displayed category over Ring

Global Instance isgraph_leftmodule {R : Ring} : IsGraph (LeftModule R)
  := Build_IsGraph _ LeftModuleHomomorphism.

Global Instance is01cat_leftmodule {R : Ring} : Is01Cat (LeftModule R)
  := Build_Is01Cat _ _ lm_homo_id (@lm_homo_compose R).

Global Instance is2graph_leftmodule {R : Ring} : Is2Graph (LeftModule R)
  := fun M N ⇒ isgraph_induced (@lm_homo_map R M N).

Global Instance is1cat_leftmodule {R : Ring} : Is1Cat (LeftModule R).
Proof.
  snrapply Build_Is1Cat'.
  - intros M N; rapply is01cat_induced.
  - intros M N; rapply is0gpd_induced.
  - intros M N L h.
    snrapply Build_Is0Functor.
    intros f g p m.
    exact (ap h (p m)).
  - intros M N L f.
    snrapply Build_Is0Functor.
    intros g h p m.
    exact (p (f m)).
  - simpl; reflexivity.
  - simpl; reflexivity.
  - simpl; reflexivity.
Defined.

Global Instance hasequivs_leftmodule {R : Ring} : HasEquivs (LeftModule R).
Proof.
  snrapply Build_HasEquivs.
  - exact LeftModuleIsomorphism.
  - intros M N; exact IsEquiv.
  - intros M N f; exact f.
  - simpl; exact _.
  - apply Build_LeftModuleIsomorphism.
  - reflexivity.
  - intros M N; apply lm_iso_inverse.
  - intros M N f; apply eissect.
  - intros M N f; apply eisretr.
  - intros M N f g fg gf.
    exact (isequiv_adjointify f g fg gf).
Defined.

Kernel of module homomorphism

Global Instance isleftsubmodule_grp_kernel {R : Ring}
  {M N : LeftModule R} (f : M $-> N)
  : IsLeftSubmodule (grp_kernel f).
Proof.
  srapply Build_IsLeftSubmodule.
  intros r m n.
  lhs nrapply lm_homo_lact.
  rhs_V nrapply (lm_zero_r r).
  apply ap.
  exact n.
Defined.

Definition lm_kernel {R : Ring} {M N : LeftModule R} (f : M $-> N)
  : LeftSubmodule M
  := Build_LeftSubmodule _ _ (grp_kernel f) _.

Image of module homomorphism

Global Instance isleftsubmodule_grp_image {R : Ring}
  {M N : LeftModule R} (f : M $-> N)
  : IsLeftSubmodule (grp_image f).
Proof.
  srapply Build_IsLeftSubmodule.
  intros r m; apply Trunc_functor; intros [n p].
  ∃ (lact r n).
  lhs nrapply lm_homo_lact.
  apply ap.
  exact p.
Defined.

Definition lm_image {R : Ring} {M N : LeftModule R} (f : M $-> N)
  : LeftSubmodule N
  := Build_LeftSubmodule _ _ (grp_image f) _.

Quotient Modules

The quotient abelian group of a module and a submodule has a natural ring action.

Global Instance isleftmodule_quotientabgroup {R : Ring}
  (M : LeftModule R) (N : LeftSubmodule M)
  : IsLeftModule R (QuotientAbGroup M N).
Proof.
  snrapply Build_IsLeftModule.
  - intros r.
    snrapply quotient_abgroup_rec.
    + refine (grp_quotient_map $o _).
      snrapply Build_GroupHomomorphism.
      × exact (lact r).
      × intros x y.
        apply lm_dist_l.
    + intros n Nn; simpl.
      apply qglue.
      apply issubgroup_in_inv_op.
      2: apply issubgroup_in_unit.
      by apply is_left_submodule.
  - intros r m n; revert m.
    snrapply Quotient_ind_hprop; [exact _ | intros m; revert n].
    snrapply Quotient_ind_hprop; [exact _ | intros n; simpl].
    rapply ap.
    apply lm_dist_l.
  - intros r s.
    snrapply Quotient_ind_hprop; [exact _| intros m; simpl].
    rapply ap.
    apply lm_dist_r.
  - intros r s.
    snrapply Quotient_ind_hprop; [exact _| intros m; simpl].
    rapply ap.
    apply lm_assoc.
  - snrapply Quotient_ind_hprop; [exact _| intros m; simpl].
    rapply ap.
    apply lm_unit.
Defined.

We can therefore form the quotient module of a module by its submodule.

Definition QuotientLeftModule {R : Ring} (M : LeftModule R) (N : LeftSubmodule M)
: LeftModule R
:= Build_LeftModule R (QuotientAbGroup M N) _.

Infix "/" := QuotientLeftModule : module_scope.

First Isomorphism Theorem

Local Open Scope module_scope.
Local Open Scope wc_iso_scope.

Definition rng_first_iso `{Funext} {R : Ring} {M N : LeftModule R} (f : M $-> N)
  : M / lm_kernel f ≅ lm_image f.
Proof.
  snrapply Build_LeftModuleIsomorphism'.
  1: rapply abgroup_first_iso.
  intros r.
  srapply Quotient_ind_hprop; intros m.
  apply path_sigma_hprop; simpl.
  apply lm_homo_lact.
Defined.

Direct products

TODO: generalise to biproducts The direct product of modules

Definition lm_prod {R : Ring} : LeftModule R → LeftModule R → LeftModule R.
Proof.
  intros M N.
  snrapply (Build_LeftModule R (ab_biprod M N)).
  snrapply Build_IsLeftModule.
  - intros r.
    apply functor_prod; exact (lact r).
  - intros r m n.
    apply path_prod; apply lm_dist_l.
  - intros r m n.
    apply path_prod; apply lm_dist_r.
  - intros r s m.
    apply path_prod; apply lm_assoc.
  - intros r.
    apply path_prod; apply lm_unit.
Defined.

Definition lm_prod_fst {R : Ring} {M N : LeftModule R} : lm_prod M N $-> M.
Proof.
  snrapply Build_LeftModuleHomomorphism.
  - apply grp_prod_pr1.
  - reflexivity.
Defined.

Definition lm_prod_snd {R : Ring} {M N : LeftModule R} : lm_prod M N $-> N.
Proof.
  snrapply Build_LeftModuleHomomorphism.
  - apply grp_prod_pr2.
  - reflexivity.
Defined.

Definition lm_prod_corec {R : Ring} {M N : LeftModule R} (L : LeftModule R)
  (f : L $-> M) (g : L $-> N)
  : L $-> lm_prod M N.
Proof.
  snrapply Build_LeftModuleHomomorphism.
  - apply (grp_prod_corec f g).
  - intros r l.
    apply path_prod; apply lm_homo_lact.
Defined.

Global Instance hasbinaryproducts_leftmodule {R : Ring}
  : HasBinaryProducts (LeftModule R).
Proof.
  intros M N.
  snrapply Build_BinaryProduct.
  - exact (lm_prod M N).
  - exact lm_prod_fst.
  - exact lm_prod_snd.
  - exact lm_prod_corec.
  - cbn; reflexivity.
  - cbn; reflexivity.
  - intros L f g p q a.
    exact (path_prod' (p a) (q a)).
Defined.