2. Notations and Preliminaries
Throughout this paper 𝕂 is a commutative ring, and all 𝕂-modules M are such that for all m∈ M, 2m = 0 implies m = 0.
Let
A,
B,
C,
etc. be algebras over ground commutative ring 𝕂. Unadorned tensor product will denote the tensor product over 𝕂. For modules
M in
A
B, symbols
M* , *
M, *
M* denote right dual, left dual and bidual of
M, and
A
B (
M,
N) denotes the 𝕂-module of (
A,
B)-bimodule maps
M →
N. In what follows we shall concentrate on right dual of
M but similar observations can be made for the left dual as well.
For all
ϕ∈
A
B (
M,
N), let
ϕ* :
N* →
M* denote the right adjoint of
ϕ i.e.,
ϕ* (
g)(
m) :=
g ◦
ϕ (
m).
We denote by (·)op : A → Aop the canonical anti-algebra isomorphism from the algebra A into its opposite Aop (which is the identity on the underlying 𝕂-modules), i.e., a = aop as module elements and (aa´)op = a´opaop for all a, a´∈ A.
The following facts are well known, but we recall them to set up the notation:
(i) If
M∈
A
B then
M* ∈
Aop
Bop with (
aopfbop)(
m) =
bf(
am).
Assume that
M∈
A
B is also finitely generated projective as a right
B-module,
i.e., there exists a dual basis
![Axioms 01 00173 i004]()
, such that for any
m∈
M,
![Axioms 01 00173 i005]()
. Then
(ii) The mapping
κM:
M →
M**, κM (
m)(
f) =
f(
m)
op is an isomorphism in
A
B, with the inverse
![Axioms 01 00173 i006]()
. In fact
κ is a natural morphism between identity functor in
A
B and the functor ()
** :
A
B →
A
B.
(iii) If
N∈
B
C then
κM,N :
M* ⊗
Bop N* → (M ⊗
B N)*, given by
κM,N (
f ⊗
Bop g)(
m ⊗
n) =
g(
f(
m)
n), is an isomorphism in
Aop
Cop with the inverse
(iv) Let
M∈
A
B,
N∈
B
C , P ∈
C
D, where
A,
B,
C,
D are algebras. Then the following diagram is commutative:
(v) Let
M∈
A
B be finitely generated projective as
B-module, with dual basis
![Axioms 01 00173 i009]()
,
i∈
I, and let
N∈
B
C be finitely generated projective as a
C-module with dual basis
![Axioms 01 00173 i010]()
,
![Axioms 01 00173 i011]()
,
i∈
J. Then
M ⊗
B N∈
A
C is finitely generated projective as a
C-module with a dual basis
The following terminology and theorems concerning corings and ring extensions are needed in this paper. For a review on coalgebras see: [
5,
6,
7]. For a review on corings see [
3].
Definition 2.1 C∈
B
B is called a
B-coring if there exist morphisms Δ
C,
εC ∈
B
B, Δ
C :
C→
C ⊗
B C,
εC :
C→
B such that
In the sequel we shall use Sweedler’s notation Δ
C(
c) =
c(1) ⊗
B c(2). Given
B-corings
C and
D, a map
ϕ ∈
B
B (
C,
D) is called a
morphism of B-corings if (
ϕ ⊗
Bϕ) ◦ Δ
C = Δ
D ◦
ϕ and
εD ◦
ϕ =
εC. The category of
B-corings is denoted by
CrgB.
Definition 2.2 Ring
![Axioms 01 00173 i015]()
is called an extension of a ring
B if there exists an injective unital ring morphism
![Axioms 01 00173 i016]()
:
B →
![Axioms 01 00173 i015]()
. Observe that
![Axioms 01 00173 i015]()
∈
B
B by
![Axioms 01 00173 i016]()
. Given ring extensions
![Axioms 01 00173 i016]()
:
B →
![Axioms 01 00173 i015]()
and
![Axioms 01 00173 i016]()
:
B →
P, a ring morphism α :
![Axioms 01 00173 i015]()
→
P is called a
morphism of ring extensions if α ◦
![Axioms 01 00173 i016]()
=
![Axioms 01 00173 i016]()
or, equivalently, if α ∈
B
B (
![Axioms 01 00173 i015]()
,
P). The category of ring extensions of
B is denoted by
RgeB.
The full subcategory of CrgB (resp. RgeB) consisting of those B-corings (resp. ring extensions of B) that are finitely generated projective as right B-modules is denoted by r.f.g.pCrgB (resp. r.f.g.pRgeB).
Lemma 2.3 (i) If C∈
CrgB then C* ∈
Bop
Bopis a ring extension of Bop with multiplication unit 1C* := εC and embedding map
(ii) If ϕ : C→D is any coring morphism then ϕ*: D* →C* is a ring extension morphism.
(iii) If ![Axioms 01 00173 i015]()
∈
r.f.g.pRgeB then
* is a Bop-coring with comultiplication and counit where
is a (finite) dual basis of
. (iv) If ϕ :
![Axioms 01 00173 i015]()
→
S is a morphism of right finitely generated projective ring extensions of B, then ![Axioms 01 00173 i015]()
:
S* →
* is a morphism of Bop-corings. (v) Functor ()**:
r.f.g.pRgeB →
r.f.g.pRgeB is equivalent to the identity functor on r.f.g.pRgeB. For all ![Axioms 01 00173 i015]()
∈
r.f.g.pRgeB, κ
R :
![Axioms 01 00173 i015]()
→
![Axioms 01 00173 i015]()
**
is a ring extension isomorphism facilitating this equivalence. (vi) Functor ()**: r.f.g.pCrgB → r.f.g.pCrgB is equivalent to the identity functor on r.f.g.pCrgB. For all C∈ r.f.g.pCrgB, κC : C→C** is a B-coring isomorphism facilitating this equivalence.
Proof. The statements (i) and (ii) are contained in Proposition 3.2 [
8], while (iii) and (v) are rephrasings of Theorem 3.7 [
8] (cf. [
3], 17.8–17.13)
(
iv) Consider any ring extension morphism
ϕ :
![Axioms 01 00173 i015]()
→
S. Let
![Axioms 01 00173 i021]()
be any finite dual basis of
![Axioms 01 00173 i015]()
, and let
![Axioms 01 00173 i022]()
be any finite dual basis of
S. For all
s∈
S*,
and
Hence ϕ* is a coring map.
(
vi) It is enough to prove that
κC, is a coring map for any
C∈
r.f.g.pCrgB. Let
C be a
B-coring, and let
![Axioms 01 00173 i025]()
, be any finite dual basis of
C. Observe that
![Axioms 01 00173 i026]()
is a dual basis of
C*. Indeed, for any
g∈
C*,
Hence, for all c∈C
and
Corollary 2.4 ()* is a duality functor between r.f.g.pRgeB and r.f.g.pCrgBop
3. An Extension for the Duality between Corings and Ring Extensions
Our aim in this section is to extend the duality between right finitely generated projective ring extensions and corings to the category of right finitely generated projective generalized Yang–Baxter structures.
We use the following terminology concerning the Yang–Baxter equation. Some references on this topic are: [
9,
10,
11],
etc.
Let B be a 𝕂-algebra. Given a (B, B)-bimodule V and a (B, B)-bilinear map R : V ⊗ B V → V ⊗ B V we write R12 = R ⊗ B id, R23 = id ⊗ B R : V ⊗ B V ⊗ B V → V ⊗ B V ⊗ B V where id : V → V is the identity map.
Definition 3.1 An invertible (B, B)-linear map R : V ⊗ B V → V ⊗ B V is called a generalized Yang–Baxter operator (or simply a generalised YB operator ) if it satisfies the equation
Definition 3.2 For an algebra B, we define the category YB strB whose objects are 4-tuples (V, φ, e, ε), where
(i) V is a (B, B)-bimodule;
(ii) φ : V ⊗ B V → V ⊗ B V is a generalized YB operator;
(iii) e∈ V such that for all b∈ B, eb = be, and for all x∈ V , φ(x ⊗ e) = e ⊗ B x, φ(e ⊗ B x) = x ⊗ B e;
(iv) ε : V → B is a (B, B)-bimodule map, such that (id ⊗ B ε) ◦φ = ε ⊗ B id, (ε ⊗ B id) ◦φ = id ⊗ B ε.
A morphism f :(V, φ, e, ε) → (V’, φ’,e’,ε’) in the category YB strB is a (B, B)-bilinear map f : V → V’ such that:
(v) (f ⊗ B f) ◦φ = φ´◦ (f ⊗ B f),
(vi) f(e) = e´,
(vii) ε´ ◦ f = ε.
Composition of morphisms is defined as the standard composition of B-linear maps. A full subcategory of YB strB consisting of all such (V, φ, e, ε) for which V is finitely generated projective as a right B-module is defined by r.f.g.pYBstrB.
Remark 3.3 Let R : V ⊗ B V → V ⊗ B V be a generalised YB operator . Then (V, R, 0, 0) is an object in the category YB strB.
Theorem 3.4 (i) There exists a functor:
Any ring extension map f is simply mapped into a (B, B) bimodule map.
(ii) F is a full and faithful embedding.
Proof. i) The proof that
φR is a generalised YB operator is left to the reader (cf. Proposition 2.1 from [
12],
![Axioms 01 00173 i032]()
). Furthermore
![Axioms 01 00173 i034]()
Hence (
R,
φR, 1
R, 0) is an object in the category
YB strB.
Let
f :
![Axioms 01 00173 i015]()
→
S be a morphism of ring extensions. Then
f(1
R)= 1
S and 0 ◦ f = 0. Moreover
Hence
![Axioms 01 00173 i036]()
is a morphism in the category
YB strB.
(
ii) If
F ![Axioms 01 00173 i015]()
=
F S, for some
![Axioms 01 00173 i015]()
,
S∈
RgeB, then obviously
![Axioms 01 00173 i015]()
=
S as (
B,
B)-bimodules, 1
S = 1
R, and the only thing which can differ is the multiplication. Denote by · the multiplication in
![Axioms 01 00173 i015]()
, and by ◦ the multiplication in
S. Then, as
φR =
φS , for all
r,
r´∈
![Axioms 01 00173 i015]()
,
hence
Multiplying tensor factors on both sides of this equation (whether using multiplication in
![Axioms 01 00173 i015]()
or
S is irrelevant) yields 2(
r ·
r´−
r ◦
r´) = 0, hence
r ·
r =
r ◦
r´, and so
![Axioms 01 00173 i015]()
=
S as algebras. Therefore
F is an embedding.
Obviously, distinct ring extension maps are also distinct as (B, B)-bimodule morphisms, hence F is a faithful functor.
Let
![Axioms 01 00173 i039]()
be a morphism in
YB strB, where
![Axioms 01 00173 i015]()
,
S∈
RgeB. Then
f is unital, and
![Axioms 01 00173 i040]()
, hence, for all
r,
r´∈
![Axioms 01 00173 i015]()
,
Multiplying factors in tensor products in both sides of the above equation yields 2(f(rr´) − f(r)f(r´)) = 0, hence f(rr´) = f(r)f(r´) and, as f is a (B, B)-bimodule map, it is a ring extension map. Therefore, F is a full functor.
Theorem 3.5 (i) There exists a functor
A coring morphism is mapped into a (B, B)-bimodule morphism.
(ii) G is a full and faithful embbeding.
Proof. i) The proof that
ψC is a generalised YB operator (cf. Proposition 2.3 from [
12]) is left to the reader (
ψC−1 =
ψC). Furthermore, for all
c∈
C,
ψC(c ⊗
B 0) = 0 = 0 ⊗
B c, ψ
C(0 ⊗
B c) = 0 = c ⊗
B 0. Moreover, for all
c,
c´∈
C,
and
Hence (C,ψC, 0,εC) is an object in YB strB. Let f : C→D be any morphism of B-corings. Then f is also a (B, B)-bimodule morphism, f(0) = 0, εD ◦ f = εC, and,
Therefore f :(C,ψC, 0,εC) → (D,ψD, 0,εD) is a morphism in YB strB.
(ii) Suppose that GC = GD for some B-corings C, D. This means that C = D as (B, B)-bimodules, εC = εD, and the only things which can differ are comultiplications. However, as ψC = ψD, we have
hence
(ΔC − ΔD) ⊗ B εC = −εC ⊗ B (ΔC − ΔD)
Composing both sides of the above equation with ΔC yields 2(ΔC − ΔD) = 0 hence ΔC = ΔD and C = D as (B, B)-corings. Hence G is an embedding.
Obviously distinct B-coring morphisms are also distinct as (B, B)-bimodule morphisms, hence G is a faithful functor.
Let f :(C,ψC, 0,εC) → (D,ψD, 0,εD), where C, D are corings, be a morphism in YB strB. Then (B, B)-bimodule morphism f : C→D is counital, i.e., εD ◦ f = εC . Furthermore, (f ⊗ B f) ◦ ψC = ψD ◦ (f ⊗ B f), and hence (f ⊗ B f) ◦ ψC ◦ ΔC = ψD ◦ (f ⊗ B f) ◦ ΔC . Observe that ψC ◦ ΔC = ΔC . Therefore
i.e., 2(f ⊗ B f) ◦ ΔC = 2ΔD ◦ f, hence (f ⊗ B f) ◦ ΔC = ΔD ◦ f, and f is a B-coring map. Therefore G is full.
Proposition 3.6 Let (V, R, e, ε) ∈ r.f.g.pYB strB. Then
where e†(f) = f(e), and
Moreover,
is a natural isomorphism in r.f.g.pYB strB.
Proof. R is invertible, hence
![Axioms 01 00173 i050]()
We shall prove that
R† satisfies the Yang–Baxter equation. Observe that
Indeed, let Γ ∈ (
V ⊗
B V )*,
f∈
V*, and let
![Axioms 01 00173 i053]()
be a dual basis of
V .
Similarly we can prove the other equality. By virtue of (17,18), we can write
By (2),
and therefore
Hence R† is a generalised YB operator .
Proofs of bilinearity of e* and centrality of ε are the same as proofs of analogues properties of duals of units and counits in Lemma 2.4. Moreover, for all f∈ V * ,
and
Furthermore, for all x = f ⊗ Bop g∈ V * ⊗ Bop V*,
and
Hence (V*,R†, ε, e†) ∈ r.f.g.pYB strBop.
Morphism
κ : () → ()** is natural in
B
B, and as V is finitely generated projective,
κV is invertible. Therefore it suffices to prove that
κV is a morphism in
r.f.g.pYB strB. To this end, observe first that
κV (e) = f → f(e)op = e†
and, for all υ∈ V ,
Note that
![Axioms 01 00173 i065]()
is a dual basis of
V*. Therefore, for all Γ ∈ (
V* ⊗
Bop V*)*,
and so, for all υ, υ´ ∈ V ,
Therefore, κV is a morphism in r.f.g.pYB strB as required.
Proposition 3.7 Let ![Axioms 01 00173 i015]()
∈
r.f.g.pRgeB, C∈
r.f.g.pCrgB. Then
i.e., Proof. From Lemma 2.4 we know that 1
R† =
εR* and 1
C* =
εC. Furthermore, for all
c,
c´∈
![Axioms 01 00173 i015]()
*,
Similarly, for all r, r´∈C* , rr´ = κC,C(r ⊗ Bop r´) ◦ ΔC, therefore for all r, r´∈C* ,
This completes the proof.
Remark 3.8 Put together the statements of Theorem 3.6, Theorem 3.5, Proposition 3.6 and Proposition 3.7, can be summarized in the following diagram:
This means that the duality between right finitely generated projective ring extensions of B and B corings extends to the category r.f.g.pYB strB.