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Axioms 2012, 1(2), 173-185; https://doi.org/10.3390/axioms1020173

Article
The Duality between Corings and Ring Extensions
1
Institute of Mathematics of the Romanian Academy, 21 Calea Grivitei Street, 010702 Bucharest, Romania
2
Department of Theoretical Physics and Computer Science, University of Łodz, Pomorska 149/153, 90-236, Łodz, Poland
*
Author to whom correspondence should be addressed.
Received: 29 June 2012; in revised form: 24 July 2012 / Accepted: 30 July 2012 / Published: 10 August 2012

Abstract

:
We study the duality between corings and ring extensions. We construct a new category with a self-dual functor acting on it, which extends that duality. This construction can be seen as the non-commutative case of another duality extension: the duality between finite dimensional algebras and coalgebra. Both these duality extensions have some similarities with the Pontryagin-van Kampen duality theorem.
Keywords:
corings; ring extension; duality; Yang–Baxter equation

Classification: MSC 16T25; 16T15

1. Introduction

Non-commutative geometry is a branch of mathematics concerned with geometric approach to non-commutative algebras, and with constructions of spaces which are locally presented by non-commutative algebras of functions. Its main motivation is to extend the commutative duality between spaces and functions to the non-commutative setting.
More specifically, in topology, compact Hausdorff topological spaces can be reconstructed from the Banach algebra of functions on the space. The Pontryagin duality theorem refers to the duality between the category of compact Hausdorff Abelian groups and the category of discrete Abelian groups. The Pontryagin–van Kampen duality theorem extends this duality to all locally compact Hausdorff Abelian topological groups by including the categories of compact Hausdorff Abelian groups and discrete Abelian groups into the category of locally compact Hausdorff Abelian topological groups (see [1]). This can be illustrated by the following diagram:
Axioms 01 00173 i001
Taking the Pontryagin–van Kampen duality theorem as a model, an extension for the duality between finite dimensional algebras and coalgebras to the category of finite dimensional Yang–Baxter structures was constructed in [2]. The resulting duality theorem can be illustrated by the following diagram:
Axioms 01 00173 i002
Our motivation in this paper is to extend the above duality to the non-commutative setting.
In Section 2, we present in a new fashion the duality between right finitely generated projective corings and ring extensions (compare with [3]).
In Section 3, we define the category of (right finitely generated projective) generalized Yang–Baxter structures. We construct full and faithful embeddings from the categories of ring extensions and corings to the category of generalized Yang–Baxter structures. We show that taking the right dual is a duality functor in the category of right finitely generated projective generalized Yang–Baxter structures. Then we conclude that the duality between right finitely generated projective corings and ring extensions can be lifted up to the category of right finitely generated projective generalized Yang–Baxter structures.
There are some more comments to be made.
  • (i) We propose as a research project the investigation of other connections between the duality of (co)algebras and the Pontryagin duality. (For example, one might try to endow the (co)algebra structures with some topological structures.)
  • (ii) At the epistemologic level, the extension of the duality of (co)algebra structures seems to be a model for the relation between interdisciplinarity, pluridisciplinarity and transdisciplinarity (see [4]).
  • (iii) This paper explains that taking the dual of some objects can be seen a “continuous” process. Let us visualize this statement by considering an example from geometry. We take a triangular prism: We can see it as two parallel triangles joint by 3 segments. In total it has 5 planar geometric figures, 9 edges and 6 vertices. The geometric dual of the triangular prism has 6 planar geometric figures, 9 edges and 5 vertices. Now, one can start with a triangular prism, “shave” its corners, and then continuously deform that figure in order to obtain the geometric dual of the triangular prism.

2. Notations and Preliminaries

Throughout this paper 𝕂 is a commutative ring, and all 𝕂-modules M are such that for all mM, 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 Axioms 01 00173 i003B, symbols M* , *M, *M* denote right dual, left dual and bidual of M, and A Axioms 01 00173 i003B (M, N) denotes the 𝕂-module of (A, B)-bimodule maps MN. 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 ϕA Axioms 01 00173 i003B (M, N), let ϕ* : N* → M* denote the right adjoint of ϕ i.e., ϕ* (g)(m) := gϕ (m).
We denote by (·)op : AAop 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 = opaop for all a, A.
The following facts are well known, but we recall them to set up the notation:
(i) If MA Axioms 01 00173 i003B then M* ∈ Aop Axioms 01 00173 i003Bop with (aopfbop)(m) = bf(am).
Assume that MA Axioms 01 00173 i003B 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 mM, Axioms 01 00173 i005. Then
(ii) The mapping κM: M M**, κM (m)(f) = f(m)op is an isomorphism in A Axioms 01 00173 i003B, with the inverse Axioms 01 00173 i006. In fact κ is a natural morphism between identity functor in A Axioms 01 00173 i003B and the functor ()** : A Axioms 01 00173 i003BA Axioms 01 00173 i003B.
(iii) If NB Axioms 01 00173 i003C then κM,N : M* ⊗ Bop N* → (M ⊗ B N)*, given by κM,N (fBop g)(mn) = g(f(m)n), is an isomorphism in Aop Axioms 01 00173 i003Cop with the inverse
Axioms 01 00173 i007
(iv) Let MA Axioms 01 00173 i003B, NB Axioms 01 00173 i003C , P ∈C Axioms 01 00173 i003D, where A, B, C, D are algebras. Then the following diagram is commutative:
Axioms 01 00173 i008
(v) Let MA Axioms 01 00173 i003B be finitely generated projective as B-module, with dual basis Axioms 01 00173 i009, iI, and let NB Axioms 01 00173 i003C be finitely generated projective as a C-module with dual basis Axioms 01 00173 i010, Axioms 01 00173 i011, iJ. Then MB NA Axioms 01 00173 i003C is finitely generated projective as a C-module with a dual basis
Axioms 01 00173 i012
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 CB Axioms 01 00173 i003B is called a B-coring if there exist morphisms ΔC, εCB Axioms 01 00173 i003B, ΔC : CCB C, εC : CB such that
Axioms 01 00173 i013
Axioms 01 00173 i014
In the sequel we shall use Sweedler’s notation ΔC(c) = c(1)B c(2). Given B-corings C and D, a map ϕB Axioms 01 00173 i003B (C, D) is called a morphism of B-corings if (ϕBϕ) ◦ ΔC = ΔDϕ and εDϕ = 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 i015B Axioms 01 00173 i003B by Axioms 01 00173 i016. Given ring extensions Axioms 01 00173 i016 : B Axioms 01 00173 i015 and Axioms 01 00173 i016 : BP, a ring morphism α : Axioms 01 00173 i015P is called a morphism of ring extensions if α ◦ Axioms 01 00173 i016 = Axioms 01 00173 i016 or, equivalently, if α ∈ B Axioms 01 00173 i003B ( 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 CCrgB then C* ∈Bop Axioms 01 00173 i003Bopis a ring extension of Bop with multiplication
Axioms 01 00173 i017
unit 1C* := εC and embedding map
Axioms 01 00173 i018
(ii) If ϕ : CD is any coring morphism then ϕ*: D* →C* is a ring extension morphism.
(iii) If Axioms 01 00173 i015r.f.g.pRgeB then Axioms 01 00173 i015* is a Bop-coring with comultiplication and counit
Axioms 01 00173 i019
Axioms 01 00173 i020
where Axioms 01 00173 i021 is a (finite) dual basis of Axioms 01 00173 i015.
(iv) If ϕ : Axioms 01 00173 i015S is a morphism of right finitely generated projective ring extensions of B, then Axioms 01 00173 i015 : S* → Axioms 01 00173 i015* is a morphism of Bop-corings.
(v) Functor ()**: r.f.g.pRgeBr.f.g.pRgeB is equivalent to the identity functor on r.f.g.pRgeB. For all Axioms 01 00173 i015r.f.g.pRgeB, κR : Axioms 01 00173 i015 Axioms 01 00173 i015** is a ring extension isomorphism facilitating this equivalence.
(vi) Functor ()**: r.f.g.pCrgBr.f.g.pCrgB is equivalent to the identity functor on r.f.g.pCrgB. For all C∈ r.f.g.pCrgB, κC : CC** 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 i015S. 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 sS*,
Axioms 01 00173 i023
and
Axioms 01 00173 i024
Hence ϕ* is a coring map.
(vi) It is enough to prove that κC, is a coring map for any Cr.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 gC*,
Axioms 01 00173 i027
Hence, for all cC
Axioms 01 00173 i028
and
Axioms 01 00173 i029
Corollary 2.4 ()* is a duality functor between r.f.g.pRgeB and r.f.g.pCrgBop
Axioms 01 00173 i030

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 : VB VVB V we write R12 = RB id, R23 = idB R : VB VB VVB VB V where id : V → V is the identity map.
Definition 3.1 An invertible (B, B)-linear map R : VB VVB V is called a generalized Yang–Baxter operator (or simply a generalised YB operator ) if it satisfies the equation
Axioms 01 00173 i074
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 : VB VVB V is a generalized YB operator;
(iii) eV such that for all bB, eb = be, and for all xV , φ(xe) = eB x, φ(eB x) = xB e;
(iv) ε : VB is a (B, B)-bimodule map, such that (idB ε) ◦φ = εB id, (εB id) ◦φ = idB ε.
A morphism f :(V, φ, e, ε) → (V’, φ’,e’,ε’) in the category YB strB is a (B, B)-bilinear map f : VV’ such that:
(v) (fB f) ◦φ = φ´◦ (fB 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 : VB VVB 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:
Axioms 01 00173 i031
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 i033 Axioms 01 00173 i034 Hence (R, φR, 1R, 0) is an object in the category YB strB.
Let f : Axioms 01 00173 i015S be a morphism of ring extensions. Then f(1R)= 1S and 0 ◦ f = 0. Moreover
Axioms 01 00173 i035
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, SRgeB, then obviously Axioms 01 00173 i015 = S as (B, B)-bimodules, 1S = 1R, 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,
Axioms 01 00173 i037
hence
Axioms 01 00173 i038
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´− rr´) = 0, hence r · r = rr´, 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, SRgeB. Then f is unital, and Axioms 01 00173 i040, hence, for all r, r´∈ Axioms 01 00173 i015,
Axioms 01 00173 i041
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
Axioms 01 00173 i042
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 cC, ψC(c ⊗ B 0) = 0 = 0 ⊗ B c, ψC(0 ⊗ B c) = 0 = c ⊗ B 0. Moreover, for all c, c´∈C,
Axioms 01 00173 i043
and
Axioms 01 00173 i044
Hence (C,ψC, 0,εC) is an object in YB strB. Let f : CD be any morphism of B-corings. Then f is also a (B, B)-bimodule morphism, f(0) = 0, εDf = εC, and,
Axioms 01 00173 i045
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
Axioms 01 00173 i046
hence
C − ΔD) ⊗ B εC = −εCBC − 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 : CD is counital, i.e., εDf = εC . Furthermore, (fB f) ◦ ψC = ψD ◦ (fB f), and hence (fB f) ◦ ψC ◦ ΔC = ψD ◦ (fB f) ◦ ΔC . Observe that ψC ◦ ΔC = ΔC . Therefore
Axioms 01 00173 i047
i.e., 2(fB f) ◦ ΔC = 2ΔDf, hence (fB f) ◦ ΔC = ΔDf, 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
Axioms 01 00173 i075
where e(f) = f(e), and
Axioms 01 00173 i048
Moreover,
Axioms 01 00173 i049
is a natural isomorphism in r.f.g.pYB strB.
Proof. R is invertible, hence Axioms 01 00173 i050 We shall prove that Rsatisfies the Yang–Baxter equation. Observe that
Axioms 01 00173 i051
Axioms 01 00173 i052
Indeed, let Γ ∈ (VB V )*, fV*, and let Axioms 01 00173 i053 be a dual basis of V .
Axioms 01 00173 i054
Similarly we can prove the other equality. By virtue of (17,18), we can write
Axioms 01 00173 i055
Axioms 01 00173 i056
By (2),
Axioms 01 00173 i057
Axioms 01 00173 i058
and therefore
Axioms 01 00173 i059
Hence Ris 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 fV * ,
Axioms 01 00173 i060
and
Axioms 01 00173 i061
Furthermore, for all x = fBop gV * ⊗ Bop V*,
Axioms 01 00173 i062
and
Axioms 01 00173 i063
Hence (V*,R, ε, e) ∈ r.f.g.pYB strBop.
Morphism κ : () → ()** is natural in B Axioms 01 00173 i003B, 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) = ff(e)op = e
and, for all υV ,
Axioms 01 00173 i064
Note that Axioms 01 00173 i065 is a dual basis of V*. Therefore, for all Γ ∈ (V* ⊗ Bop V*)*,
Axioms 01 00173 i066
and so, for all υ, υ´ V ,
Axioms 01 00173 i067
Therefore, κV is a morphism in r.f.g.pYB strB as required.
Proposition 3.7 Let Axioms 01 00173 i015r.f.g.pRgeB, Cr.f.g.pCrgB. Then Axioms 01 00173 i068i.e.,
Axioms 01 00173 i069
Axioms 01 00173 i070
Proof. From Lemma 2.4 we know that 1R = εR* and 1C* = εC. Furthermore, for all c, c´∈ Axioms 01 00173 i015*,
Axioms 01 00173 i071
Similarly, for all r, r´∈C* , rr´ = κC,C(rBop r´) ◦ ΔC, therefore for all r, r´∈C* ,
Axioms 01 00173 i072
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:
Axioms 01 00173 i073
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.

4. Conclusions

We extended the duality between right finitely generated projective ring extensions and right finitely generated projective corings to the category of right finitely generated projective generalized Yang–Baxter structures. This duality and its extension could be seen as a more general construction. For example, at the epistemologic level, the extension of the duality of (co)algebra structures seems to be a model for the relation between interdisciplinarity, pluridisciplinarity and transdisciplinarity (see [4]). It would be interesting to interpret this construction in terms of particle interactions.
The relationships between sub(co)algebras and (co)ideals are well-known, and the term of YB ideal was proposed for the first time in [11]. The following question arises: What are the relationships between sub(co)rings, (co)ideals and generalized Yang–Baxter structures?
We think that there are more connections between the Pontryagin–van Kampen duality and the above extension of the duality of (co)algebra structures.

Acknowledgments

We would like to thank Tomasz Brzezi´nski for helpful remarks. The first author thanks for a Marie Curie Research Fellowship, HPMF-CT-2002-01782 at Swansea University. The work of BZ was supported by the EPSRC grant GR/S01078/01.

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