A Galois-Grothendieck-type correspondence for groupoid actions
In this paper we present a Galois-Grothendiecktype correspondence for groupoid actions. As an application a Galois-type correspondence is also given.
Збережено в:
| Дата: | 2014 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут прикладної математики і механіки НАН України
2014
|
| Назва видання: | Algebra and Discrete Mathematics |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/152342 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | A Galois-Grothendieck-type correspondence for groupoid actions / A. Paques, T. Tamusiunas // Algebra and Discrete Mathematics. — 2014. — Vol. 17, № 1. — С. 80–97. — Бібліогр.: 9 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-152342 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1523422019-06-11T01:25:19Z A Galois-Grothendieck-type correspondence for groupoid actions Paques, A. Tamusiunas, T. In this paper we present a Galois-Grothendiecktype correspondence for groupoid actions. As an application a Galois-type correspondence is also given. 2014 Article A Galois-Grothendieck-type correspondence for groupoid actions / A. Paques, T. Tamusiunas // Algebra and Discrete Mathematics. — 2014. — Vol. 17, № 1. — С. 80–97. — Бібліогр.: 9 назв. — англ. 1726-3255 2010 MSC:13B02, 13B05, 16H05, 18B40. http://dspace.nbuv.gov.ua/handle/123456789/152342 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
In this paper we present a Galois-Grothendiecktype correspondence for groupoid actions. As an application a Galois-type correspondence is also given. |
| format |
Article |
| author |
Paques, A. Tamusiunas, T. |
| spellingShingle |
Paques, A. Tamusiunas, T. A Galois-Grothendieck-type correspondence for groupoid actions Algebra and Discrete Mathematics |
| author_facet |
Paques, A. Tamusiunas, T. |
| author_sort |
Paques, A. |
| title |
A Galois-Grothendieck-type correspondence for groupoid actions |
| title_short |
A Galois-Grothendieck-type correspondence for groupoid actions |
| title_full |
A Galois-Grothendieck-type correspondence for groupoid actions |
| title_fullStr |
A Galois-Grothendieck-type correspondence for groupoid actions |
| title_full_unstemmed |
A Galois-Grothendieck-type correspondence for groupoid actions |
| title_sort |
galois-grothendieck-type correspondence for groupoid actions |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2014 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/152342 |
| citation_txt |
A Galois-Grothendieck-type correspondence for groupoid actions / A. Paques, T. Tamusiunas // Algebra and Discrete Mathematics. — 2014. — Vol. 17, № 1. — С. 80–97. — Бібліогр.: 9 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT paquesa agaloisgrothendiecktypecorrespondenceforgroupoidactions AT tamusiunast agaloisgrothendiecktypecorrespondenceforgroupoidactions AT paquesa galoisgrothendiecktypecorrespondenceforgroupoidactions AT tamusiunast galoisgrothendiecktypecorrespondenceforgroupoidactions |
| first_indexed |
2025-07-13T02:51:51Z |
| last_indexed |
2025-07-13T02:51:51Z |
| _version_ |
1837498489476481024 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 17 (2014). Number 1. pp. 80 – 97
c© Journal “Algebra and Discrete Mathematics”
A Galois-Grothendieck-type correspondence for
groupoid actions
Antonio Paques, Thaísa Tamusiunas
Communicated by V. V. Kirichenko
Abstract. In this paper we present a Galois-Grothendieck-
type correspondence for groupoid actions. As an application a
Galois-type correspondence is also given.
1. Introduction
S. U. Chase, D. K. Harrison and A. Rosenberg developed in [3] a Galois
theory for commutative ring extensions R ⊃ K under the assumption that
R is a strongly separable K-algebra and the elements of the Galois group
G are pairwise strongly distinct K-automorphisms of R . Among the main
results of that paper, Theorem 2.3 states a one-to-one correspondence
between the subgroups of the group G and the K-subalgebras of R which
are separable and G-strong.
The Galois theory due to Grothendieck, in its total generality, is
contextualized in the language of schemes (see [7]). A version of this
theory in the specific context of fields has been presented by A. Dress in
[4] (see also [2]). Dress showed that a simplification of the Galois theory
for groups acting on fields is possible by combining Dedekind’s lemma
with some elementary facts on G-sets, in the case that G is a group.
Dedekind’s lemma states that for a field extension L of a field K the set
AlgK(A,L) of all K-algebra homomorphisms of a K-algebra A into L is a
linearly independent subset of the L-vector space HomK(A,L). It turns
out that strongly distinct algebra homomorphisms of separable algebras
are a kind of homomorphisms which satisfy a version of Dedekind’s lemma.
2010 MSC: 13B02, 13B05, 16H05, 18B40.
Key words and phrases: groupoid action, G-set, Galois-Grothendieck equiva-
lence, Galois correspondence.
A. Paques, T. Tamusiunas 81
In [5], M. Ferrero and the first author showed that the same approach
used by Dress can be adopted in Galois theory for groups acting on
commutative rings, and, as a natural sequel of this method, they obtained
some new results.
The goal of this paper is to develop a Galois theory for groupoids
acting on commutative rings using the original viewpoints of Grothendieck
and Dress. We start by introducing a new version of Dedekind’s lemma
(section 2) we will need for our purposes, and standard notions and basic
facts concerning to groupoid actions on sets and algebras (section 3). The
Galois-Grothendieck-type correspondence for an action β of a groupoid
G on a K-algebra R, given in the section 4, establishes an equivalence
between the category of all finite G-split sets and the category of all R-split
K-algebras, under the assumption that R is a β-Galois extension of K. As
an application of this result we present in the section 5 a generalization of
the Galois-type correspondence given by Chase, Harrison and Rosenberg
in [3].
Throughout, K is a fixed commutative ring with identity and algebras
over K are always commutative and unital. Ring homomorphisms are
assumed to be unitary, and unadorned ⊗ means ⊗K .
2. Dedekind’s Lemma revisited
We start by recalling that a K-algebra R is said to be separable if R
is a projective R ⊗ R-module. This is equivalent to the existence of an
element υ =
∑
i xi ⊗ yi ∈ R ⊗ R, which turns out to be an idempotent,
unique such that
∑
i xiyi = 1R and rυ = υr, for every r ∈ R. If, in
addition, R is projective and finitely generated as a K-module, we say
that R is a strongly separable K-algebra, or, if R is also faithful over K,
a strongly separable extension of K. Any faithful, projetive and finitely
generated K-module is called faithfully projective.
Let f, g : T −→ S be ring homomorphisms. We say that f and g
are strongly distinct if, for every nonzero idempotent π ∈ S, there exists
x ∈ T such that f(x)π 6= g(x)π.
Lemma 2.1. [5, Lemma 1.2] Let T be a separable K-algebra, and f : T →
K a T -algebra homomorphism. Then, there exists a unique idempotent
π ∈ T such that f(π) = 1 and xπ = f(x)π, for all x ∈ T . Furthermore,
if {fj | j ∈ J} is a nonempty set of pairwise strongly distinct K-algebra
homomorphisms from T into K, then the corresponding idempotents
πj , j ∈ J, are pairwise orthogonal and fi(πj) = δij1K , for all i, j ∈ J .
The next results are slight extensions of similar results given in [5,
Section 2].
82 A Galois-Grothendieck-type correspondence
Proposition 2.2. Suppose that T and R are K-algebras with T separable
over K, and V is a nonempty set of homomorphisms of K-algebras v :
T −→ Ev, where Ev = R1v and {1v}v∈V is a set of nonzero idempotents
of R. Then, the following statements are equivalent:
(i) For each v ∈ V , the elements of Vv = {u ∈ V | 1u = 1v} are pairwise
strongly distinct.
(ii) For each u ∈ Vv there exist xiu ∈ Ev, yiu ∈ T , 1 ≤ i ≤ mu, such
that
∑mu
i=1 xiuu
′(yiu) = δu,u′1v, for every u′ ∈ Vv.
(iii) For each v ∈ V , Vv is free over Ev in HomK(T,Ev).
Proof. (i) ⇒ (ii) Since T is separable over K, for each v ∈ V , Ev ⊗ T is
separable over Ev. Also, for all u ∈ Vv the mappings
fu : Ev ⊗ T −→ Ev,
x⊗ y 7−→ xu(y)
are pairwise strongly distinct homomorphisms. Then, by Lemma 2.1, there
exists πu =
∑mv
i=1 xiu ⊗ yiu ∈ Ev ⊗T such that fu′(πu) = δu,u′1v, for every
u, u′ ∈ Vv, and (ii) follows.
(ii) ⇒ (iii) Assume that V ′
v is a finite subset of Vv and
∑
u′∈V ′
v
ru′u′ = 0
in HomK(T,Ev), where ru′ ∈ Eu′ = Ev. Hence, for u ∈ V ′
v , we have
ru = (
∑
u′∈V ′
v
δu,u′1v)ru′ =
∑
u′∈V ′
v
(
∑mu
i=1 xiuu
′(yiu))ru′ =∑mu
i=1 xiu(
∑
u′∈V ′
v
u′(yiu)ru′) = 0,
showing that Vv is free over Ev.
(iii) ⇒ (i) Immediate.
Corollary 2.3. Assume that T is a strongly separable extension of K,
R is a K-algebra and V is a nonempty set of homomorphisms of K-
algebras v : T −→ Ev, where Ev = R1v and {1v}v∈V is a set of nonzero
idempotents of R. Suppose that for each v ∈ V , the elements of Vv = {u ∈
V | 1u = 1v} are pairwise strongly distinct. Then, #Vv ≤ rankKp
Tp, for
every prime ideal p of K.
Proof. It follows from Proposition 2.2 that Vv is free over Ev in
HomK(T,Ev). Then, we have via localization that (Vv)p is free over
(Ev)p in HomKp
(Tp, (Ev)p), for every prime ideal p of K.
Furthermore, notice that T is a faithfully projetive K-module. So, if
n = rankKp
Tp, then Tp ≃ (Kp)
n as Kp-modules and HomKp
(Tp, (Ev)p) ≃
((Ev)p)
n as (Ev)p-modules. Consequently, #Vv = #(Vv)p ≤ n.
A. Paques, T. Tamusiunas 83
Lemma 2.4. Assume that T and R are K-algebras and V is a non-empty
finite set of homomorphisms of K-algebras v : T −→ Ev, where Ev = R1v
and {1v}v∈V is a set of nonzero idempotents of R. Suppose that K is
isomorphic to a direct summand of R as K-modules and Ev is a faithfully
projective K-module, for each v ∈ V . Then, the following statements are
equivalent:
(i) T is a strongly separable extension of K, for each v ∈ V the ele-
ments of Vv = {u ∈ V | 1u = 1v} are pairwise strongly distinct and
rankKT = #Vv.
(ii) T is faithfully projective over K, for each v ∈ V there exist xiv ∈ Ev,
yiv ∈ T , 1 ≤ i ≤ mv, such that
∑mv
i=1 xivu(yiv) = δu,v1v, for every
u ∈ Vv, and rankKT = #Vv.
(iii) For each v ∈ V , the mapping ϕv : Ev ⊗ T −→
∏
u∈Vv
Eu given by
ϕv(r ⊗ t) = (ru(t))u∈Vv , is an isomorphism of R-algebras.
Proof. (i)⇒(ii) Clearly, T is faithfully projective over K, and the rest of
the assertion follows from Proposition 2.2.
(ii)⇒(iii) Take v ∈ V . The mapping ϕv is clearly an R-algebra homo-
morphism. ϕv is also surjective since for any r = (ru)u∈Vv ∈
∏
u∈Vv
Eu,
there is z =
∑
u∈Vv
∑mu
i=1 ruxiu ⊗ yiu ∈ Ev ⊗ T and ϕv(z) = r. Fur-
thermore, rankEv (
∏
u∈Vv
Eu) = rankEv (Ev)#Vv = #Vv = rankKT =
rankEv (Ev ⊗ T ). Thus, it follows, by [8, Corollaire I.2.4], that ϕv is an
isomorphism.
(iii) ⇒ (i) Since, for each v ∈ V , ϕv is an isomorphism, it follows
that (rankKp
(Eg)p)(rankKp
Tp) = rankKp
(Eg ⊗ T )p = rankKp
En
g =
n(rankKp
(Eg)p), thus rankKp
Tp = n, for all prime ideal p of K. Hence,
rankKT = n, so T is faithful over K.
In the sequel we will prove that T is a strongly separable extension ofK.
It follows from the assumptions on R and Ev that T ≃ K ⊗T ≃ K1v ⊗T
is isomorphic to a direct summand of Ev ⊗ T ≃
∏
u∈Vv
Eu = (Ev)n,
where n = #Vv. Therefore, T is a finitely generated and projective K-
module. Furthermore, by [8, Proposition III.1.7 (c)] (Ev)n =
∏
u∈Vv
Eu is
Ev-separable. So, by [8, Proposition III.2.2], T is separable over K.
It remains to show that the elements of Vv are pairwise strongly distinct.
Given u ∈ Vv, take s = (δl,u1l)l∈Vv
∈
∏
u∈Vv
Eu. Then, there exists z =∑mu
i=1 riu ⊗ tiu ∈ Ev ⊗T such that ϕv(z) = s. Thus, (
∑mu
i=1 riul(tiu))l∈Vv
=
(δl,u1l)l∈Vv
, that implies
∑mu
i=1 riul(tiu) = δl,u1l for each l ∈ Vv, and the
assertion follows by Proposition 2.2.
84 A Galois-Grothendieck-type correspondence
3. Groupoid actions on sets and algebras
The axiomatic version of groupoid that we adopt in this paper was
taken from [9]. A groupoid is a nonempty set G, equipped with a partially
defined binary operation (which will be denoted by concatenation), where
the usual group axioms hold whenever they make sense, that is:
(i) For every g, h, l ∈ G, g(hl) exists if and only if (gh)l exists and in
this case they are equal;
(ii) For every g, h, l ∈ G, g(hl) exists if and only if gh and hl exist;
(iii) For each g ∈ G, there exist (unique) elements d(g), r(g) ∈ G such
that gd(g) and r(g)g exist and gd(g) = g = r(g)g;
(iv) For each g ∈ G there exists g−1 ∈ G such that d(g) = g−1g and
r(g) = gg−1.
An element e ∈ G is called an identity of G if e = d(g) = r(g−1), for
some g ∈ G. We will denote by G0 the set of all the identities of G and
by G2 the set of all the pairs (g, h) such that the product gh is defined.
The statements of the following lemma are straightforward from the
above definition. Such statements will be freely used along this paper.
Lemma 3.1. Let G be a groupoid. Then,
(i) for every g ∈ G, the element g−1 is unique satisfying g−1g = d(g)
and gg−1 = r(g),
(ii) for every g ∈ G, d(g−1) = r(g) and r(g−1) = d(g),
(iii) for every g ∈ G, (g−1)−1 = g,
(iv) for every g, h ∈ G, (g, h) ∈ G2 if and only if d(g) = r(h),
(v) for every g, h ∈ G, (h−1, g−1) ∈ G2 if and only if (g, h) ∈ G2 and,
in this case, (gh)−1 = h−1g−1,
(vi) for every (g, h) ∈ G2, d(gh) = d(h) and r(gh) = r(g),
(vii) for every e ∈ G0, d(e) = r(e) = e and e−1 = e,
(viii) for every (g, h) ∈ G2, gh ∈ G0 if and only if g = h−1,
(ix) for every g, h ∈ G, there exists l ∈ G such that g = hl if and only
if r(g) = r(h),
(x) for every g, h ∈ G, there exists l ∈ G such that g = lh if and only
if d(g) = d(h).
Given a groupoid G and H a nonempty subset of G, we say that H is
a subgroupoid of G if it satisfies the following conditions:
(i) For every g, h ∈ H, if there exists gh then gh ∈ H.
(ii) For every g ∈ H, g−1 ∈ H.
A. Paques, T. Tamusiunas 85
If, in addition, H0 = G0, we say that H is an wide subgroupoid.
An action of a groupoid G on a nonempty set X is a collection γ of
subsets Xg = Xr(g) of X and bijections γg : Xg−1 −→ Xg (g ∈ G) such
that:
(i) γe is the identity map IdXe of Xe, for every e ∈ G0,
(ii) γg ◦ γh(x) = γgh(x), for every (g, h) ∈ G2 and x ∈ Xh−1 = X(gh)−1 .
In this case, we also say that X is a G-set. If, in addition, the union of the
subsets Xe, e ∈ G0, is disjoint and equal to X (shortly X =
⋃̇
e∈G0
Xe)
we say that X is a G-split set.
Example 3.2. A groupoid G is a G-split set. In fact, for X = G, take
Xg = r(g)G = {r(g)l | r(l) = r(g)} = Xr(g) and γg : Xg−1 → Xg
given by γg(d(g)l) = gd(g)l (= gl = r(g)gl), for all g ∈ G. Notice that
G =
⋃̇
e∈G0
Xe by construction.
Example 3.3. Consider H an wide subgroupoid of G. Take the equiva-
lence relation ≡H defined by: for every a, b ∈ G, a ≡H b if and only if there
exists b−1a and b−1a ∈ H . Notice that g = gd(g) ∈ gH = {gh | r(h) =
d(g)}, for every g ∈ G, for H is wide. Then, the set G
H
= {gH | g ∈ G}
is a G-split set. Indeed, for X = G
H
, it is enough to take Xg = {lH ∈ G
H
|
r(l) = r(g)} = Xr(g) and to define γg : Xg−1 → Xg by γg(lH) = glH,
for all g ∈ G. As in the previous example, also here G
H
=
⋃̇
e∈G0
Xe by
construction.
An action of a groupoid G on a K-algebra R [1] is a collection β of
ideals Eg = Er(g) of R and algebra isomorphisms βg : Eg−1 → Eg (g ∈ G),
such that R is a G-set via β. In this case, the set
Rβ := {r ∈ R | βg(rx) = rβg(x), for all g ∈ G and x ∈ Eg−1}
is indeed a K-subalgebra of R, called the subalgebra of the invariants of
R under the action β. If each Eg is unital, with identity element 1g, then
it is immediate to see that r ∈ Rβ if and only if βg(r1g−1) = r1g, for all
g ∈ G.
Let R, G and β = {βg : Eg−1 → Eg}g∈G be as above. Accordingly
to [1], the skew groupoid ring R ⋆β G corresponding to β is defined as the
direct sum
R ⋆β G =
⊕
g∈G
Egδg
86 A Galois-Grothendieck-type correspondence
in which the δg’s are symbols, with the usual addition, and multiplication
determined by the rule
(xδg)(yδh) =
{
xβg(y)δgh if (g, h) ∈ G2
0 otherwise,
for all g, h ∈ G, x ∈ Eg and y ∈ Eh. It is straightforward to check that
this multiplication is well defined and that R ⋆β G is associative. If G0
is finite and each Ee, e ∈ G0, is unital, then R ⋆β G is also unital [6],
with identity element given by
∑
e∈G0
1eδe, where 1e denotes the identity
element of Ee.
Hereafter, in this section,
• G is a finite groupoid,
• γ = {γg : Xg−1 → Xg}g∈G is an action of G on a fixed nonempty
and finite set X such that X =
⋃̇
e∈G0
Xe, that is, X is a finite
G-split set.
• and β = {βg : Eg−1 → Eg}g∈G is an action of G on a fixed faithful
K-algebra R such that each Ee (e ∈ G0) is unital with identity
element 1e, R =
⊕
e∈G0
Ee, and Rβ = K.
In this context, any left R ⋆β G-module M is also an R-module via
the imbedding r 7→
∑
e∈G0
r1eδe, for all r ∈ R. We put
MG = {x ∈ M | (1gδg)x = 1gx, for all g ∈ G}
to denote the K-module of the invariants of M under G. Notice that
the K-algebra R is also a left R ⋆β G-module via the action (rgδg)x =
rgβg(x1g−1), for all x ∈ R, g ∈ G and rg ∈ Eg, and RG = Rβ = K.
Now, consider the set
Map(X,R) = {f : X → R | f(Xg) ⊆ Eg, for all g ∈ G},
which clearly is an R-algebra (in particular, a K-algebra) under the usual
pointwise operations, whose identity element is
∑
e∈G0
1′
e, where 1′
g is
defined by
1′
g(x) =
{
1g if x ∈ Xg
0, otherwise
for every g ∈ G.
Furthermoremore, it is straightforward to check that
• Mg = Map(X,R)g = {f ∈ Map(X,R) | f(Xh) = 0, if Xh 6= Xg}
is an ideal of Map(X,R) with identity element 1′
g;
A. Paques, T. Tamusiunas 87
• Mg = Mr(g);
• αg : Mg−1 → Mg, given by
αg(f1′
g−1)(x) =
{
βg ◦ f1′
g−1 ◦ γg−1(x) if x ∈ Xg
0 otherwise,
is an isomorphism of K-algebras;
• α = {αg : Mg−1 → Mg}g∈G is an action of G on Map(X,R);
• Map(X,R) =
⊕
e∈G0
Me;
• Map(X,R) is a leftR⋆βG-module via the action (rgδg)f = rgαg(f1′
g−1).
We will denote by A(X) the K-subalgebra of the invariants of
Map(X,A) under α, as well as under G, that is, A(X) = Map(X,R)α =
{f ∈ Map(X,R) | αg(f1′
g−1) = f1′
g, for all g ∈ G} = Map(X,R)G. No-
tice that if f ∈ A(X), then βg(f(x)) = f(γg(x)), for every x ∈ Xg−1 .
For g ∈ G and every x ∈ Xg set Ex = Eg. For g ∈ G and x ∈ X, let
ρx : A(X) → Ex be the algebra homomorphism given by ρx(f) = f(x), for
every f ∈ A(X). Set Vg(X) := {ρx | x ∈ Xg}. Clearly, Vg(X) = Vr(g)(X).
Lemma 3.4. Assume that K is a direct summand of R as K-modules
and Eg is a faithfully projective K-module, for each g ∈ G. Then the
following conditions are equivalent:
(i) For every g ∈ G, the elements of Vg(X) are pairwise strongly distinct,
rankKA(X) = #Vg(X) and A(X) is a strongly separable extension
of K;
(ii) For every g ∈ G, the map ϕg : Eg ⊗ A(X) →
∏
x∈Xg
Ex, given by
ϕg(r ⊗ f) = (rf(x))x∈Xg , is an isomorphism of R-algebras.
Proof. It is an immediate consequence of Lemma 2.4.
Following [1] R is a β-Galois extension of Rβ = K if there exist
elements ri, si ∈ R, 1 ≤ i ≤ m, such that
∑
1≤i≤m xiβg(si1g−1) = δe,g1e,
for all e ∈ G0 and g ∈ G. The elements xi, yi are called the β-Galois
coordinates of R over Rβ. It is immediate to see that, in this case, the
trace map
tβ : R → R, given by tβ(r) =
∑
g∈G
βg(r1g−1),
is a K-linear map, and tβ(R) = K by [1, Lemma 4.2 and Corollary 5.4].
Hence, K is a direct summand of R as K-modules.
88 A Galois-Grothendieck-type correspondence
Lemma 3.5. Assume that R is a β-Galois extension of K. Then, for
each g ∈ G, the map ϕg : Eg ⊗A(X) →
∏
x∈Xg
Ex, given by ϕg(r ⊗ f) =
(rf(x))x∈Xg , is an isomorphism of R-algebras.
Proof. Since Map(X,R)G = A(X), it follows from [1, Theorem 5.3]
that the map µ : R ⊗ A(X) → Map(X,R) given by µ(r ⊗ f) = rf is
an isomorphism of R-algebras, which clearly induces an isomorphism
µg : Eg ⊗ A(X)) → Map(Xg, Eg). On the other hand, Map(Xg, Eg) ≃∏
x∈Xg
Ex, as R-algebras, via the map ηg : f 7→ (f(x))x∈Xg . Since ϕg =
ηgµg, the result follows.
4. The Galois-Grothendieck-type correspondence
We start recalling that G, R, X, β and γ are as in the previous section.
Let V (X) =
⋃
e∈G0
Ve(X) = {ρx|x ∈ Xe, e ∈ G0} = {ρx|x ∈ Xg, g ∈ G}.
Let Y and W be G-sets via the actions ε = {εg : Yg−1 → Yg}g∈G and
ϑ = {ϑg : Wg−1 → Wg}g∈G, respectively. A map ψ : Y → W is said an
isomorphism of G-sets if the following conditions are satisfied:
(i) ψ is a bijection;
(ii) ψ(Yg) = Wg, for all g ∈ G;
(iii) ψ(εg(y)) = ϑg(ψ(y)), for all y ∈ Yg−1 and g ∈ G.
Lemma 4.1. Assume that R is a β-Galois extension of K. Then:
(i) V (X) is a G-split set;
(ii) The elements of Vg(X) are pairwise strongly distinct, for every
g ∈ G,;
(iii) The map ω : X → V (X), given by ω(x) = ρx, is an isomorphism
of G-sets.
Proof. (i) Take σ = {σg : Vg−1(X) → Vg(X)}g∈G, where σg(ρx)(f) =
βg(f(x)), for every x ∈ Xg−1 . Observe that f ∈ A(X), hence σg(ρx)(f) =
βg(f(x)) = f(γg(x)) = ργg(x)(f) and, consequently, σg(ρx) ∈ Vg(X),
showing that the map σg is well-defined. Moreover, σg is a bijection with
inverse σg−1 , for every g ∈ G. It is immediate to check that σ is an action
of G on V (X), and V (X) =
⋃̇
e∈G0
Ve(X) by construction.
(ii) It follows from Lemma 3.5 that, for every g ∈ G, the map ϕg : Eg ⊗
A(X) →
∏
x∈Xg
Ex, given by ϕg(r⊗ f) = (rf(x))x∈Xg , is an isomorphism
of R-algebras. Thus, for each x ∈ Xg, there exist rix ∈ Eg and fix ∈
A(X), 1 ≤ i ≤ mx, such that (
∑mx
i=1 rixfix(y))y∈Xg = (δx,y1g)y∈Xg . Hence,∑mx
i=1 rixρy(fix) =
∑mx
i=1 rixfix(y) = δx,y1g, for every y ∈ Xg, and the
assertion follows by Proposition 2.2.
A. Paques, T. Tamusiunas 89
(iii) Consider the surjective map ωg : Xg → Vg(X) given by ωg(x) = ρx,
for every x ∈ Xg. Indeed, ωg is a bijection. If ρx = ρy, for x, y ∈ Xg, then
f(x) = f(y), for every f ∈ A(X).
On the other hand, the map ηg : Map(Xg, Eg) →
∏
x∈Xg
Ex, given by
ηg(f) = (f(x))x∈Xg , is an isomorphism of R-algebras, whose inverse is
the map η′
g :
∏
x∈Xg
Ex → Map(Xg, Eg) given by η′
g(r)(x) = rx, where
r = (rx)x∈Xg ∈
∏
x∈Xg
Ex. Furthermore, the map ϕg : Eg ⊗ A(X) →∏
x∈Xg
Ex, given by ϕg(r ⊗ f) = (rf(x))x∈Xg , is also an isomorphism of
R-algebras, by Lemma 3.5.
Thus, Eg ⊗ A(X) ≃
∏
x∈Xg
Ex ≃ Map(Xg, Eg), and so, for every
p ∈ Map(Xg, Eg), there exists λ =
∑
1≤i≤m ri ⊗ fi ∈ Eg ⊗ A(X) such
that p = η′
g ◦ ϕg(λ). Consequently,
p(x) = (η′
g ◦ ϕg(λ))(x) = η′
g((
∑
1≤i≤m
rifi(z))z∈Xg )(x)
=
∑
1≤i≤m
rifi(x) =
∑
1≤i≤m
rifi(y) = p(y),
for every p ∈ Map(Xg, Eg). So, x = y.
Therefore, the map ω : X → V (X), given by ω(x) = ωg(x) if x ∈ Xg,
is also a bijection, and ω(Xg) = Vg(X).
Finally, ω commutes with the actions σ and γ. Indeed, for x ∈ Xg−1
and f ∈ A(X), we have
ω(γg(x))(f) = ργg(x)(f) = f(γg(x))
= βg(f(x)) = σg(ρx)(f) = σg(ω(x))(f),
which concludes the proof.
For any K-algebras B and C, we will denote by AlgK(B,C) the set
of all K-algebra homomorphisms from B into C.
Lemma 4.2. Let B be a K-algebra and g ∈ G. Suppose that Eg is
faithfully projective and there exists an isomorphism of Eg-algebras ϕg :
Eg ⊗B → (Eg)ng , ng ≥ 1. Then:
(i) B is faithfully projective over K with constant rank ng;
(ii) B is a strongly separable extension of K;
(iii) There exist ϕ(g,1), . . . , ϕ(g,ng) ∈ AlgK(B,Eg) such that ϕg(r ⊗ b) =
(rϕ(g,i)(b))1≤i≤n for every r ∈ Eg and b ∈ B;
(iv) The elements of Vg(B) = {ϕ(g,i)| 1 ≤ i ≤ ng} are pairwise strongly
distinct;
(v) Vg(B) = AlgK(B,Eg) whenever the elements of AlgK(B,Eg) are
pairwise strongly distinct.
90 A Galois-Grothendieck-type correspondence
Proof. The assertions (i) and (ii) follows by the same arguments used in
the proof of Lemma 2.4((iii)⇒(i)).
(iii) Denote by ηg : B → Eg ⊗ B the map given by b 7→ 1g ⊗ b, and
by π(g,i) : (Eg)n → Eg the ith-projection, for every 1 ≤ i ≤ ng. Clearly,
the maps ϕ(g,i) := π(g,i)ϕgηg are in AlgK(B,Eg) and it is easy to see that
ϕg(r ⊗ b) = (rϕ(g,i)(b))1≤i≤ng , for all r ∈ Eg and b ∈ B.
(iv) Since ϕg is an isomorphism, for each 1 ≤ i ≤ ng, there ex-
ist ril ∈ Eg and bil ∈ B, 1 ≤ l ≤ mg, such that ϕg(
∑mg
l=1 ril ⊗ bil) =
(
∑mg
l=1 rilϕ(g,j)(bil))1≤j≤ng = (δi,j1g)1≤j≤ng , that is,
∑mg
l=1 silϕ(g,j)(bil) =
δi,j1g, for every 1 ≤ j ≤ ng. Consequently, the elements of Vg(B) are
pairwise strongly distinct, by (ii) and Proposition 2.2.
(v) Suppose that the elements of AlgK(B,Eg) are pairwise strongly
distinct. Then, by (i), (ii) and Corollary 2.3, #AlgK(B,Eg) ≤ rankKB =
ng = #Vg(B) ≤ #AlgK(B,Eg). Thus, Vg(B) = AlgK(B,Eg).
The next lemma provide us a necessary and sufficient condition for the
set V (B) =
⋃
e∈G0
Ve(B) to be a G-set. Again here, this union is disjoint
and finite by construction.
Lemma 4.3. Let B, Eg, ϕg and Vg(B) (g ∈ G), be as in Lemma 4.2.
Then the following assertions are equivalent:
(i) V (B) is a G-set via ξ = {ξg : Vg−1(B) → Vg(B)}g∈G, with
ξg(ϕ(g−1,i))(b) = βg(ϕ(g−1,i)(b)), for every b ∈ B;
(ii) For every g, h ∈ G with r(g) = r(h) and Vg−1(B) = Vh−1(B), the
elements ξg(ϕ(g−1,i)) and ξh(ϕ(g−1,j)) are strongly distinct for all
1 ≤ i, j ≤ ng.
Proof. (i) ⇒ (ii) It is enough to notice that if r(g) = r(h) then Vg(B) =
Vh(B). Now, the assertion follows from Lemma 4.2(iv).
(ii) ⇒ (i) It is enough to show that each ξg, g ∈ G, is a bijection for the
conditions (i)-(ii) of the definiton of a groupoid action are straightforward.
Also, each ξg is injective by construction, thus it is enough to prove that
it is surjective.
We start by noticing that the elements of Vg−1(B) are pairwise strongly
distinct, by Lemma 4.2. Consequently, the elements of ξg(Vg−1(B)) are
pairwise strongly distinct and it follows from the assumption that also the
elements of Yg(B) =
⋃
{h∈G|r(h)=r(g)} ξh(Vg−1(B)) are pairwise strongly
distinct.
Clearly, Yg(B) ⊆ Vg(B), and noting that r(r(g)) = r(g) and Vg(B) =
Vr(g)(B) = Vr(g)−1(B) = ξr(g)(Vr(g)−1(B)), we have that Vg(B) ⊆ Yg(B),
for every g ∈ G.
A. Paques, T. Tamusiunas 91
Furthermore, ξg(Vg−1(B)) ⊆ Yg(B) = Vg(B) and by Lemma 4.2
#ξg(Vg−1(B)) = #Vg−1(B) = ng−1 = rankKB = ng = #Vg(B). Hence,
ξg(Vg−1(B)) = Vg(B), and ξg is a bijection.
Assume that S =
⊕n
j=1 Sj is a K-algebra, where Sj = S1j and
{1j}1≤j≤n are pairwise orthogonal central idempotents in S, for some
n ≥ 1. An K-algebra T is said to be S-split if:
(i) For each 1 ≤ j ≤ n, there exists an isomorphism of K-algebras
φj : Sj ⊗ T → (Sj)m, for some given m ≥ 1;
(ii) V (T ) =
⋃n
j=1 Vj(T ) is aG-set, where Vj(T ) is defined as in Lemma 4.2.
Notice that (i) is equivalent to say that S ⊗ T ≃ Sm and, in particular,
V (T ) is a finite G-split set.
Lemma 4.4. Let B, Eg, ϕg and Vg(B) (g ∈ G) be as in Lemma 4.2.
Assume that R is a β-Galois extension of K and V (B) is a G-set via
ξ = {ξg : Vg−1(B) → Vg(B)}g∈G. Then, the mapping ν : B → A(V (B)),
given by ν(b)(ϕ(g,i)) = ϕ(g,i)(b), for b ∈ B and ϕ(g,i) ∈ V (B), is an
isomorphism of K-algebras.
Proof. We start by checking that ν is a well defined. Indeed, for g ∈ G,
b ∈ B and ϕ(g,i) ∈ V (B), we have
αg(ν(b)1′
g−1)(ϕ(g,i)) = βg ◦ ν(b)1′
g−1 ◦ ξg−1(ϕ(g,i)) = βg(ν(b)(ξg−1(ϕ(g,i)))1g−1)
= βg(ξg−1(ϕ(g,i))(b)1g−1) = βg(βg−1(ϕ(g,i)(b)1g)1g−1)
= βr(g)(ϕ(g,i)(b)1r(g)) = ϕ(g,i)(b)1r(g)
= ϕ(g,i)(b)1g = ν(b)(ϕ(g,i))1g = ν(b)1′
g(ϕ(g,i)),
showing that ν(b) ∈ A(V (B)). Clearly, ν is an algebra homomorphism.
It remains to check that it is a bijection.
Given a, b ∈ B, if a 6= b, then ϕg(1g ⊗ a) 6= ϕg(1g ⊗ b), since for each
g ∈ G, Eg is faithful over K and ϕg is an isomorphism. Thus, there exists
1 ≤ i ≤ ng such that ν(a)(ϕ(g,i)) = ϕ(g,i)(a) 6= ϕ(g,i)(b) = ν(b)(ϕ(g,i)). So,
ν(a) 6= ν(b) and ν is injective.
By Lemmas 3.5 and 4.2, the K-algebras A(V (B)) and B are faithfully
projective and separable, and rankKA(V (B)) = #Vg(B) = rankKB.
Since, ν(B) ≃ B as K-algebras, it follows from [5, Lemma 1.1] that
ν(B) = A(V (B)), so ν is surjective.
Let R−splitAlg denote the category whose objects are the R-split K-
algebras and whose morphisms are algebra homomorphisms. Also, let
G−splitFinSet denote the category whose objects are finite G-split sets and
whose morphisms are G-maps (i.e, maps that commute with the action of
92 A Galois-Grothendieck-type correspondence
G). Let θ :G−split FinSet →R−split Alg and θ′ :R−split Alg →G−split FinSet
be the maps given by X 7→ A(X) and B 7→ V (B), respectively.
Theorem 4.5 (The Galois-Grothendieck equivalence). Assume that R is
a β-Galois extension of K and Eg is faithfully projective, for every g ∈ G.
Then, θ is a contravariant functor that induces an equivalence between
the categories G−splitFinSet and R−splitAlg, with inverse θ′.
Proof. By Lemma 3.5, given a finite G-split set X, the map ϕg : Eg ⊗K
A(X) −→
∏
x∈Xg
Ex defined by ϕg(r ⊗K f) = (rf(x))x∈Xg is an isomor-
phism of R-algebras, for every g ∈ G. Thus, it is immediate, from the
definitions, that Vg(A(X)) = Vg(X), for every g ∈ G. Indeed, it is enough
to see that
ϕ(g,i)(f) = π(g,i)ϕgηg(f) = π(g,i)(ϕg(1g ⊗K f))
= π(g,i)((f(x))x∈Xg ) = f(x) = ρx(f),
for all f ∈ A(X) and 1 ≤ i ≤ ng. Hence V (X) = V (A(X)).
Finally, recall that X ≃ V (X) as G-sets, and B ≃ A(V (B)) as Rβ-
algebras, by Lemmas 3.5, 4.1 and 4.4. Hence, X ≃ V (A(X)) = θ′(θ(X))
and B ≃ A(V (B)) = θ(θ′(B)).
5. The Galois-type correspondence
Let R, G and β = {βg : Eg−1 → Eg | g ∈ G} be as in the previous
section, and H ⊆ G an wide subgroupoid of G. Then, βH = {βh : Eh−1 →
Eh | h ∈ H} is an action of H on R. Furthermore, recall from Example 3.3
that G
H
= {gH |g ∈ G} is a finite G-set via the action γ = {γg : Xg−1 →
Xg}g∈G, where Xg = {lH ∈ G
H
| r(l) = r(g)} = Xr(g) and γg(lH) = glH,
for all g ∈ G. Recall also that G
H
=
⋃̇
e∈G0
Xe.
Lemma 5.1. A( G
H
) ≃ RβH as K-algebras, for every wide subgroupoid H
of G.
Proof. We start by noticing that
∑
e∈G0
f(eH) ∈ RβH , for every f ∈
A( G
H
). Indeed, recall that f(eH) ∈ Ee, for all e ∈ G0, βh−1(f(lH)) =
f(γh−1(lH)) = f(h−1lH), for all lH ∈ Xh, and hH = r(h)H, for all
h ∈ H. So,
βh(
∑
e∈G0
f(eH)1h−1) =
∑
e∈G0
βh(f(eH)1h−1) = βh(f(d(h)H))
= βh(f(h−1hH)) = βh(βh−1(f(hH))) = βr(h)(f(hH))
A. Paques, T. Tamusiunas 93
= f(hH) = f(r(h)H) = f(r(h)H)1r(h) = f(r(h)H)1h
=
∑
e∈G0
f(eH)1h.
Therefore, the map
θ : A( G
H
) −→ RβH
f 7−→
∑
e∈G0
f(eH).
is well defined.
Conversely, given g1, g2 ∈ G and r ∈ RβH , if g1H = g2H then
βg1(r1
g−1
1
) = βg2(r1
g−1
2
). Indeed, from g1H = g2H it follows that for
any h1 ∈ H there exists h2 ∈ H such that g1h1 = g2h2. So, g1 =
g1d(g1) = g1r(h1) = g1h1h
−1
1 = g2h2h
−1
1 . Furthermore, E(g2h2h−1
1 )−1 =
Eh1 = E(h2h−1
1 )−1 and E
g−1
2
= E
h2h−1
1
. Thus,
βg1(r1
g−1
1
) = β
g2h2h−1
1
(r1
h1h−1
2 g−1
2
) = βg2(β
h2h−1
1
(r1
h1h−1
2 g−1
2
))
= βg2(β
h2h−1
1
(r1
h1h−1
2 g−1
2
)β
h2h−1
1
(1
h1h−1
2 g−1
2
))
= βg2(β
h2h−1
1
(r1
h1h−1
2
)β
h2h−1
1
(1
h1h−1
2
)) = βg2(r1
h2h−1
1
)
= βg2(r1
g−1
2
)
Hence, the map
θ′ : RβH −→ Map( G
H
, R),
r 7−→ θ′
r
where θ′
r(lH) = βl(r1l−1), is well defined. In fact, θ′
r(gH) ∈ A( G
H
) since
αg(θ′
r1′
g−1)(lH) = βg(θ′
r1′
g−1(γg−1(lH))) = βg(θ′
r(g−1lH)1g−1)
= βg(βg−1l(r1l−1g)) = βg(βg−1(βl(r1l−1)))
= βr(g)(βl(r1l−1)) = βl(r1l−1)
= βl(r1l−1)1g = θ′
r1′
g(lH),
for all g ∈ G such that r(g) = r(l). If r(g) 6= r(l) then αg(θ′
r1′
g−1)(lH) =
0 = θ′
r1′
g(lH).
Clearly, θ and θ′ are homomorphisms of K-algebras. Furthermore,
θ ◦ θ′(r) = θ(θ′
r) =
∑
e∈G0
θ−1
r (eH)
=
∑
e∈G0
βe(r1e) =
∑
e∈G0
r1e = r,
94 A Galois-Grothendieck-type correspondence
for every r ∈ R, and
θ′ ◦ θ(f)(gH) = θ′∑
e∈G0
f(eH)
(gH) = βg(
∑
e∈G0
f(eH)1g−1)
= βg(f(d(g)H)) = βg(βg−1(f(gH)))
= βr(g)(f(gH)) = f(gH),
for every f ∈ A( G
H
) and g ∈ G. The proof is complete.
For any K-subalgebra T of R put HT = {g ∈ G | βg(t1g−1) =
t1g, for all t ∈ T}. It is easy to check that HT is an wild subgroupoid of
G. We say that T is β-strong if for every g, h ∈ G such that r(g) = r(h)
and g−1h /∈ HT , and, for every nonzero idempotent e ∈ Eg = Eh, there
exists an element t ∈ T such that βg(t1g−1)e 6= βh(t1h−1)e.
Lemma 5.2. For each gH ∈ G
H
, let ρgH : A( G
H
) → Er(g) the homomor-
phism of K-algebras given by ρgH(f) = f(gH), for every f ∈ A( G
H
). If
the elements of VgH = {ρlH | r(l) = r(g)} are pairwise strongly distinct,
then RβH is β-strong.
Proof. By the Lemma 5.1, A( G
H
) ≃ RβH via the map θ. Consider φgH :=
ρgH ◦ θ−1 : RβH → Er(g). Since the elements of VgH are pairwise strongly
distinct, it is easy to see that the elements of ṼgH = {φlH | r(l) = r(g)}
are also pairwise strongly distinct.
Let T = RβH and take g, h ∈ G such that r(g) = r(h) and g−1h /∈ HT .
Given a nonzero idempotent e ∈ Eg = Eh, there exists r ∈ RβH such that
φgH(r)e 6= φhH(r)e. Thus,
βg(r1g−1)e = θ−1(r)(gH)e = ρgH(θ−1(r))e = φgH(r)e
6= φhH(r)e = ρhH(θ−1(r))e = θ−1(r)(gH)e
= βh(r1h−1)e.
Therefore, RβH is β-strong.
Lemma 5.3. Assume that R is a β-Galois extension of K and suppose
that T is a subalgebra of R which is separable over K and β-strong. Then
there exist elements xi, yi ∈ T , 1 ≤ i ≤ m, such that
∑m
i=1 xiβg(yi1g−1) =
δe,g1e, for all e ∈ G0. In particular, T is a faithfully projective K-module.
Proof. Let υ =
∑n
i=1 xi ⊗ yi ∈ T ⊗ T be the separability idempotent of T
over K and µ : T ⊗ T the multiplication map. For g ∈ G, define
ψg : T ⊗ T → T ⊗ Eg
x⊗ y 7→ x⊗ βg(y1g−1).
A. Paques, T. Tamusiunas 95
and take υg = µ(ψg(e)) =
∑n
i=1 xiβg(yi1g−1) ∈ Eg. Clearly, υg is an idem-
potent of Eg, for µ and ψ are K-algebra homomorphisms. In particular,
υe = 1e, for all e ∈ G0.
Moreover, µ and ψg are T ⊗K-linear. Thus, for every t ∈ T ,
tυg = tµ(ψg(e)) = (t⊗ 1R).µ(ψg(e)) = µ(ψg((t⊗ 1R)e))
= µ(ψg((1R ⊗ t)e)) = µ(ψg((1R ⊗ t))µ(ψg(e))
= βg(t1g−1)υg.
Since T is β-strong, if g /∈ G0, then υg = 0, that is,
∑n
i=1 xiβg(yi1g−1) = 0.
For the second part, it is enough to take the maps fi ∈ HomK(T,K)
given by fi(t) = trβ(yit), 1 ≤ i ≤ m, and to see that
n∑
i=1
fi(t)xi =
n∑
i=1
∑
g∈G
βg(yit1g−1)xi =
∑
e∈G0
1et = 1Rt = t,
for every t ∈ T .
Lemma 5.4. Assume that R is a β-Galois extension of K and let T be
a subalgebra of R. Then the following conditions are equivalents:
(i) T is separable over K and β-strong;
(ii) T = RβHT .
In particular, in this case, T is R-split.
Proof. (i) ⇒ (ii) By Lemma 5.3, T is projective and finitely gener-
ated as K-module. Since T ⊆ RβHT , we have Tp ⊆ (RβHT )p, and thus
rankKp
Tp ≤ rankKp
(RβHT )p, for every prime ideal p of K. We shall prove
that indeed rankKp
Tp = rank(Kp
(RβHT )p for every prime ideal p of K,
and, consequently, T = RβHT , by [5, Lemma 1.1].
Let {gi ∈ G |1 ≤ i ≤ n} be a left tranversal of HT in G. Define
fi : T −→ Egi
t 7−→ βgi
(t1
g−1
i
).
Clearly, the fi’s are K-algebra homomorphisms and the elements of
Vgi
= {fj | 1gj
= 1gi
} are pairwise strongly distinct, for T is β-strong.
Therefore, by Corollary 2.3, #Vgi
≤ rank(Rβ)pTp, for every prime ideal p
of K.
By Lemma 3.5, we have that Egi
⊗ RβHT ≃
∏
x∈( G
HT
)gi
Ex, thus
(Egi
)p ⊗Kp
(RβHT )p ≃
∏
x∈( G
HT
)gi
(Ex)p. Recall from Example 3.3 that
96 A Galois-Grothendieck-type correspondence
( G
HT
)gi
= {lHT | r(l) = r(gi)}. Then, #Vgi
= #( G
HT
)gi
. Therefore,
rankKp
(RβHT )p = rank(Egi
)p((Egi
)p ⊗Kp
(RβHT )p)
= rank(Egi
)p
∏
x∈( G
HT
)gi
(Ex)p
= #( G
HT
)gi
= #Vgi
≤ rank(Rβ)pTp,
and so rankKp
Tp = rankKp
(RβHT )p.
(ii) ⇒ (i) By Lemmas 3.5 and 2.4, T = RβHT ≃ A( G
HT
) is separable
over K. Furthermore, by Lemma 4.1 the elements of VgHT
are pairwise
strongly distinct. Hence, T is β-strong, by Lemma 5.2.
The last assertion follows from Lemmas 3.5 and 4.1.
Theorem 5.5 (The Galois correspondence). Assume that R is a β-Galois
extension of K and Eg is faithfully projective, for every g ∈ G. Then
the correspondence H 7→ RβH is one-to-one between the set of all the
wide subgroupoids of G and the set of all the subalgebras of R which are
separable over K and β-strong.
Proof. Let wsg(G) be the set of the wide subgroupoids H of G, quot(G)
the set of the quotients sets G
H
of G and sss(R) the set of the separable
and β-strong K-subalgebras of R. The bijection between wsg(G) and
quot(G) is obvious. The bijection between quot(G) and sss(R) follows
from Lemma 5.4 and Theorem 4.5.
6. A final remark
Again, R, G and β are as in the previous sections. In almost all
results in the two last sections the assumption that Eg is a faithful
K-module was required. So, it is natural to ask under what conditions
such an assumption occurs. To answer this question it is necessary to
have a description of the elements in Rβ = K. An easy calculus shows
that an element x =
∑
e∈G0
xe ∈ R =
⊕
e∈G0
Ee is in Rβ if and only if
xr(g) = βg(xd(g)), for all g ∈ G. It is an immediate consequence of this
fact that, given x ∈ K and g ∈ G, x1g = 0 if and only if xr(g) = 0 if and
only if xd(g) = 0. Therefore, given x ∈ K and g ∈ G, x1g = 0 implies
x = 0 if and only if, for all h ∈ G, either d(h±1) = d(g) or d(h±1) = r(g).
From these considerations we have the following lemma.
Lemma 6.1. For each g ∈ G, Eg is faithful over K if and only if either
d(h±1) = d(g) or d(h±1) = r(g), for all h ∈ G.
The following two examples illustrate the above lemma. Notice that
both of them are also examples of β-Galois extensions.
A. Paques, T. Tamusiunas 97
Examples 6.2. (1) Consider R = Sv1 ⊕ Sv2 ⊕ Sv3 ⊕ Sv4, where S is a
ring and v1, v2, v3 and v4 are pairwise orthogonal central idempotents of R,
with sum 1R. Let G = {g, g−1, d(g), r(g)} be a groupoid and β the action
of G on R given by: Eg = Er(g) = Sv3 ⊕ Sv4, Eg−1 = Ed(g) = Sv1 ⊕ Sv2,
βr(g) = IEr(g)
, βd(g) = IEd(g)
, βg(av1 + bv2) = av3 + bv4, βg−1(av3 + bv4) =
av1 +bv2, for all a, b ∈ S. It is easy to see that R is a β-Galois extension of
K = Rβ = S(v1 +v3)⊕S(v2 +v4), with β-Galois coordinates xi = vi = yi,
1 ≤ i ≤ 4. Furthermore, it is immediate that xEg = 0 = xEg−1 if and
only if x = 0, for all x ∈ K.
(2) Let R = Sv1 ⊕ Sv2 ⊕ Sv3 ⊕ Sv4 ⊕ Sv5 ⊕ Sv6, where S is a ring
and vi, 1 ≤ i ≤ 6, are pairwise orthogonal central idempotents of R, with
sum 1R. Take the groupoid G = {g, g−1, d(g), r(g), h = h−1, d(h) = r(h)}
and β = {βl : El−1 → El}l∈G, where El and βl, for l = d(g), r(g), g, g−1,
are as in the example (1), Eh = Er(h) = Sv5 + Sv6, βr(h) = IEr(h)
,
and βh(av5 + bv6) = av6 + bv5. Again, R is a β-Galois extension of
K = Rβ = S(v1 +v3) ⊕S(v2 +v4) ⊕S(v5 +v6), with β-Galois coordinates
xi = vi = yi, 1 ≤ i ≤ 6. Nevertheless, in this case we have, for instance,
xEh = 0 for x = v1 + v3 ∈ K.
References
[1] D. Bagio and A. Paques, Partial groupoid actions: globalization, Morita theory
and Galois theory, Comm. Algebra 40 (2012), 3658-3678.
[2] F. Borceux and G. Janelidze, Galois Theories, Cambridge Univ. Press, 2001.
[3] S. Chase; D. K. Harrison and A. Rosenberg, Galois Theory and Galois Cohomology
of Commutative Rings, Mem. AMS 52 (1968), 1-19.
[4] A. Dress, One More Shortcut to Galois Theory, Adv. Math. 110 (1995), 129-140.
[5] M. Ferrero and A. Paques, Galois Theory of Commutative Rings Revisited, Beiträge
zur Algebra und Geometrie, 38 (1997), n◦2, 399-410.
[6] D. Flôres and A. Paques, Duality for groupoid (co)actions, Comm. Algebra 42
(2014), 637-663.
[7] A. Grothendieck, Revêtements étales et groupe fondamental, SGA 1, exposé V,
LNM 224, Sringer Verlag (1971).
[8] M. A. Knus and M. Ojanguren, Théorie de la Descente et Algèbres d’Azumaya,
Lecture Notes in Math. 389, Springer-Verlag (1974).
[9] M. V. Lawson, Inverse Semigroups. The Theory of Partial Symmetries, World
Scientific Pub. Co, London (1998).
Contact information
A. Paques,
T. Tamusiunas
Instituto de Matemática, Universidade Federal
do Rio Grande do Sul, 91509-900, Porto Alegre,
RS, Brazil
E-Mail: paques@mat.ufrgs.br,
trtamusiunas@yahoo.com.br
Received by the editors: 25.01.2014
and in final form 25.01.2014.
|