Projectivity and flatness over the graded ring of semi-coinvariants
Let k be a field, C a bialgebra with bijective antipode, A a right C-comodule algebra, G any subgroup of the monoid of grouplike elements of C. We give necessary and sufficient conditions for the projectivity and flatness over the graded ring of semi-coinvariants of A. When A and C are commutative a...
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irk-123456789-1546192019-06-16T01:31:54Z Projectivity and flatness over the graded ring of semi-coinvariants Guedenon, T. Let k be a field, C a bialgebra with bijective antipode, A a right C-comodule algebra, G any subgroup of the monoid of grouplike elements of C. We give necessary and sufficient conditions for the projectivity and flatness over the graded ring of semi-coinvariants of A. When A and C are commutative and G is any subgroup of the monoid of grouplike elements of the coring A⊗C, we prove similar results for the graded ring of conormalizing elements of A. 2010 Article Projectivity and flatness over the graded ring of semi-coinvariants / T. Guedenon // Algebra and Discrete Mathematics. — 2010. — Vol. 10, № 1. — С. 43–56. — Бібліогр.: 13 назв. — англ. 1726-3255 http://dspace.nbuv.gov.ua/handle/123456789/154619 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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Let k be a field, C a bialgebra with bijective antipode, A a right C-comodule algebra, G any subgroup of the monoid of grouplike elements of C. We give necessary and sufficient conditions for the projectivity and flatness over the graded ring of semi-coinvariants of A. When A and C are commutative and G is any subgroup of the monoid of grouplike elements of the coring A⊗C, we prove similar results for the graded ring of conormalizing elements of A. |
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Guedenon, T. |
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Guedenon, T. Projectivity and flatness over the graded ring of semi-coinvariants Algebra and Discrete Mathematics |
| author_facet |
Guedenon, T. |
| author_sort |
Guedenon, T. |
| title |
Projectivity and flatness over the graded ring of semi-coinvariants |
| title_short |
Projectivity and flatness over the graded ring of semi-coinvariants |
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Projectivity and flatness over the graded ring of semi-coinvariants |
| title_fullStr |
Projectivity and flatness over the graded ring of semi-coinvariants |
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Projectivity and flatness over the graded ring of semi-coinvariants |
| title_sort |
projectivity and flatness over the graded ring of semi-coinvariants |
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Інститут прикладної математики і механіки НАН України |
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2010 |
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http://dspace.nbuv.gov.ua/handle/123456789/154619 |
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Projectivity and flatness over the graded ring of semi-coinvariants / T. Guedenon // Algebra and Discrete Mathematics. — 2010. — Vol. 10, № 1. — С. 43–56. — Бібліогр.: 13 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT guedenont projectivityandflatnessoverthegradedringofsemicoinvariants |
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2025-07-14T06:39:51Z |
| last_indexed |
2025-07-14T06:39:51Z |
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1837603426579513344 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 10 (2010). Number 1. pp. 42 – 56
c© Journal “Algebra and Discrete Mathematics”
Projectivity and flatness
over the graded ring of semi-coinvariants
T. Guédénon
Communicated by V. Mazorchuk
Abstract. Let k be a field, C a bialgebra with bijective
antipode, A a right C-comodule algebra, G any subgroup of the
monoid of grouplike elements of C. We give necessary and sufficient
conditions for the projectivity and flatness over the graded ring of
semi-coinvariants of A. When A and C are commutative and G is
any subgroup of the monoid of grouplike elements of the coring
A⊗C, we prove similar results for the graded ring of conormalizing
elements of A.
Introduction
In the theory of Hopf-Galois extension, it is often important to know
whether certain modules over the ring of coinvariants are projective or flat.
These properties reflect the notions of principal bundles and homogeneous
spaces in a noncommutative setting. In [4], with S. Caenepeel, we gave
sufficient conditions for the projectivity over the subring of coinvariants
of an H-comodule algebra, where H is a Hopf algebra. In [6], we gave
necessary and sufficient conditions for the projectivity and flatness over
the endomorphism ring of a finitely generated module. In [9], necessary
and sufficient conditions for the projectivity and flatness over the endomor-
phism ring of a finitely generated comodule over coring have been studied.
In [10], these results have been extended to the colour endomorphism ring
of a finitely generated G-graded comodule over a G-graded coring, where
G is an abelian group with a bicharacter. To establish all these results
the methods and techniques are inspired from [7]. In the present paper, C
is a bialgebra, A is a C-comodule algebra and G is any subgroup of the
monoid of grouplike elements of C. We consider the G-graded ring S(A) of
T. Guédénon 43
semi-coinvariants of A which is a subring of A containing the subalgebra
of coinvariants of A. We adapt to the graded set-up the methods and
techniques of [7] and [9] to give necessary and sufficient conditions for the
projectivity and flatness over the graded ring S(A).
In an appendix, when A and C are commutative and G is any subgroup
of the monoid of grouplike elements of the coring A⊗C, we give necessary
and sufficient conditions for the projectivity and flatness over the graded
ring N (A) of conormalizing elements of A which is a subring of A.
Throughout we will be working over a field k. All algebras and coalge-
bras are over k. Background information on comodules over coalgebras
and comodules over corings can be found in [1], [2], [3] and [11]. Except
where otherwise stated, all unlabelled tensor products and Hom are tensor
products and Hom over k. We denote by M the category of vector spaces.
1. Preliminary results
We will use the following well-known results of graded ring theory [12].
Let G be a group, B a G-graded ring and Mgr−B, the category of right
G-graded B-modules.
— Let N be a right G-graded B-module. For every x in G, N(x) is
the graded B-module obtained from N by a shift of the gradation by x.
As vector spaces, N and N(x) coincide, and the actions of B on N and
N(x) are the same, but the gradations are related by N(x)y = Nxy for all
y ∈ G.
— An object of Mgr−B is projective (resp. flat) in Mgr−B if and only
if it is projective (resp. flat) in MB, the category of right B-modules.
— An object of Mgr−B is free in Mgr−B if it has a B-basis consisting
of homogeneous elements or equivalently, if it is isomorphic to some
⊕i∈IN(xi), where (xi, i ∈ I) is a family of elements of G.
— Any object of Mgr−B is a quotient of a free object in Mgr−B , and
any projective object in Mgr−B is isomorphic to a direct summand of a
free object.
— An object of Mgr−B is flat in Mgr−B if and only if it is the inductive
limit of finitely generated free objects in Mgr−B.
We will recall some preliminaries on corings and comodules over corings.
Let A be a k-algebra. An A-coring C is an (A,A)-bimodule together with
two (A,A)-bimodule maps ∆C : C → C ⊗A C and ǫC : C → A such that
the usual coassociativity and counit properties hold. Let C be an A-coring.
A right C-comodule is a right A-module M together with a right A-linear
map ρM,C :M →M ⊗A C such that
(idM⊗A ǫC)◦ρM,C = idM , and (idM⊗A∆C)◦ρM,C = (ρM,C⊗A idC)◦ρM,C .
44 Projectivity and flatness over the graded ring
We will use Sweedler-Heyneman notation but we will omit the symbol
∑
:
∆C(c) = c1 ⊗A c2 ρ(M,C)(m) = m0 ⊗A m1.
The algebra A is an A-coring called the trivial A coring. Any k-coalgebra
is a k-coring. A morphism of right C-comodules f : M → N is a right
A-linear map such that ρN,C ◦ f = (f ⊗A idM ) ◦ ρM,C ; or equivalently, a
right A-linear map such that f(m)0⊗A f(m)1 = f(m0)⊗Am1. We denote
the set of comodule morphisms between M and N by HomC(M,N), by
MC the category formed by right C-comodules and morphisms of right
C-comodules. By [2], the category MC has direct sum.
We write ∗C = AHom(AC,AA), the left dual ring of C. Then ∗C is an
associative ring with unit ǫC (see [2, 17.8]): its multiplication is defined by
f#g = f ◦ (idC ⊗A g) ◦∆C , or equivalently f#g(c) = f(c1g(c2))
for all left A-linear maps f , g: C → A and c ∈ C. We will denote by
∗CM the category of left ∗C-modules. Any right C-comodule M is a left
∗C-module: the action is defined by f.m = m0f(m1) (see [2, 19.1]).
A grouplike element of C is an element X ∈ C such that ∆C(X) =
X ⊗AX and ǫC(X) = 1A. We know from [1] that if C contains a grouplike
element X, then A becomes a right C-comodule: the coaction is defined
by ρA,X(a) = Xa. So we have a0 ⊗A a1 = a0a1 = Xa. The algebra A
equipped with this structure of a right C-module will be denoted AX .
Lemma 1.1. Assume that C contains a grouplike element X. Then AX
is a cyclic left ∗C-module under the action defined by f.a = f(Xa) for all
f ∈ ∗C and a ∈ A.
Proof. We already noticed that AX is a left ∗C-module with the given
∗C-action. By [2], there is a ring anti-morphism i : A → ∗C defined
by i(a)(c) = aǫC(c); a ∈ A, c ∈ C. Now for every a ∈ AX , we have
i(a).1A = i(a)(X) = aǫC(X) = a.
Let M be a right C-comodule. We define MX = {m ∈M |ρM,C(m) =
m⊗A X}. We have AX = {a ∈ A|ρA,C(a) = a⊗A X = aX}. An element
m ∈MX is called a X-coinvariant element in ([2], section 28.4) and will
be called a conormal element in this paper.
Lemma 1.2. For every right C-comodule M , MX = HomC(AX ,M).
Proof. (See [2], section 28.4).
T. Guédénon 45
Assume that C is projective as a left A-module. By [2, 18.14], MC is
a Grothendieck category and by [2, 19.3], it is a full subcategory of ∗CM;
i.e.,
HomC(M,N) = ∗C Hom(M,N) for any M,N ∈ MC .
As a consequence, an object of MC that is projective in ∗CM is projective
in MC . From now on all comodules are right comodules.
Let C be a bialgebra with comultiplication ∆C and counit ǫC . We will
write ∆C(c) = c1 ⊗ c2 for all c ∈ C. If M is a C-comodule, we write
ρM,C(m) = m(0) ⊗m(1) for every m ∈ M . Let A be an algebra. We say
that A is a right C-comodule algebra if A is a C-comodule and the unit
and the multiplication are right C-colinear; i.e.,
ρA,C(aa
′) = (aa′)(0)⊗(aa′)(1) = a(0)a
′
(0)⊗a(1)a
′
(1) and ρA,C(1A) = 1A⊗1C .
By [2] or [3], C = A⊗ C is an A-coring with A-multiplications
a′(a⊗ c)a′′ = a′aa′′(0) ⊗ ca′′(1)
and comultiplication idA ⊗∆C . The category MC is isomorphic to the
category MC
A of relative right-right (A,C) Hopf modules, that is the
category of right A-modules M which are also C-comodules such that
ρM,C(ma) = m(0)a(0) ⊗m(1)a(1).
Note that for M ∈ MC we have m0 ⊗A m1 = m(0) ⊗A (1A ⊗m(1)).
The morphisms of MC are just the A-linear maps which are also C-
colinear maps. We will use the notation MC instead of MC
A. The left
dual ∗C of C is anti-isomorphic to the Koppinen smash product #(C,A);
i.e., the vector space Hom(C,A) endowed with the product f#g(c) =
f(c2)(0)g(c1f(c2)(1)) and unit ι ◦ ǫC , where ι is the unit of A. Every
grouplike element x of C induces a grouplike element 1A ⊗ x of C. So the
coring C contains 1A ⊗ 1C as a grouplike element, therefore A is an object
of MC .
2. Main results
We keep the notations and conventions of the preceding paragraph, A
is an algebra, C is a bialgebra, C = A ⊗ C and MC is the category of
C-comodules.
Let us denote by G any subgroup of the monoid of grouplike elements
of C and by kG the group algebra of G. Let x ∈ G, and let M be a right
C-comodule. Set M1⊗x =Mx. So
Mx = {m ∈M |ρM,C(m) = m0 ⊗A (1A ⊗ x) = m⊗ x = ρM,C(m)}.
46 Projectivity and flatness over the graded ring
When x = 1C , M1C =M coC is the subspace of C-coinvariants of M and
A1C = AcoC is the subring of C-coinvariants of A. An element m ∈ Mx
will be called a semi-coinvariant element. We set S(M) = ⊕x∈GMx, so
S(A) = ⊕x∈GAx. It is easy to see that S(A) is a G-graded algebra
called the subalgebra of semi-coinvariants of A and S(M) is a right G-
graded S(A)-module called the submodule of semi-coinvariants of M .
When C is a Hopf algebra and G = G(C), the algebra S(A) is called
the semi-invariant subalgebra of A in [13]. We will denote by Mgr−S(A),
the category of right G-graded S(A)-modules. The morphisms of this
category are the graded morphisms of degree 1C . Recall that Mgr−S(A) =
MkG
S(A), the category of relative (S(A), kG)-Hopf modules. For any object
N ∈ Mgr−S(A), N ⊗S(A) A is an object of MC : the A-module structure
is the obvious one, while the C-coaction comes from both N and A;
i.e., ρN,C(n ⊗S(A) a) = nx ⊗S(A) a(0) ⊗ xa(1) for every n ∈ Nx, x ∈ G,
a ∈ A, where ρN,kG(n) = nx ⊗ x. To each x ∈ G, we associate the functor
(−)x : MC → M; M 7→Mx. We also have the semi-coinvariant functor
S(−) : MC → Mgr−S(A), M 7→ S(M) = ⊕xMx
and an induction functor
F (−) = −⊗S(A) A : Mgr−S(A) → MC ; N 7→ F (N) = N ⊗S(A) A.
It is easy to show that (F (−),S(−)) is an adjoint pair of functors; in other
words: for any M ∈ MC and N ∈ Mgr−S(A), Hom
C(N ⊗S(A) A,M) ∼=
Homgr−S(A)(N,S(M)). The unit and counit of the pair (F (−),S(−)) are
the following: for N ∈ Mgr−S(A) and M ∈ MC :
uN : N → S(N ⊗S(A) A), uN (n) = n⊗S(A) 1
cM : S(M)⊗S(A) A→M, cM (m⊗S(A) a) = ma.
The adjointness property means that we have
S(cM ) ◦ uS(M) = idS(M), cF (N) ◦ F (uN ) = idF (N) (⋆).
Let x ∈ G, and let M be a C-comodule. We can define (see [13, page
346], where C is a Hopf algebra and G = G(C)) a new C-comodule Mx,
the underlying A-module of which is the same as that of M , while the
C-coaction is new and is given by
ρM,x(m) = m(0)⊗xm(1) = m(0)⊗A(1A⊗xm(1)) = m(0)⊗A(1A⊗x)(1A⊗m(1)).
We call Mx the twisted C-comodule obtained from M and x. Note that
M1C is exactly M with its original C-comodule structure. Note also that
T. Guédénon 47
Ax is A with the C-coaction defined by the grouplike element 1⊗ x of C,
that is, Ax = A1A⊗x. So A1C is exactly the C-comodule A.
By Lemma 1.1, Ax is a cyclic left ∗C-module, so [6] or [9] gives nec-
essary and sufficient conditions for the projectivity and flatness over the
endomorphism ring HomC(Ax, Ax) = ∗C Hom(Ax, Ax).
We have (Mx)y = Mxy, (Mx)y = Mx−1y and Ax ⊗M = Mx, for all
x, y ∈ G. To each element x ∈ G, we associate an equivalent functor
(−)x : MC → MC ; M 7→Mx,
which has inverse (−)x
−1
. Lemma 1.2 implies that the functor (−)x is
isomorphic to HomC(Ax,−).
Let us recall that over any ring A, a left module Λ is called finitely
presented if there is an exact sequence Am → An → Λ → 0 for some
natural integers m and n. If A is left noetherian, every finitely generated
left A-module is finitely presented.
Lemma 2.1. The functor S(−) commutes with direct sums; it commutes
with direct limits if ∗C is left noetherian.
Proof. Let {Mi}i∈I be a family of objects in MC . By Lemma 1.1, every
Ax is a cyclic ∗C-module. So the functor HomC(Ax,−) = ∗C Hom(Ax,−)
commutes with direct sums in MC . We have
S(⊕iMi) = ⊕xHom
C(Ax,⊕iMi)
= ⊕x ⊕i Hom
C(Ax,Mi)
= ⊕i ⊕x Hom
C(Ax,Mi)
= ⊕iS(Mi)
and we get the first assertion. Assume that ∗C is left noetherian, and let
{Mi}i∈I be a directed family of objects in MC . Then every Ax is a finitely
presented left ∗C-module since Ax is a finitely generated left ∗C-module
and ∗C is left noetherian. So the functor HomC(Ax,−) = ∗C Hom(Ax,−)
commutes with direct limits in MC , and
S(lim−→Mi) = ⊕xHom
C(Ax, lim−→Mi)
= ⊕x lim−→HomC(Ax,Mi)
= lim−→⊕xHom
C(Ax,Mi)
= lim−→S(Mi)
Lemma 2.2. Let M be a C-comodule. Then
(1) S(M)(x) = S(Mx−1
) for every x ∈ G
(2) The k-linear map f : S(Ax)⊗S(A) A→ Ax; u⊗S(A) a 7→ ua is an
isomorphism in MC for all u ∈ S(Ax) and a ∈ A.
48 Projectivity and flatness over the graded ring
Proof. (1) We have S(M)(x) = ⊕y∈GMxy and S(Mx−1
) = ⊕y∈G(M
x−1
)y.
On the other hand, m ∈Mxy if and only if ρM,C(m) = m⊗xy if and only
if m(0)⊗m(1) = m⊗xy if and only if m(0)⊗x−1m(1) = m⊗ y if and only
if ρ
Mx−1
,C
(m) = m⊗ y if and only if m ∈ (Mx−1
)y.
(2) Assume that u is homogeneous of degree y. Note that u⊗S(A) a =
1 ⊗S(A) ua for every a ∈ A. Then f is an A-linear isomorphism: its
inverse is defined by a 7→ 1⊗S(A) a. Now we have ρAx,C(u) = u⊗ y; i.e.,
u(0) ⊗ xu(1) = u⊗ y; i.e., u(0) ⊗ u(1) = u⊗ x−1y. It follows that
(ua)(0) ⊗ x(ua)(1) = ua(0) ⊗ xx−1ya(1)
= ua(0) ⊗ ya(1)
= f((u⊗S(A) a)(0))⊗ ((u⊗S(A) a)(1)
So f is C-colinear.
Let A be projective in MC . Then each Ax is also projective in MC .
Therefore Lemma 1.2 implies that the functor (−)x is exact for every
x ∈ G. It follows that the functor S(−) is exact. We refer the reader to
[13, Proposition 1.3] for necessary and sufficient conditions for A to be
projective in MC if C is a Hopf algebra.
In the remainder of this section, (xi, i ∈ I) is a family of elements of
G.
Lemma 2.3. For every index set I,
(1) c
⊕i∈IA
x
−1
i
is an isomorphism;
(2) u⊕i∈IS(A)(xi) is an isomorphism;
(3) if A is projective in MC, then u is a natural isomorphism; in other
words, the induction functor F = (−)⊗S(A) A is fully faithful.
Proof. (1) It is straightforward to check that the canonical isomorphism
⊕i∈IS(A)(xi)⊗S(A) A ≃ ⊕i∈IA
x−1
i is just c
⊕i∈IA
x
−1
i
◦ (κ⊗ idA),
where κ is the isomorphism ⊕i∈IS(A)(xi) ∼= S(⊕i∈IA
x−1
i ), (see Lemmas
2.1 and 2.2). So c
⊕i∈IA
x
−1
i
is an isomorphism.
(2) Putting M = ⊕i∈IA
x−1
i in (⋆), we find
S(c
⊕i∈IA
x
−1
i
) ◦ u
S(⊕i∈IA
x
−1
i )
= id
S(⊕i∈IA
x
−1
i )
.
From Lemmas 2.1 and 2.2, we get
S(c
⊕i∈IA
x
−1
i
) ◦ u⊕i∈IS(A)(xi) = id⊕i∈IS(A)(xi).
T. Guédénon 49
From (1), S(c
⊕i∈IA
x
−1
i
) is an isomorphism, hence u⊕i∈IS(A)(xi) is an iso-
morphism.
(3) Take a free resolution ⊕j∈JS(A)(xj) → ⊕i∈IS(A)(xi) → N → 0 of
a right graded S(A)-module N . Since u is natural, we have a commutative
diagram
⊕j∈JS(A)(xj) //
u⊕j∈JS(A)(xj)
��
⊕i∈IS(A)(xi) //
u⊕i∈IS(A)(xi)
��
N //
uN
��
0
S(⊕j∈JA
x−1
j ) // S(⊕i∈IA
x−1
i ) // S(N ⊗S(A) A) // 0
The top row is exact. The bottom row is exact, since the sequence
⊕j∈JA
x−1
j → ⊕i∈IA
x−1
i → N ⊗S(A) A → 0 is exact in MC (because
−⊗S(A)A is right exact) and S(−) is an exact functor. By (2), u⊕i∈IS(A)(xi)
and u⊕j∈JS(A)(xj) are isomorphisms. It follows from the five lemma that
uN is an isomorphism.
We can now give equivalent conditions for projectivity and flatness of
P ∈ Mgr−S(A).
Theorem 2.4. For P ∈ Mgr−S(A), we consider the following statements.
(1) P ⊗S(A) A is projective in MC and uP is injective;
(2) P is projective as a right graded S(A)-module;
(3) P ⊗S(A)A is a direct summand in MC of some ⊕i∈IA
x−1
i , and uP
is bijective;
(4) there exists Q ∈ MC such that Q is a direct summand of some
⊕i∈IA
x−1
i , and P ∼= S(Q) in Mgr−S(A);
(5) P ⊗S(A) A is a direct summand in MC of some ⊕i∈IA
x−1
i .
Then (1) ⇒ (2) ⇔ (3) ⇔ (4) ⇒ (5).
If A is projective in MC, then (5) ⇒ (3) ⇒ (1).
Proof. (2) ⇒ (3). If P is projective as a right graded S(A)-module, then
we can find an index set I and P ′ ∈ Mgr−S(A) such that ⊕i∈IS(A)(xi) ∼=
P⊕P ′. Then obviously ⊕i∈IA
x−1
i ∼= ⊕i∈IS(A)(xi)⊗S(A)A
∼= (P⊗S(A)A)⊕
(P ′ ⊗S(A) A). Since u is a natural transformation, we have a commutative
diagram:
⊕i∈IS(A)(xi)
∼= //
u⊕i∈IS(A)(xi)
��
P ⊕ P ′
uP⊕uP ′
��
S(⊕i∈IA
x−1
i ) ∼=
// S(P ⊗S(A) A)⊕ S(P ′ ⊗S(A) A)
50 Projectivity and flatness over the graded ring
From the fact that u⊕i∈IS(A)(xi) is an isomorphism, it follows that uP (and
uP ′) are isomorphisms.
(3) ⇒ (4). Take Q = P ⊗S(A) A.
(4) ⇒ (2). Let f : ⊕i∈IA
x−1
i → Q be a split epimorphism in MC .
Then the map S(f) : S(⊕i∈IA
x−1
i ) ∼= ⊕i∈IS(A)(xi) → S(Q) ∼= P is
split surjective in Mgr−S(A), hence P is projective as a right graded
S(A)-module.
(4) ⇒ (5). We already proved that (2) ⇔ (3) ⇔ (4). Since (5) is
contained in (3), we get (4) ⇒ (5).
(1) ⇒ (2). Take an epimorphism f : ⊕i∈IS(A)(xi) → P in Mgr−S(A).
Then
F (f) =: ⊕i∈IS(A)(xi)⊗S(A) A
∼= ⊕i∈IA
x−1
i → P ⊗S(A) A
is surjective, and splits in MC since P ⊗S(A) A is projective in MC .
Consider the commutative diagram
⊕i∈IS(A)(xi)
f //
u⊕i∈IS(A)(xi)
��
P //
uP
��
0
S(⊕i∈IA
x−1
i )
SF (f)
// S(P ⊗S(A) A) // 0
The bottom row is split exact, since any functor, in particular S(−)
preserves split exact sequences. By Lemma 2.3(2), u⊕i∈IS(A)(xi) is an
isomorphism. A diagram chasing argument tells us that uP is surjective.
By assumption, uP is injective, so uP is bijective. We deduce that the
top row is isomorphic to the bottom row, and therefore splits. Thus
P ∈ Mgr−S(A) is projective.
(5) ⇒ (3). Under the assumption that A is projective in MC , (5) ⇒
(3) follows from Lemma 2.3(3).
(3) ⇒ (1). By (3), P ⊗S(A)A is a direct summand of some ⊕i∈IA
x−1
i . If
A is projective in MC , then ⊕i∈IA
x−1
i is projective in MC . So P ⊗S(A) A
being a direct summand of a projective object of MC is projective in
MC .
Theorem 2.5. Assume that ∗C is left noetherian. For P ∈ Mgr−S(A),
the following assertions are equivalent.
(1) P is flat as a right graded S(A)-module;
(2) P ⊗S(A) A = lim−→Qi, where Qi
∼= ⊕j≤ni
Ax−1
ij in MC for some
positive integer ni, and uP is bijective;
T. Guédénon 51
(3) P ⊗S(A)A = lim−→Qi, where Qi ∈ MC is a direct summand of some
⊕j∈IiA
x−1
ij in MC, and uP is bijective;
(4) there exists Q = lim−→Qi ∈ MC, such that Qi
∼= ⊕j≤ni
Ax−1
ij for
some positive integer ni and S(Q) ∼= P in Mgr−S(A);
(5) there exists Q = lim−→Qi ∈ MC, such that Qi is a direct summand
of some ⊕j∈IiA
x−1
ij in MC, and S(Q) ∼= P in Mgr−S(A).
If A is projective in MC, these conditions are also equivalent to condi-
tions (2) and (3) without the assumption that uP is bijective.
Proof. (1) ⇒ (2). P = lim−→Ni, with Ni = ⊕j≤ni
S(A)(xij). Take Qi =
⊕j≤ni
Ax−1
ij , then
lim−→Qi
∼= lim−→(Ni ⊗S(A) A) ∼= (lim−→Ni)⊗S(A) A
∼= P ⊗S(A) A.
Consider the following commutative diagram:
P = lim−→Ni
lim(uNi
)
//
uP
��
lim−→S(Ni ⊗S(A) A)
f
��
S((lim−→Ni)⊗S(A) A) ∼=
// S(lim−→(Ni ⊗S(A) A))
By Lemma 2.3(2), the uNi
are isomorphisms. By Lemma 2.1, the natural
homomorphism f is an isomorphism. Hence uP is an isomorphism.
(2) ⇒ (3) and (4) ⇒ (5) are obvious.
(2) ⇒ (4) and (3) ⇒ (5). Put Q = P ⊗S(A) A. Then uP : P →
S(P ⊗S(A) A) is the required isomorphism.
(5) ⇒ (1). We have a split exact sequence 0 → Ni → Pi = ⊕j∈IiA
x−1
ij →
Qi → 0 in MC . Consider the following commutative diagram:
0 // FS(Ni) //
cNi
��
FS(Pi) //
cPi
��
FS(Qi) //
cQi
��
0
0 // Ni
// Pi
// Qi
// 0
We know from Lemma 2.3(1) that cPi
is an isomorphism. Both rows
in the diagram are split exact, so it follows that cNi
and cQi
are also
isomorphisms. Next consider the commutative diagram:
(lim−→S(Qi))⊗S(A) A
f⊗idA // S(Q)⊗S(A) A
cQ
��
lim−→(S(Qi)⊗S(A) A)
h
OO
lim cQi // Q
52 Projectivity and flatness over the graded ring
where h is the natural homomorphism and f is the isomorphism lim−→S(Qi) ∼=
S(lim−→(Qi)) (see Lemma 2.1). h is an isomorphism, because the functor
(−)⊗S(A)A preserves inductive limits. limcQi
is an isomorphism, because
every cQi
is an isomorphism. It follows that cQ is an isomorphism, hence
S(cQ) is an isomorphism. From (⋆), we get S(cQ) ◦ uS(Q) = idS(Q). It
follows that uS(Q) is also an isomorphism. Since S(Q) ∼= P , uP is an
isomorphism. Consider the isomorphisms
P ∼= S(P ⊗S(A) A) ∼= S(S(Q)⊗S(A) A) ∼= S(Q) ∼= lim−→S(Qi);
where the first isomorphism is uP , the third is S(cQ) and the last one is f .
By Lemmas 2.1 and 2.2, each S(Pi) ∼= ⊕j∈IS(A)(xij) is projective as a
right graded S(A)-module, hence each S(Qi) is also projective as a right
graded S(A)-module, and we conclude that P ∈ Mgr−S(A) is flat. The
final statement is an immediate consequence of Lemma 2.3(3).
3. Appendix
The notations introduced in the preceding sections will be retained through-
out. We want to extend the results of section 2 to the more general type
of grouplike elements of C = A⊗ C. Assume that A and C are commu-
tative. Then the coring C = A ⊗ C becomes a commutative associative
algebra with identity element 1A ⊗ 1C : the multiplication in C is given by
(a⊗ c)(a′ ⊗ c′) = aa′ ⊗ cc′. Denote by G(C) the set of grouplike elements
of C. Let ai ∈ A and ci ∈ C. Then
∑
(ai ⊗ ci) is an element of G(C) if
and only if
∑
(ai ⊗ ci1 ⊗ ci2) =
∑
(aiaj(0) ⊗ ciaj(1) ⊗ cj) and
∑
aiǫC(ci) = 1A.
The product of two grouplike elements of C is a grouplike element.
Let X =
∑
(ai⊗ci) be an element of G(C), and let M be a C-comodule.
We have ρM,C(m) = m0⊗Am1 = m(0)⊗A (1A⊗m(1)). For every X ∈ G(C),
we can define a new C-comodule MX , the underlying A-module of which
is the same as that of M , while the C-coaction is new and is given
by ρM,X(m) = m0 ⊗A m1 = m(0) ⊗A X(1A ⊗ m(1)); i.e.; ρM,X(m) =∑
m(0)ai ⊗ cim(1), where X =
∑
(ai ⊗ ci) ∈ G(C). We call MX the
twisted C-comodule obtained from M and X. Note that M1A⊗1C is exactly
M with its original C-comodule structure. For every a ∈ AX , we have
ρA,X(a) = a(0)⊗AX(1A⊗a(1)). So AX is exactly the one we have defined
in section 1. We have (MX)Y = MXY and AX ⊗A M = MX , for all
X,Y ∈ G(C).
From now on we assume that G is any subgroup of the monoid G(C).
We have (MX)Y =MX−1Y for every X ∈ G. We set N (M) = ⊕X∈GMX ,
T. Guédénon 53
so N (A) = ⊕X∈GAX . Then N (A) is a commutative G-graded algebra
called the subalgebra of conormalizing elements of A and N (M) is a right
G-graded N (A)-module called the submodule of conormalizing elements
of M . We will denote by Mgr−N (A) the category of G-graded N (A)-
modules. The morphisms of this category are the graded morphisms of
degree 1A ⊗ 1C . Recall that Mgr−N (A) is the category MkG
N (A) of relative
right-right (N (A), kG)-Hopf modules. For any object N ∈ Mgr−N (A),
N ⊗N (A)A is an object of MC : the A-module structure is the obvious one
and the C-coaction comes from both N and A; i.e., ρN,C(n ⊗N (A) a) =
nX ⊗N (A) a(0) ⊗A X(1A ⊗ a(1)) for every n ∈ NX ;X ∈ G, a ∈ A, where
ρN,kG(n) = nX ⊗X. We have an induction functor,
G = −⊗N (A) A : Mgr−N (A) → MC ; N 7→ N ⊗N (A) A.
To each element X ∈ G, we associate an equivalent functor
(−)X : MC → MC ; M 7→MX ,
which has inverse (−)X
−1
. We also associate to each X ∈ G a functor
(−)X : MC → Mgr−N (A); M 7→MX .
We define the conormalizing functor
N (−) : MC → Mgr−N (A), M 7→ N (M) = ⊕XMX .
Lemma 3.1. (−⊗N (A)A, N (−)) is an adjoint pair of functors; in other
words, for any M ∈ MC and N ∈ Mgr−N (A), Hom
C(N ⊗N (A) A,M) ∼=
Homgr−N (A)(N,N (M)).
Proof. Let N be an object of Mgr−N (A), M an object of MC and f ∈
HomC(N⊗N (A)A,M). Let n be a homogeneous element of N of degree X,
then n⊗N (A)1A is an element of (N ⊗N (A) A)X and f(n⊗N (A)1A) ∈MX .
Let us define k-linear maps
φ : HomC(N ⊗N (A) A,M) → Hom(N,N (M))
by φ(f)(n) = f(n⊗N (A) 1A) and
ψ : Homgr−N (A)(N,N (M)) → Hom(N ⊗N (A) A,M)
by ψ(g)(n⊗N (A) a) = g(n)a. It is easy to show that
φ(f) ∈ Homgr−N (A)(N,N (M)), ψ(g) ∈ HomC(N ⊗N (A) A,M)
and that φ is a bijection with inverse ψ.
54 Projectivity and flatness over the graded ring
Let us denote by F ′ the functor −⊗N (A) A. The unit and counit of
the adjunction pair (F ′, N (−)) are the following: for N ∈ Mgr−N (A)
and M ∈ MC :
uN : N → N (N ⊗N (A) A), uN (n) = n⊗N (A) 1A
cM : N (M)⊗N (A) A→M, cM (m⊗N (A) a) = ma.
The adjointness property means that we have
N (cM ) ◦ uN (M) = idN (M), cF ′(N) ◦ F
′(uN ) = idF ′(N) (⋆⋆).
The proofs of the following results are similar to those of the preceding
section and we omit them.
Lemma 3.2. The functor N (−) commutes with direct sums; it commutes
with direct limits if ∗C is left noetherian.
Let A be projective in MC . Then each AX is projective in MC . So by
Lemma 1.2, the functor (−)X is exact for every X ∈ G. It follows that
the functor N (−) is exact.
Lemma 3.3. Let M be a C-comodule. Then
(1) N (M)(X) = N (MX−1
) for every X ∈ G;
(2) The k-linear map f : N (AX)⊗N (A) A→ AX ; u⊗N (A) a 7→ ua is
an isomorphism in MC.
Lemma 3.4. For every index set I,
(1) c
⊕i∈IA
X
−1
i
is an isomorphism;
(2) u⊕i∈IN (A)(Xi) is an isomorphism;
(3) if A is projective in MC, then u is a natural isomorphism; in other
words, the induction functor F ′ = (−)⊗N (A) A is fully faithful.
Theorem 3.5. For P ∈ Mgr−N (A), we consider the following statements.
(1) P ⊗N (A) A is projective in MC and uP is injective;
(2) P is projective as a graded N (A)-module;
(3) P ⊗N (A) A is a direct summand in MC of some ⊕i∈IA
X−1
i , and
uP is bijective;
(4) there exists Q ∈ MC such that Q is a direct summand of some
⊕i∈IA
X−1
i , and P ∼= N (Q) in Mgr−N (A);
(5) P ⊗N (A) A is a direct summand in MC of some ⊕i∈IA
X−1
i .
Then (1) ⇒ (2) ⇔ (3) ⇔ (4) ⇒ (5).
If A is projective in MC, then (5) ⇒ (3) ⇒ (1).
T. Guédénon 55
Theorem 3.6. Assume that ∗C is left noetherian. For P ∈ Mgr−N (A),
the following assertions are equivalent.
(1) P is flat as a graded N (A)-module;
(2) P ⊗N (A) A = lim−→Qi, where Qi
∼= ⊕j≤ni
AX−1
ij in MC for some
positive integer ni, and uP is bijective;
(3) P ⊗N (A)A = lim−→Qi, where Qi ∈ MC is a direct summand of some
⊕j∈IiA
X−1
ij in MC, and uP is bijective;
(4) there exists Q = lim−→Qi ∈ MC, such that Qi
∼= ⊕j≤ni
AX−1
ij for
some positive integer ni and N (Q) ∼= P in Mgr−N (A);
(5) there exists Q = lim−→Qi ∈ MC, such that Qi is a direct summand
of some ⊕j∈IiA
X−1
ij in MC, and N (Q) ∼= P in Mgr−N (A).
If A is projective in MC, these conditions are also equivalent to condi-
tions (2) and (3), without the assumption that uP is bijective.
We conclude the paper by the following remarks:
Remarks 3.7. By [8, Propostion 2.3], if C is a finite-dimensional Hopf
algebra, then G(A⊗ C) is a group. If G(A⊗ C) = {1⊗ c; c ∈ G(C)},
then G(A ⊗ C) is obviously a group isomorphic to G(C). In this case,
the conormal elements and the semi-coinvariant elements are exactly the
same. By [5, Proposition 5.1], this can happen in the following situation:
k is algebraically closed, A is a finitely generated normal k-algebra and
C is the affine coordinate ring of a connected algebraic group G acting
rationally on A. More precisely, in this situation, we have
G(A⊗ C) = {1⊗ φ; φ ∈ G(C) = χ(G)},
where χ(G) is the group of characters of G.
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Contact information
T. Guédénon 110, Penworth Drive S.E., Calgary, AB,
Canada T2A 5H4
E-Mail: guedenth@yahoo.ca
Received by the editors: 09.09.2009
and in final form 09.09.2009.
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