Annales Mathématiques Blaise Pascal

Dominique Manchon

Charles Torossian

ANNALES MATHÉMATIQUES
BLAISE PASCAL
Dominique Manchon, Charles Torossian
Erratum : Cohomologie tangente et cup-produit pour
la quantification de Kontsevich
Volume 11, no1 (2004), p. 129-130.
<http://ambp.cedram.org/item?id=AMBP_2004__11_1_129_0>
© Annales mathématiques Blaise Pascal, 2004, tous droits réservés.
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Annales Mathematiques Blaise Pascal 11, 129-130 (2004)
Erratum : Cohomologie tangente et
cup-produit pour la quantification de
Kontsevich
Dominique Manchon
Charles Torossian
Quelques erreurs de signe se sont glissées dans la proposition 4.1
et le théorème 4.6 de l’article récemment publié (Annales Mathématiques
Blaise Pascal 10, 75-106 (2003)). Elles proviennent d’une confusion dans
l’application du lemme 4.2 : dans la proposition 4.1 et le théorème 4.6, α est
un k1-champ de vecteurs, β est un k2-champ de vecteurs et γ est le 2-tenseur
de Poisson. En revanche, l’entier k2 qui apparaît dans le lemme 4.2 est égal
à 2 lorsqu’on l’applique à l’intégration sur une strate de type 1, car k2 va
s’appliquer ici au 2-tenseur de Poisson γ. Le signe (−1)(k1−1)k2 vaut donc 1.
Quant au signe (−1)k1(k2−1) il doit être remplacé par (−1)k1 . En effet
l’argument concernant les strates de type 2 à la fin de la démonstration du
lemme 4.2 est défaillant et doit être corrigé par le suivant : on échange les
positions 1 associée à α et 2 associée à β, ce qui fait apparaître un signe
(−1)k1k2 dû au poids, puis on échange à nouveau les positions après avoir
remplacé β par [β, γ], ce qui fait apparaître cette fois le signe (−1)(k2+1)k1 .
Le produit de ces deux signes est bien (−1)k1 . Voici donc les énoncés corrigés
de la proposition 4.1 et du théorème 4.6 :
Proposition 4.1: Soit γ un 2-tenseur de Poisson formel, soit ∗ l’étoile-
produit construit à partir de γ à l’aide du L∞-quasi-isomorphisme U , soient
α un k1-champ de vecteurs et β un k2-champ de vecteurs. Alors on a, avec
129
D. Manchon, Ch. Torossian
m = k1 + k2 :∑
n≥0
~n
n!
∑
Γ∈Gn+2,m
W ′′ΓBΓ(α⊗ β ⊗ γ ⊗ · · · ⊗ γ) =
=
∑
n≥0
~n
n!
∑
∆∈Gn+2,m−1
W̃∆[∗,B∆(α⊗ β ⊗ γ ⊗ · · · ⊗ γ)]
−
∑
n≥0
~n
n!
∑
∆∈Gn+1,m̃
W∆
(
B∆([α, γ]⊗ β ⊗ γ ⊗ · · · ⊗ γ)
+ (−1)k1B∆(α⊗ [β, γ]⊗ γ ⊗ · · · ⊗ γ)
)
, (4.1)
où W̃∆ désigne l’intégrale de ω∆ sur l’image réciproque de ξ([0, 1]) par l’application
oubli F : Cn+2,m−1 → C2,0 ou F : Cn+1,m → C2,0.
Théorème 4.6: Soit α un k1-champ de vecteurs et soit β un k2-champ de
vecteurs. Alors on a, avec m = k1 + k2 :
U ′~γ(α ∪ β)− U ′~γ(α) ∪ U ′~γ(β) =∑
n≥0
~n
n!
∑
∆∈Gn+2,m−1
W̃∆[∗,B∆(α⊗ β ⊗ γ ⊗ · · · ⊗ γ)]
−
∑
n≥0
~n
n!
∑
∆∈Gn+1,m̃
W∆
(
B∆([α, γ]⊗ β ⊗ γ ⊗ · · · ⊗ γ)+
(−1)k1B∆(α⊗ [β, γ]⊗ γ ⊗ · · · ⊗ γ)
)
. (4.2)
Signalons également que dans l’expression du crochet de Schouten modi-
fié, qui figure dans la preuve du lemme 4.2, le signe (−1)k1k2 doit être rem-
placé par (−1)(k1−1)(k2−1). Ceci entraîne une modification de signe quelques
lignes plus loin dans l’expression de Q12(γ1.γ2), où le signe (−1)r+s+k2 doit
être remplacé par (−1)r+s+k1−1.
Dominique Manchon Charles Torossian
Université Blaise Pascal Ecole Normale Supérieure
CNRS - UMR 6620 CNRS - UMR 8553
24 avenue des Landais 45 rue d’Ulm
63177 Aubière Cedex 75230 Paris Cedex 05
FRANCE FRANCE
manchon@math.univ-bpclermont.fr torossia@ens.fr
130


Erratum : Cohomologie tangente et cup-produit pour la quantification de Kontsevich

Susumu Oda

ANNALES MATHÉMATIQUES
BLAISE PASCAL
Susumu Oda
On finitely generated birational flat extensions of
integral domains
Volume 11, no1 (2004), p. 35-40.
<http://ambp.cedram.org/item?id=AMBP_2004__11_1_35_0>
© Annales mathématiques Blaise Pascal, 2004, tous droits réservés.
L’accès aux articles de la revue « Annales mathématiques Blaise Pascal »
(http://ambp.cedram.org/), implique l’accord avec les conditions générales
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copie ou impression de ce fichier doit contenir la présente mention de copy-
right.
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Clermont-Ferrand — France
cedram
Article mis en ligne dans le cadre du
Centre de diffusion des revues académiques de mathématiques
http://www.cedram.org/
Annales Mathematiques Blaise Pascal 11, 35-40 (2004)
On finitely generated birational flat
extensions of integral domains
Susumu Oda
1 Introduction
In this paper, all rings and their extensions are commutative with a unit
element.
It is well-known that birational, integral, flat extensions of integral do-
mains are trivial.
Our objective is to extend this fact to a result that birational, finitely
generated, flat extensions of integral domains are open-immersions. In addi-
tion, we show that their complementary closed sets are of grade one if not
empty.
We use the following notation unless otherwise specified: R is an integral
domain with quotient field K and A is a birational extension of R in K.
2 Results
Lemma 2.1: Assume that A is flat over R and that the canonical morphism
Spec(A) → Spec(R) is surjective. Then R = A.
Proof: Take a ∈ A with a = y/x (x, y ∈ R). A is faithfully flat over R by
[3, (7.2)]. So it follows that y = ax ∈ xA ∩ R = xR (cf. [3, (7.5)]). Hence
a = y/x ∈ R. Therefore R = A.
Proposition 2.2: Let A be a birational extension of R. If A is integral and
flat over R, then A = R.
Proof: Since A is integral over R, the canonical map Spec(A) → Spec(R)
is surjective. Hence our conclusion follows from Lemma 2.1.
35
S. Oda
Lemma 2.3: Assume that A is flat over R. Then for any P ∈ Spec(A),
AP = RP∩R. Moreover, AP∩R = RP∩R.
Proof: Put p = P ∩R. Then Rp → AP is flat. As a flat extension of rings
satisfies the Going-Down Theorem (cf. [2, (5.D)]), Spec(AP ) → Spec(Rp) is
surjective. Hence AP = Rp by Lemma 2.1. The last statement follows from
the factorization Rp ↪→ Ap ↪→ AP .
Theorem 2.4: Let R be an integral domain with quotient field K and let A
be a birational extension of R in K. Put
IR(A) := {a ∈ R | a 6= 0, A[1/a] = R[1/a]} ∪ {0}.
Assume that A is finitely generated over R. Then
(i) IR(A) is a radical ideal of R and IR(A) 6= (0).
(ii) For p ∈ Spec(R), IR(A) 6⊆ p ⇐⇒ Ap = Rp.
Proof: Put A = R[α1, . . . , αn] and let αi = ai/b (ai, b ∈ R, b 6= 0).
(i) Since A[1/b] = R[1/b], we have b ∈ IR(A) and so IR(A) 6= (0). Let
a, b ∈ IR(A). Since A[1/a] = R[1/a] and A[1/b] = R[1/b], there exists an
integer ` >> 0 such that a`αi ∈ R and b`αi ∈ R for every 1 ≤ i ≤ n. Thus
we have (a + b)2`αi ∈ R and hence A[1/(a + b)] = R[1/(a + b)], which shows
that a + b ∈ IR(A). For any r(6= 0) ∈ R, it is obvious that ra ∈ IR(A).
Therefore, IR(A) is a non-zero ideal of R. The ideal IR(A) is a radical ideal
by definition.
(ii) If there exists a ∈ IR(A) \ p with p ∈ Spec(R), then Ap = A[1/a]p =
R[1/a]p = Rp. Conversely, suppose that Ap = Rp for p ∈ Spec(R). Put
αi = ci/ti, ci ∈ R, ti ∈ R \ p and let t = t1 · · · tn ∈ R \ p. Since αi ∈ R[1/t],
we have A ⊆ R[1/t], that is, A[1/t] = R[1/t] with t 6∈ p, which means that
t ∈ IR(A) but t 6∈ p. Thus IR(A) 6⊆ p.
Theorem 2.5: Let R be an integral domain with quotient field K and let A be
a birational extension of R in K. If (0) 6= IR(A) 6= R, then grade(IR(A)) = 1,
i.e. IR(A) contains only a regular sequence of one element.
Proof: Suppose that there exists a regular sequence {x, y} in IR(A). Take
an element α ∈ A \R (such an element exists because IR(A) 6= R). Then for
a large integer ` ∈ N, we have x`α = a ∈ R and y`α = b ∈ R. Then in K,
x`/y` = a/b, that is, x`b = y`a in R. Since {x`, y`} is also a regular sequence,
36
On finitely generated birational flat extensions
we have a = x`c for some c ∈ R. So we have x`α = a = x`c. Since R is an
integral domain, we have α = c ∈ R, which is a contradiction.
Remark 2.6: A is finitely generated over R =⇒ IR(A) 6= (0), as was seen
above, but the reverse implication does not always hold. Let R = k[X, Y ]
be a polynomial ring over a field k and let A = k[Y, {X/Y `}`∈N]. Then it
is obvious that A is a birational, infinitely generated extension of R. But
R[1/Y ] = A[1/Y ] and hence IR(A) 3 Y .
Theorem 2.7: Let R be an integral domain with quotient field K and let
A be an extension of R. Assume that A is a birational, finitely generated
extension of R in K and that A is flat over R. Then the canonical morphism
Spec(A) → Spec(R) is an open-immersion. Moreover, Spec(A) ∼= Spec(R) \
V (IR(A)) is canonically an isomorphism of schemes.
Proof: We claim that IR(A)A = A. In fact, suppose that there exists
P ∈ Spec(A) such that IR(A)A ⊆ P . Put p = P ∩ R so that p ⊇ IR(A).
Now AP is faithfully flat over Rp. Thus AP = Ap = Rp by Lemma 2.3.
Hence IR(A) 6⊆ p by Theorem 2.4, a contradiction. Therefore, we have
shown IR(A)A = A and Spec(A) → Spec(R) \ V (IR(A)) is defined. Next we
will show that this map is surjective by using the fact that a flat birational
extension of integral domains R → A verifies the following property: if p
is a prime ideal of R, then either pA = A or Rp → Ap is an isomorphism.
Suppose that there exists p ∈ Spec(R) \ V (IR(A)) such that pA = A. Then
IR(A) 6⊆ p implies that there exists a(6= 0) ∈ IR(A) \ p. So R[1/a] =
A[1/a]. Thus Rp = R[1/a]p = A[1/a]p = Ap, which is a contradiction.
Thus Spec(A) → Spec(R) \ V (IR(A)) is surjective. Now let P, P ′ ∈ Spec(A)
with P ∩ R = P ′ ∩ R := p. Then AP = Rp = AP ′ , all of which are local
rings. Hence P = P ′. So Spec(A) → Spec(R) \ V (IR(A)) is injective. Since
for any P ∈ Spec(A), AP = RP∩R, Spec(A) → Spec(R) \ V (IR(A)) is a
homeomorphism. Hence Spec(A) → Spec(R) is an open-immersion.
Corollary 2.8: Let R be an integral domain with quotient field K and let
A be an extension of R. Assume that A is a birational, finitely generated
extension of R in K and that A is flat over R. Let (P) be any local-global
property (e.g. regular, normal, . . . ). If R has (P), so does A.
37
S. Oda
Proof: Since Spec(A) → Spec(R) is an open-immersion by Theorem 2.7,
our conclusion is obvious.
Lemma 2.9: Let R be a UFD, and let P be a prime ideal of R with
grade(P ) = 1. Then ht(P ) = 1.
Proof: Suppose that ht(P ) ≥ 2. Then there exists a prime element x ∈ P .
Since P 6= (x), take y ∈ P \ (x). Then {x, y} is a regular sequence in P ,
which means that grade(P ) ≥ 2, a contradiction. Hence ht(P ) = 1.
Proposition 2.10: Let R be a UFD, and let I be an ideal of R with
grade(I) = 1. Then ht(I) = 1 and hence V (I) = V (a) ⊆ Spec(R) for
some a ∈ R.
Proof: Let P be a minimal prime ideal containing I. Then
√
IRP = PRP .
Since grade(I) = 1 implies grade(
√
I) = 1, we have 1 = grade(
√
IRP ) =
grade(PRP ). Noting that RP is a UFD, ht(PRP ) = 1 by Lemma 2.9. Since
I ⊆ P , ht(I) = 1. Since R is a UFD,
√
I = P1∩ · · ·∩Pn for some ht(Pi) = 1.
Indeed, if I is an ideal whose minimal prime ideals are finitely generated, then
I has only finitely many minimal prime ideals [1, Theorem]. Put Pi = (ai)
with ai ∈ I. Hence in Spec(R), V (I) = V (
√
I) = V (P1 ∩ · · · ∩ Pn) =
V (P1 · · ·Pn) = V (a1 · · · an) = V (a), where a = a1 · · · an.
Theorem 2.11: Let R be an integral domain with quotient field K and let
A be an extension of R. Assume that A is a birational, finitely generated
extension of R in K and that A is flat over R. If R is a UFD (a unique
factorization domain), then A = R[1/a] for some a ∈ R.
Proof: We may assume that IR(A) 6= R. Then grade(IR(A)) = 1 by
Theorem 2.5. Since R is a UFD, V (IR(A)) = V (a) for some a ∈ R. So by the
last statement of Theorem 2.7, Spec(A) ∼= Spec(R) \ V (a) = Spec(R[1/a]).
Therefore, A = R[1/a].
Theorem 2.12: Let R be an integral domain with quotient field K and let
A be an extension of R. Assume that A is a birational, finitely generated
extension of R in K and that A is flat over R. If R is a UFD, then A is also
a UFD.
Proof: If R is a UFD, then a localization R[1/a] with a ∈ R \ (0) is a
UFD. So our conclusion follows from Theorem 2.11.
38
On finitely generated birational flat extensions
Added in Proof.
Professor Gabriel Picavet informed the author of the following in a letter.
We write it here with his permission:
“ During the sixties, Pr. Samuel organized in Paris a seminar about epi-
morphisms of the category of commutative rings. Daniel Lazard was finishing
his thesis about flatness whose reference is “ Autour de la platitude, Bull.
Soc. Math. France, (97), 1969, 81-128 ”.
A classical example of flat epimorphism is a localization with respect to
a multiplicative subset. Hence, if A is an integral domain with quotient field
K, then A → K is a flat epimorphism.
Now there is a fundamental result (Corollaire 3.2) in chapter IV of Lazard’s
paper: let A → C → B a composite of ring morphisms be such that A → B
is a flat epimorphism and C → B is injective. Then C → B is a flat epimor-
phism and if A → C is flat, then A → C is an epimorphism.
Hence, if R → A is birational and flat, R → A is a flat epimorphism.
Now faithfully flat epimorphisms are isomorphisms (Lazard, Lemme 1.2)
and we find Lemma 2.1 and Proposition 2.2. Moreover, if A → B is a ring
morphism, then A → B is a flat epimorphism if and only if AP → BQ is
an isomorphism for each Q ∈ Spec(B) and P := f−1(Q) and Spec(B) →
Spec(A) is injective.
Thus the local-global principle (Corollary 2.8) is an easy consequence.
If we read the E.G.A of Dieudonné and Grothendieck, we may see that
an open immersion of affine schemes is nothing but a flat epimorphism of
finite presentation (as an algebra). Moreover, by a paper of Michel Raynaud
and Laurent Gruson,“ Critères de platitude et projectivité, Invent. Math.
(13), 1971, 1-89 ”, a flat ring morphism of finite type A → B where A is an
integral domain is of finite presentation.
Now, Chevalley’s theorem states that a flat morphism of finite presenta-
tion A → B is Zariski open (even if A and B are not noetherian).
It follows that we have a result more general than Theorem 2.7. More
generally, if A → B is a flat epimorphism, Spec(B) → Spec(A) is a homeo-
morphism onto its image (Lazard, Corollaire 2.2). ”
Thus the part of Theorem 2.7 that Spec(A) → Spec(R) is an open-
immersion has been known. But we would like to emphasize that our proof
is elementary and simpler because we do not use the notion of epimorphism.
39
S. Oda
Moreover, note the explicit computation of the open set introduced in The-
orem 2.7 and the interesting property of its complement V (IR) (cf. Theo-
rem 2.5).
Finally the author wishes to express his deep appreciation for Professor
Picavet’s kind co-operation.
References
[1] D. D. Anderson. A note on minimal prime ideals. Proc. Amer. Math.
Soc., 122:13–14, 1994.
[2] H. Matsumura. Commutative Algebra , 2-nd edition. Benjamin, Reading,
1980.
[3] H. Matsumura. Commutative Ring Theory. Cambridge University Press,
New York, 1986.
Susumu Oda
Kochi University,
Faculty of Education
Department of Mathematics
2-5-1 Akebono-cho,
Kochi 780-8520
JAPAN
ssmoda@cc.kochi-u.ac.jp
40


On finitely generated birational flat extensions of integral domains

Telemachos Hatziafratis

On an integral formula of Berndtsson related to the inversion of the Fourier-Laplace transform of $\bar{\partial }$-closed $(n,n-1)$-forms

Clara Lucía Aldana Domínguez

Representation of a gauge group as motions of a Hilbert space

Andres Rodriguez

ANNALES MATHÉMATIQUES
BLAISE PASCAL
Andres Rodriguez
Notes on prequantization of moduli of G-bundles with
connection on Riemann surfaces
Volume 11, no2 (2004), p. 181-186.
<http://ambp.cedram.org/item?id=AMBP_2004__11_2_181_0>
© Annales mathématiques Blaise Pascal, 2004, tous droits réservés.
L’accès aux articles de la revue « Annales mathématiques Blaise Pascal »
(http://ambp.cedram.org/), implique l’accord avec les conditions générales
d’utilisation (http://ambp.cedram.org/legal/). Toute utilisation commerciale
ou impression systématique est constitutive d’une infraction pénale. Toute
copie ou impression de ce fichier doit contenir la présente mention de copy-
right.
Publication éditée par le laboratoire de mathématiques
de l’université Blaise-Pascal, UMR 6620 du CNRS
Clermont-Ferrand — France
cedram
Article mis en ligne dans le cadre du
Centre de diffusion des revues académiques de mathématiques
http://www.cedram.org/
Annales Mathematiques Blaise Pascal 11, 181-186 (2004)
Notes on prequantization of moduli of
G-bundles with connection on Riemann
surfaces
Andres Rodriguez
Abstract
Let X → S be a smooth proper family of complex curves (i.e. fam-
ily of Riemann surfaces), and F a G-bundle over X with connection
along the fibres X → S. We construct a line bundle with connection
(LF ,∇F ) on S (also in cases when the connection on F has regular
singularities). We discuss the resulting (LF ,∇F ) mainly in the case
G = C∗. For instance when S is the moduli space of line bundles
with connection over a Riemann surface X, X = X × S, and F is the
Poincaré bundle over X , we show that (LF ,∇F ) provides a prequan-
tization of S.
1 Introduction
Of special interest in physics are line bundles with connection over various
moduli spaces of G-bundles with connection over Riemann surfaces. Such
line bundles are used to construct conformal field theories, which for ex-
ample produce interesting 3-manifold (topological) invariants. We consider
the problem of constructing such line bundles as a problem of constructing
Deligne cohomology classes.
Recall the Deligne complexes on an algebraic variety S, roughly
Z(n) = ...0→ Z→ O → Ω1 → ...→ Ωn → 0→ ...
In particular Z(1) = ...0 → Z → O → 0... ' O∗[1], and hence classes in
H2(S, Z(1)) correspond to isomorphism classes of line bundles on S. Further,
classes in H2(S, Z(2)) correspond to isomorphism classes of line bundles with
connection on S. So our objective is to construct classes in H2(S, Z(2)) where
S is one of the moduli spaces being considered.
181
A. Rodriguez
In topology, the theory of characteristic classes constructs cohomology
classes on spaces which are equiped with say G-bundles over them. Let us
review how this is done. Recall that a G-bundle on a space Y is described
by a classifying map Y → BG (defined up to homotopy). Also H (BG, C) ∼=
(Symg∗)G as graded algebras, with elements of g∗ being of degree 2. So
a G-bundle on Y yields for instance classes in H4(Y, C) corresponding to
G-invariant bilinear forms on g.
In the setting of objects with algebraic structure, one may carry out an
analogous procedure to construct classes in Deligne cohomology.
Suppose Y is an algebraic variety, G a reductive algebraic group, and F
a G-bundle over Y . Instead of a classifying map we have
Y ← F ×G ∆G→ BG,
where ∆G is the standard simplicial G-scheme model of EG and BG :=
∆G/G . But since ∆G ∼ pt. as a simplicial scheme,
H2m(Y, Z(m)) ∼→ H2m(F ×G ∆G, Z(m))← H2m(BG, Z(m)).
And there is a natural map H2m(BG, Z(m))→ H2mtop (BG, Z) which is actually
an isomorphism [1]. So again, we have classes in H4(Y, Z(2)) corresponding
to certain bilinear G-invariant forms on g.
Consider the particular situation of a proper smooth family of complex
curves (i.e. family of compact Riemann surfaces) X → S, and a G-bundle
F on X . We may construct classes in H4(X , Z(2)) as above. And fur-
ther, classes in H4(X , Z(2)) may be integrated down to a obtain classes in
H2(S, Z(1)).
We consider arbitrary X → S as above, and F with (relative) connection;
and fix a class in H4(BG, Z(2)). By following the same construction with
slightly different complexes we produce a class in H2(S, Z(2)). i.e. a line
bundle with connection on S. In the cases relevant to physics the curvature
is the natural 2-form which the considered moduli spaces carry; which makes
our objects prime candidates for physical applications.
Here we discuss the complexes involved in the construction, and compute
the curvature of the resulting objects in the case when G = C∗.
The ideas discussed here grew out of conversations with A. Beilinson, who
in particular suggested a version of our main construction.
182
Prequantization of moduli of G-torsors
2 Construction.
2.1 Regular case.
Consider X π→ S a family of proper smooth curves, F a G-bundle on X with
connection along the fibres. Put ∆F := F ×G ∆G; and denote q : ∆F → X ,
p : ∆F → BG.
We will construct complexes Z∆(n), Zπ(n) on ∆F , X respectively, for
which a diagram of the form
H4(BG, Z(2))→ H4(∆F , Z∆(2)) ∼← H4(X , Zπ(2))
tr→ H2(S, Z(2))
holds.
First of all recall that for any complex C  on S, there is a natural
Hn(X , π∗(C )) tr→ Hn−2(S, C ),
so put Zπ(n) = π∗(Z(n)).
We shall now construct Z∆ on ∆F such that p : ∆F → BG, q : ∆F → X
induce
H4(BG, Z(2))→ H4(∆F , Z∆(2)) ∼← H4(X , Zπ(2)).
Suppose U is a neighbourhood in X over which the relative connection
F can be extended to a total flat connection and further there is a (flat)
trivialization FU
∼→ U × G. Which yields a trivialization ∆FU
∼→ U ×∆G,
and a map tU : ∆FU → S ×∆G. Put Z∆∆FU (n) = t
∗
U(Z(n)), which defines a
complex Z∆(n) on ∆F .
∆F p→ BG induces p∗ : p∗(Z(n))→ Z∆(n) because p factors through tU .
Also ∆F q→ X induces q∗ : q∗(Zπ(n))→ Z∆(n).
Claim: H4(∆F , Z∆(2)) q
∗
← H4(X , Zπ(2)) is an isomorphism.
We will first check that q∗(Zπ)(n) → Z∆(n) is a quasi-isomorphism by
showing that that is the case locally.
Consider U as in the construction of Z∆(n), and tU : ∆FU → S ×∆G as
before. Since ∆G ' pt., Z(n) ' p∗1(Z(n)) on S ×∆G. Hence
t∗U(Z(n)) ' tU ∗ (p∗1(Z(n))) ' (p1 ◦ tU)∗(Z(n)) = (π ◦ q)∗(Z(n))
' q∗(π∗(Z(n))) = q∗(Zπ(n)).
183
A. Rodriguez
Finally, the map induced by p∗ on cohomology is
Hm(X , Zπ(n))
∼→ Hm(∆F , q∗(Z(n))) ∼→ Hm(∆F , Z∆(n)),
with the first map being an isomorphism again because ∆G ' pt. 
H4(BG, Z(2)) p
∗
→ H4(∆F , Z∆(2)) ∼← H4(X , Zπ(2))
tr→ H2(S, Z(2))
now yields an isomorphism class (LF ,∇F).
2.2 Case of regular singularities.
We shall discuss the case of G compact, and then comment about the case
of general G.
X → S,F , ∆F as before, and non-intersecting sections S σi→ X , along
which the connection on F has regular singularities. Assume that the isomor-
phism type of the Fs at the marked points is constant, and fix trivializations
of the underlying bundle of F at the σi(s)’s.
Notation: Ω(n) := 0 → Ω0 → ... → Ωn−1 → 0... (Ω0 = O being in
position 0).
Ωπ(n) := π
∗(ΩS(n)),
Ω,G∆G(n) := (Ω

∆G(n))
G ' Ω∆G(n).
Remark: Let Y have a G-torsor H, and suppose G acts freely on E.
Then there is a canonical map Ω,GE ↪→ Ω∗Y×HE. A special case of this is when
E = G, in which case g∗ ∼→ Ω1,GG → Ω1Y×HG maps c ∈ g
∗ to c(σ), with σ
being the connection 1-form of H. Let ΩY,H,E(' Ω
,G
E ) denote the image.
Put Ω,∆(n) := q∗(Ωπ(n))⊗C ΩX ,F ,∆G(↪→ Ω∆F),
and
Zπ(n) := π∗(ZS(n)) ' 0→ π∗(Z)→ Ω0π → ...→ Ωnπ → 0...
' 0→ Z→ Ω0π → ...→ Ωnπ → 0...,
Z∆(n) := 0→ Z→ Ω0,∆ → ...→ Ωn,∆ → 0→ ...
For general G, replace Ω complexes for BG, ∆G, by appropriate Ω[logD]
complexes.
184
Prequantization of moduli of G-torsors
3 Curvature in the regular case with G = C∗
Let S= infinitesimal point with closed point 0, X = X × S for X a smooth
proper complex curve. Denote BC∗ by P.
Consider
H2(P, Z) ∼← H2(P, Z(1)) p
∗
→ H2(∆F , Z∆(1)) q
∗
← H2(X , Zπ(1)).
Let c ∈ H2(P, Z) be the canonical generator, and c∆, cπ the corresponding
classes in H2(∆F , Z∆(1)), H2(X , Zπ(1)) respectively.
Consider Zπ(1) ' π∗(O∗S)[1]. Then F is described by a class [F ]π ∈
H2(X , Zπ(1)).
Claim: cπ = [F ]π.
Let {Ui} be open cover of P s.t. (Ui, fij) describe c; and {Vk} open cover
of X s.t. over each Vk there is a flat trivialization of F , so it is described
by (Vk, gkl) with the gkl being locally constant along fibres X → S. Denote
φ : ∆C∗ → P. Since φ∗(c) is trivial, there are Fi on φ−1(Ui) such that
fij = FjF
−1
i and a∗(Fi) = a−1Fi for any a ∈ C∗ acting on φ−1(Ui).
Note that (Vk × Ui, Fi) describes a chain with values in Z∆(1). But the
boundary of (Vk×Ui, Fi) is exactly {(Vk×Ui, gkl)}−p∗(c), so p∗(c) = q∗([F ]π)
in H2(∆F , Z∆(1)). 
Notation: d : Z(n)→ Ωn[n], and the same letter will denote the induced
map on cohomology.
For v ∈ T0S, v : H2(X , π∗(Ω1)[1])→ H2(X , π∗(C)[1]) is that induced by
contraction by v.
Claim: d(cπ)v ∈ H2(X, C[1]) ∼= H1(X, C) is actually the class describing
the infinitesimal deformation of F0 along S in the direction v.
By considering classes in H2(X , Zπ(1)) ∼= H1(X , π∗(O∗S)) as represented
by Čech cocycles, the current claim follows from the previous one. 
Recall that c2 ∈ H4(BG, Z(2)) is the canonical generator. Then (c∆)2, c2π
are the corresponding classes in H4(∆F , Z∆(2)), H4(X , Zπ(2)) respectively.
Denote the curvature of (LF ,∇F) by ωF .
Claim: for v, w ∈ T0S, ωF(v, w) = tr(d(cπ)v  d(cπ)w).
185
A. Rodriguez
d : Zπ(2) → π∗(Ω2) on X is the pullback of d : Z(2) → Ω2 on S, so the
naturality of tr implies that
d([(LF ,∇F)])(v, w) = tr(d(c2π)v,w).
But d actually induces a ring map on cohomology, then
tr(d(c2π)v,w) = tr(d(cπ)v  d(cπ)w).
(v,w : π∗(Ω2)→ C is induced by contraction by v, w). 
Consider S being the moduli space of line bundles with connection over
X and F the Picard bundle over S. For any s ∈ S, TsS can be identified
with H1(X, C). So H1(X, C) ⊗ H1(X, C) → H2(X, C) → C defines a non-
degenerate bilinear form on TsS, which actually endows S with a symplectic
structure. Denote the symplectic form by ωS.
Proposition. ωS = ωF . i.e. (LF ,∇F) is a prequantization of S.
This is a direct consequence of the previous two claims. 
References
[1] P. Deligne. Théorie de Hodge. III. Inst. Hautes Ètudes Sci. Publ. Math.,
44:5–77, 1974.
Andres Rodriguez
University of Chicago
Department of Mathematics
5734 S. University Avenue
Chicago, Illinois 60637
USA
andresr@math.uchicago.edu
186


Notes on prequantization of moduli of $G$-bundles with connection on Riemann surfaces

Nikolaos A. Papadopoulos

Mario Paschke

Andrés Reyes

Florian Scheck

The spin-statistics relation in nonrelativistic quantum mechanics and projective modules

Mohamed Akkouchi

Abdellah Bounabat

Manfred Goebel

Optimality Conditions for a Nonlinear Boundary Value Problem Using Nonsmooth Analysis

Tibor Šalát

Vladimír Toma

ANNALES MATHÉMATIQUES
BLAISE PASCAL
Tibor Šalát, Vladimír Toma
A Classical Olivier’s Theorem and Statistical
Convergence
Volume 10, no2 (2003), p. 305-313.
<http://ambp.cedram.org/item?id=AMBP_2003__10_2_305_0>
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Annales Mathematiques Blaise Pascal 10, 305-313 (2003)
A Classical Olivier’s Theorem and Statistical
Convergence
Tibor Šalát
Vladimír Toma
Abstract
L. Olivier proved in 1827 the classical result about the speed of
convergence to zero of the terms of a convergent series with positive
and decreasing terms. We prove that this result remains true if we
omit the monotonicity of the terms of the series when the limit op-
eration is replaced by the statistical limit, or some generalizations of
this concept.
Résumé. L. Olivier démontrait en 1827 un résultat classique sur
la vitesse de convergence vers zéro d’une série convergente à termes
positifs décroissants. Nous démontrons que ce résultat reste valable
si nous omettons la monotonie des termes de la série, en remplaçant
l’opération limite par la limite statistique ou encore par des générali-
sations de ce concept.
1 Introduction
The above mentioned result of L. Olivier was published in [7] p. 39 (see also
[5] p. 125) and claims that if an ≥ an+1 > 0, (n = 1, 2, . . . ) and
∞∑
n=1
an < +∞
then lim
n→∞
nan = 0. Simple examples show that without the monotonicity con-
dition an ≥ an+1, (n = 1, 2, . . . ), the sequence (nan)n≥1 need not converge
to zero.
Example 1. Let an = 1n if n is a square i.e. n = k
2, (k = 1, 2, . . . ),
and an = 1n2 otherwise. Then an > 0 (n = 1, 2, . . . ),
∞∑
n=1
an < +∞, but
lim
n→∞
nan 6= 0 since k2ak2 = 1, (k = 1, 2, . . . ).
305
T. Šalát and V. Toma
Remark: As the unknown referee pointed out the example can be strength-
ened taking an = log nn if n = k
2, in which case the sequence (nan)n≥1 is not
bounded.
The notion which allows us to describe the behavior of the sequence
(nan)n≥1 is the notion of statistical convergence introduced in paper [2], (see
also [3], [10], and [9])
Definition: We say that a sequence (xn)n≥1 (of real or complex numbers)
statistically converges to a number L, and we write lim-stat xn = L, if for
each ε > 0 the set A(ε) := {n : |xn − L| ≥ ε} has zero asymptotic density,
i.e. the limit
d(A(ε)) := lim
n→∞
1
n
n∑
k=1
χA(ε)(k)
exists and is equal to zero. Here χA is the characteristic function of a set A.
In what follows we will show that the sequence (nan)n≥1 statistically
converges to 0 if
∑∞
n=1 an < +∞, (an > 0, n = 1, 2, . . . ) without the mono-
tonicity assumption on the sequence (an)n≥1.
The notion of statistical convergence was generalized using the concept
of an admissible ideal = of subsets of positive integers N = {1, 2, . . . }, that
is = ⊆ P(N) and = is additive (i.e. A, B ∈ = ⇒ A∪B ∈ =), hereditary (i.e.
B ⊂ A ∈ = ⇒ B ∈ =), containing all singletons and not containing N.
Definition: (See [6].) We say that a sequence (xn)n≥1 =-converges to a num-
ber L and we write =-lim xn = L, if for each ε > 0 the set A(ε) := {n :
|xn − L| ≥ ε} belongs to the ideal =.
An admissible ideal is for example:
=f := {A ⊆ N : A is finite set}.
Let us note that =f -lim xn = L means the same as lim
n→∞
xn = L. Some
admissible ideals can be obtained using various concepts of density of sets
A ⊆ N. Using the asymptotic density defined above we obtain the ideal
=d := {A ⊆ N : d(A) = 0}.
Obviously =d -convergence means the statistical convergence. Another type
of density is the logarithmic density defined by means of lower and upper
306
Olivier’s theorem and statistical convergence
logarithmic density of a set A ⊆ N:
δ(A) := lim inf
n→∞
∑n
k=1 χA(k)
1
k
log n
, δ(A) := lim sup
n→∞
∑n
k=1 χA(k)
1
k
log n
.
If δ(A) = δ(A) =: δ(A) the number δ(A) is called the logarithmic density of
the set A. Using the logarithmic density we can define the ideal
=δ := {A ⊆ N : δ(A) = 0}.
A little bit more complicated is the concept of the uniform density. For any
A ⊆ N, t ≥ 0, s ≥ 1 denote by A(t + 1, t + s) the number of elements of the
set A ∩ [t + 1, t + s]. Put
βs := lim inf
t→∞
A(t + 1, t + s), βs := lim sup
t→∞
A(t + 1, t + s).
Then there exist
u(A) := lim
s→∞
βs
s
, u(A) := lim
s→∞
βs
s
called the lower and upper uniform density of A, respectively. We prove the
existence of lims→∞ βss only, since the proof for lims→∞
βs
s
is similar. Let us
choose a fixed p ∈ N. Then for any s ∈ N there exists ts ≥ 1 such that
βs = A(ts + 1, ts + s) and simultaneously βp ≤ A(t + 1, t + p) for every
t ≥ ts. For any s ∈ N there exist unique integer numbers ks, rs ≥ 0 such
that s = ksp + rs with 0 ≤ rs ≤ p − 1. Then we have (with the convention
A(x, y) = 0 if y < x)
βs
s
=
1
ksp + rs
A(ts + 1, ts + ksp + rs) =
=
1
ksp + rs
[
ks∑
i=1
A(ts + 1 + (i− 1)p, ts + ip) + A(ts + ksp + 1, ts + ksp + rs)]
Hence βs
s
≥ ksβp
ksp+rs
and when s →∞ then also ks →∞ and so for fixed p we
have limks→∞
ksβp
ksp+rs
= βp
p
. Therefore u(A) = lim inf
s→∞
βs
s
≥ βp
p
and consequently
u(A) ≥ sup
p≥1
βp
p
. Since obviously lim sup
s→∞
βs
s
≤ sup
p≥1
βp
p
, we can conclude that
there exists the limit lims→∞ βss = sup
p≥1
βp
p
. If u(A) = u(A) =: u(A) then the
307
T. Šalát and V. Toma
common value u(A) is called the uniform density of A (cf.[1]). Using the
uniform density we can define the admissible ideal
=u := {A ⊆ N : u(A) = 0}.
To compare the above defined ideals let us remember the lower and upper
asymptotic density defined by
d(A) := lim inf
n→∞
1
n
n∑
k=1
χA(k), d(A) := lim sup
n→∞
1
n
n∑
k=1
χA(k).
The following relations between these densities can be verified (cf.[1],[4]):
0 ≤ u(A) ≤ d(A) ≤ δ(A) ≤ δ(A) ≤ d(A) ≤ u(A) ≤ 1.
Consequently we get the chain of inclusions for the above defined ideals:
=f ⊆ =u ⊆ =d ⊆ =δ. (1.1)
The ideal which will play an important role in the main theorem is the
following one
=c := {A ⊆ N :
∑
a∈A
a−1 < +∞}.
It is well known (see [8]) that
∑
a∈A
a−1 < +∞ implies d(A) = 0. So the
following inclusion holds
=c ⊆ =d. (1.2)
2 Main Results
The above mentioned Olivier’s result can be formulated in the terms of =f -
convergence as follows. If
an > 0 (n = 1, 2, . . . ),
∞∑
n=1
an < +∞
and
a1 ≥ a2 ≥ · · · ≥ an ≥ an+1 ≥ . . .
308
Olivier’s theorem and statistical convergence
then
=f -lim nan = 0.
In the sequel we are going to study the ideals = with the following property:
If an > 0 (n = 1, 2, . . . ),
∞∑
n=1
an < +∞ then =-lim nan = 0. (2.1)
From Example 1 we can conclude that the ideal =f does not have the property
2.1. Let us denote by S(T ) the class of all admissible ideals =, with the
property 2.1. So we have that =f /∈ S(T ). The following theorem claims a
more useful fact.
Theorem 2.1: Ideal =c is an element of S(T ).
Proof: We proceed by contradiction. Let
=c := {A ⊆ N :
∑
a∈A
a−1 < +∞} /∈ S(T ).
Then there exist numbers an > 0 (n = 1, 2, . . . ) with
∞∑
n=1
an < +∞ such
that the equality =c-lim nan = 0 does not hold. This means that there exists
ε0 > 0 for which A(ε0) = {n : nan ≥ ε0} /∈ =c. Hence from the definition of
the ideal =c we get
∑
n∈A(ε0)
n−1 = +∞. For n ∈ A(ε0) we have nan ≥ ε0 and
so
an ≥
ε0
n
for every n ∈ A(ε0).
Using this and the comparison criterion for infinite series we get∑
n∈A(ε0)
an ≥ ε0
∑
n∈A(ε0)
n−1 = +∞.
So it must be also
∞∑
n=1
an = +∞ and this is a contradiction.
The claim in the following lemma is a trivial fact about preservation of
the limit.
Lemma 2.2: Let =1,=2 be admissible ideals such that =1 ⊆ =2. If =1- lim xn =
L, then also =2-limxn = L.
An obvious consequence of Lemma 2.2 is the following theorem.
309
T. Šalát and V. Toma
Theorem 2.3: If =1 ⊆ =2 are two admissible ideals and =1 ∈ S(T ) then
=2 ∈ S(T ).
Corollary 2.4: If = is an admissible ideal and = ⊇ =c then = ∈ S(T ).
Theorem 2.5: The ideal =c is the smallest element in the class S(T ) par-
tially ordered by inclusion.
Proof: Let = ∈ S(T ). We prove that for any set M = {m1 < m2 < . . . } ∈
=c we have M ∈ =. We can suppose that M is an infinite set because if it
were finite it belongs to = as it is an admissible ideal, hence contains all finite
sets. Since M ∈ =c following the definition of the ideal =c we have
∞∑
k=1
1
mk
< +∞
Let us define numbers an (n = 1, 2, . . . ) as follows
amk =
1
mk
(k = 1, 2, . . . ),
an =
1
n2 + n
for n ∈ N \M.
Then obviously an > 0 (n = 1, 2, . . . ) and
∑∞
n=1 an < +∞ by the definition
of numbers an. Since = ∈ S(T ) we have
=-lim nan = 0.
This implies that for each ε > 0 we have
A(ε) = {n : nan ≥ ε} ∈ =,
specifically M = A(1) ∈ =.
The problem of characterization of the class of all ideals having the prop-
erty 2.1 was formulated orally by the second author in a discussion in the
Seminar on real functions theory in Bratislava. The above results allow us
to give such a characterization.
Theorem 2.6: The class S(T ) consists of all admissible ideals = ⊆ P(N)
such that = ⊇ =c.
310
Olivier’s theorem and statistical convergence
Proof: 1) If = ∈ S(T ) then after Theorem 2.5 we have = ⊇ =c.
2) Let = be an admissible ideal and =c ⊆ =. Due to Theorem 2.1 we have
=c ∈ S(T ) and the Corollary 2.4 yields = ∈ S(T ).
Remark: Referring to inclusions 1.1 and 1.2 we can claim that S(T ) contains
as elements the ideals =d and =δ. Since the =d-convergence is in fact the
statistical convergence we get
∞∑
n=1
an < +∞, (an > 0) =⇒ lim-stat nan = 0.
In connection with Theorem 2.6 it is important to know how many ideals
have the property 2.1 and how many don’t have this property. We know yet
that the ideal =f which yields usual convergence is not an element of S(T ).
We give some more such examples.
Example 2. For every integer j ≥ 0 let Dj := {2j(2n + 1) : n = 0, 1, . . . };
then obviously
∞⋃
j=0
Dj = N. Let =] be the set of all A ⊆ N which intersect
only finite number of the sets D0, D1, . . . . Then =] is obviously an ideal
which does not contain the set A = {1, 2, 22, . . . , 2j, . . . } since A meets every
Dj but A is obviously an element of =c. Hence =c 6⊆ =].
Example 3. The ideal =u := {A ⊂ N : u(A) = 0} where u(A) is the uniform
density of the set A, is not an element of the class S(T ). To prove this let
us choose A =
∞⋃
k=1
Ak with Ak = {22k + 1, . . . , 22k + 2k}, (k = 1, 2, . . . ).
Obviously A ∈ =c. To prove that A /∈ =u let us determine the upper uniform
density of the set A (see [1]). To this end, consider the sets A ∩ [t + 1, t + s]
with t ≥ 0, s ∈ N. If we take tk = 22k, sk = 2k, (k = 1, 2, . . . ), then A(tk +
1, tk + sk) = sk, (k = 1, 2, . . . ) and β(sk) := lim sup
t→∞
A(t + 1, t + sk) = sk and
consequently u(A) = limk→∞ β
(sk)
sk
= 1. So we have proved that A /∈ =u and
hence =c 6⊆ =u
The next two propositions give an idea of how many admissible ideals are
appropriate for the analog of Olivier’s theorem and how many can not be
used to obtain this analog.
311
T. Šalát and V. Toma
Proposition 2.7: There are infinitely many admissible ideals which are not
elements of the class S(T ).
Proof: For any infinite set M ⊆ N we enlarge the ideal =f by adjunction
of the set M defining a new ideal =M := {A ∪ B : A ∈ =f , B ⊆ M}. If we
choose M such that N\M is also infinite then =M is an admissible ideal. We
can choose moreover M such that
∑
m∈M m
−1 < +∞ and then =M ⊂ =c.
Applying Theorem 2.5 we conclude that =M is not an element of S(T ). To
see that there are infinitely many ideals of the type =M it is sufficient to
observe that =M 6= =M ′ if and only if the symmetric difference M4M ′ is an
infinite set.
Remark: P. Kostyrko observed that for each q, 0 < q < 1, the admissible
ideal =(q)c := {A ⊆ N :
∑
a∈A a
−q < +∞} is a proper subset of the ideal =c.
So we get again by Theorem 2.5 infinitely many admissible ideals that do
not belong to S(T ).
Proposition 2.8: There are infinitely many admissible ideals which are
elements of the class S(T ).
Proof: Let us take a set M with
∑
m∈M
m−1 = +∞ such that N \ M is
infinite. Then the ideal
=∗M = {A ∪B : A ∈ =c, B ⊆ M}
is an admissible ideal such that =c ⊂ =∗M and consequently =∗M ∈ S(T ). To
see that there are infinitely many such ideals =∗M let us write M = {m1 <
m2 < . . . } with
∞∑
k=1
m−1k = +∞. Then by comparison test we have also
∞∑
k=1
m−12k−1 = +∞ =
∞∑
k=1
m−12k . If we take M
′ = {m2k−1 : k = 1, 2 . . . } then
=c ⊂ =∗M ′ ⊂ =∗M and in this way we construct a decreasing chain of infinitely
many admissible ideals of the class S(T ).
References
[1] T. C. Brown and A. R. Freedman. The uniform density of sets of integers
and Fermat’s last theorem. C. R. Math. Rep. Acad. Sci. Canada, XII:1–
6, 1990.
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Olivier’s theorem and statistical convergence
[2] H. Fast. Sur la convergence statistique. Coll. Math., 2:241–244, 1951.
[3] J. A. Fridy. On statistical convergence. Analysis, 5:301–313, 1985.
[4] H. Halberstam and K. F. Roth. Sequences I. Oxford University Press,
Oxford, 1966.
[5] K. Knopp. Theorie und Anwendung der unendlichen Reihen 3. Aufl.
1931.
[6] P. Kostyrko, T. Šalát, and W. Wilczński. I-convergence. Real Anal.
Exch., 26:669–689, 2000-2001.
[7] L. Olivier. Remarques sur les séries infinies et leur convergence. J. reine
angew. Math., 2:31–44, 1827.
[8] B. J. Powel and T. Šalát. Convergence of subseries of the harmonic
series and asymptotic densities of sets of positive integers. Publ. Inst.
Math.(Beograd), 50(64):60–70, 1991.
[9] I. J. Schoenberg. The integrability of certain functions and related
summability methods. Amer. Math. Monthly, 66:361–375, 1959.
[10] T. Šalát. On statistically convergent sequences of real numbers. Math.
Slovaca, 30:139–150, 1980.
Tibor Šalát Vladimír Toma
Comenius University Comenius University
Department of Mathematics Department of Mathematics
Mlynská dolina Mlynská dolina
84248 Bratislava 84248 Bratislava
SLOVAK REPUBLIC SLOVAK REPUBLIC
toma@fmph.uniba.sk
313


A Classical Olivier’s Theorem and Statistical Convergence

David E. Dobbs

Gabriel Picavet

On Strong Going-Between, Going-Down, And Their Universalizations, II

P.N. Natarajan

ANNALES MATHÉMATIQUES
BLAISE PASCAL
P.N. Natarajan
Product Theorems for Certain Summability Methods
in Non-archimedean Fields
Volume 10, no2 (2003), p. 261-267.
<http://ambp.cedram.org/item?id=AMBP_2003__10_2_261_0>
© Annales mathématiques Blaise Pascal, 2003, tous droits réservés.
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Annales Mathematiques Blaise Pascal 10, 261-267 (2003)
Product Theorems for Certain Summability
Methods in Non-archimedean Fields
P.N. Natarajan
Abstract
In this paper, K denotes a complete, non-trivially valued, non-
archimedean field. Sequences and infinite matrices have entries in K.
The main purpose of this paper is to prove some product theorems
involving the methods M and (N, pn) in such fields K.
AMS subject classification: 40, 46.
Keywords and phrases: regular summability methods, M, (N, pn) meth-
ods, product theorems, consistency, analytic functions.
Throughout the present paper, K denotes a complete, non-trivially val-
ued, non-archimedean field. Sequences and infinite matrices have entries in
K.
Given an infinite matrix A = (ank), n, k = 0, 1, 2, · · · and a sequence
x = {xk}, k = 0, 1, 2, · · · , by the A-transform of x = {xk}, we mean the
sequence A(x) = {(Ax)n}, where
(Ax)n =
∞∑
k=0
ankxk, n = 0, 1, 2, · · · ,
it being assumed that the series on the right converge. If lim
n→∞
(Ax)n = l, we
say that x = {xk} is A-summable to l. If lim
n→∞
(Ax)n = l whenever lim
k→∞
xk =
l, we say that the matrix method A is regular. Necessary and sufficient
conditions for A to be regular in terms of its entries are well-known (see [2]).
The (N, pn) methods (or Nörlund methods) were introduced in K and
some of their properties were studied earlier by Srinivasan (see [6]). A more
detailed study of the (N, pn) methods was taken up by the author later and
published in a series of articles (for instance, see [4], [5]).
261
P.N. Natarajan
The (N, pn) method is defined by the matrix A = (ank), where
ank =
pn−k
Pn
, k ≤ n;
= 0, k > n,
where p0 6= 0, |pj| < |p0|, j = 1, 2, · · · and Pn =
n∑
k=0
pk, n = 0, 1, 2, · · ·. The
following result is known ([4], Theorem 1).
Theorem 1.1: The (N, pn) method is regular if and only if
lim
n→∞
pn = 0.
Let {λn} be a sequence in K such that lim
n→∞
λn = 0. Let M = (bnk), where
bnk = λn−k, k ≤ n;
= 0, k > n.
In this context, we note that the M method reduces to the Y method of
Srinivasan (see [6]), when K = Qp, the p-adic field for a prime p, λ0 = λ1 =
1
2
, λn = 0, n ≥ 2.
We need the following definition in the sequel.
Definition: Two matrix methods A = (ank), B = (bnk) are said to be consis-
tent, if whenever x = {xk} is A-summable to s and B-summable to t, then
s = t.
It is clear that the relation “matrices A and B are consistent" is an equiv-
alence relation.
We now recall that a product theorem means the following: given regular
methods A, B, does x = {xk} ∈ (A) imply B(x) ∈ (A), limits being the
same, where (A) is the convergence field of A? i.e., does “A(x) converges"
imply “A(B(x)) converges to the same limit"?
The main purpose of this paper is to prove some product theorems involv-
ing the M, (N, pn) methods in K. In the sequel, we suppose that the (N, pn)
methods are regular.
Theorem 1.2: If (N, pn)(x) converges to l, then (N, pn)(M(x)) converges
to l
(
∞∑
n=0
λn
)
.
262
Product Theorems for Certain Summability Methods
Proof: Let
τn =
p0xn + p1xn−1 + · · ·+ pnx0
Pn
,
tn = λnx0 + λn−1x1 + · · ·+ λ0xn, n = 0, 1, 2, · · · .
By hypothesis, lim
n→∞
τn = l. Since lim
n→∞
pn = 0 and p0 6= 0, lim
n→∞
Pn = P, P 6= 0.
Now,
τ ′n = (N, pn) ({tn})
=
p0tn + p1tn−1 + · · ·+ pnt0
Pn
=
1
Pn
[p0(λnx0 + λn−1x1 + · · ·+ λ0xn)
+p1(λn−1x0 + λn−2x1 + · · ·+ λ0xn−1)
+ · · ·+ pn(λ0x0)]
=
1
Pn
[λ0(p0xn + p1xn−1 + · · ·+ pnx0)
+λ1(p0xn−1 + p1xn−2 + · · ·+ pn−1x0)
+ · · ·+ λn(p0x0)]
=
1
Pn
[λ0Pnτn + λ1Pn−1τn−1 + · · ·+ λnP0τ0]
=
1
Pn
[{λ0Pn(τn − l) + λ1Pn−1(τn−1 − l) + · · ·
+λnP0(τ0 − l)}+ l {λ0Pn + λ1Pn−1 + · · ·+ λnP0}]
=
1
Pn
[{λ0Pn(τn − l) + λ1Pn−1(τn−1 − l) + · · ·+ λnP0(τ0 − l)}
+l {λ0(Pn − P ) + λ1(Pn−1 − P ) + · · ·+ λn(P0 − P )}
+ lP {λ0 + λ1 + · · ·+ λn}] .
Using Theorem 1 of [3] and the fact that |Pn| = |P0|, n = 0, 1, 2, · · · , it follows
that
lim
n→∞
1
Pn
[λ0Pn(τn − l) + λ1Pn−1(τn−1 − l) + · · ·+ λnP0(τ0 − l)] = 0
263
P.N. Natarajan
and
lim
n→∞
1
Pn
[λ0(Pn − P ) + λ1(Pn−1 − P ) + · · ·+ λn(P0 − P )] = 0,
so that
lim
n→∞
τ ′n = l
∞∑
n=0
λn,
i.e., (N, pn)(M(x)) converges to l
∞∑
n=0
λn, completing the proof of the
theorem.
Corollary 1.3: If we want to get the same limit l, we have to choose the
sequence {λn} such that
∞∑
n=0
λn = 1, an example being the Y method of Srini-
vasan.
Corollary 1.4: The Y and (N, pn) methods are consistent.
We make use of well-known properties of analytic elements (a general
reference in this direction is [1]) to prove our next result.
Theorem 1.5: Let |λn| ≤ |λ0|, n = 0, 1, 2, · · ·. If M({an}) converges to l,
then M((N, pn)({an})) converges to l too.
Proof: Let F be a complete, algebraically closed extension of K; let U
be the disk |x| ≤ 1 in F ; let H(U) be the F -algebra of analytic elements in
U, which is known as the set of restricted power series with coefficients in F.
Let A be the algebra of analytic functions in the disk D of F : |x| < 1.
Let φ(x) =
∞∑
n=0
λnx
n. We note that φ ∈ H(U) and φ in invertible in A,
since |λn| ≤ |λ0|, n = 0, 1, 2, · · · .
Let M̂ be the linear mapping defined by M in the space of power series:
if M({an}) = {cn}, then
M̂(f) =
∞∑
n=0
cnx
n.
It is easily seen that
M̂(f) = φ(x)f(x).
264
Product Theorems for Certain Summability Methods
Since M({an}) has limit l,
φ(x)f(x) =
∞∑
n=0
(l + εn)x
n
=
l
1− x
+ ε,
where ε =
∞∑
n=0
εnx
n ∈ H(U), since lim
n→∞
εn = 0. Since φ is invertible in A, we
have
f(x) =
l
(1− x)φ
+
ε
φ
.
But
l
(1− x)φ
+
ε
φ
∈ A. Thus f ∈ A and therefore is bounded because so
are
l
1− x
,
1
φ
, ε and so
l
(1− x)φ
+
ε
φ
. Consequently the sequence {an} is also
bounded (see, for instance, [1]).
Let π(x) =
∞∑
n=0
pnx
n. π ∈ H(U), since lim
n→∞
pn = 0.
Let (N̂, pn)(f) =
∞∑
n=0
cnx
n. Since {Pn} converges to a limit P 6= 0 and
since {an} is bounded,
∞∑
n=0
cnx
n =
1
P
{π(x)f(x) + θ(x)},
where θ(x) =
∞∑
n=0
θnx
n, θn =
(
1
Pn
− 1
P
) n∑
k=0
pkan−k, n = 0, 1, 2, · · ·.
Noting that lim
n→∞
θn = 0, we have θ ∈ H(U).
Thus, we have,
PM̂((N̂, pn)(f)) = φ(x)(π(x)f(x) + θ(x))
=
lπ(x)
1− x
+ ε(x)π(x) + φ(x)θ(x)
=
lπ(1)
1− x
+
l(π(x)− π(1))
1− x
+ ε(x)π(x) + φ(x)θ(x).
265
P.N. Natarajan
It is well-known (see [1]) that x− 1 divides π(x)− π(1) in H(U) and so
PM̂((N̂, pn)(f)) =
lπ(1)
1− x
+ τ(x),
where τ(x) ∈ H(U). Since π(1) = P,
M̂((N̂, pn)(f)) =
l
1− x
+
1
P
τ(x),
which proves that M((N, pn)({an})) has limit l. This completes the proof of
the theorem.
Acknowledgement. I thank the referee for his constructive suggestions,
which enabled me to present the material in a much better form.
References
[1] A. Escassut. Analytic elements in p-adic Analysis. World Scientific Pub-
lishing Co., 1995.
[2] A.F. Monna. Sur le théorème de Banach-Steinhaus. Indag. Math.,
25: 121–131, 1963.
[3] P.N. Natarajan. Multiplication of series with terms in a non-archimedean
field. Simon Stevin, 52: 157–160, 1978.
[4] P.N. Natarajan. On Nörlund method of summability in non-archimedean
fields. J.Analysis, 2: 97–102, 1994.
[5] P.N. Natarajan and V.Srinivasan. Silvermann-Toeplitz theorem for dou-
ble sequences and series and its application to Nörlund means in non-
archimedean fields. Ann.Math. Blaise Pascal, 9: 85–100, 2002.
[6] V.K. Srinivasan. On certain summation processes in the p-adic field.
Indag. Math., 27: 368–374, 1965.
266
Product Theorems for Certain Summability Methods
P.N. Natarajan
Ramakrishna Mission Vivekananda
College
Department of Mathematics
Mylapore
Chennai 600 004
INDIA
pinnangudinatarajan@hotmail.com
267


Product Theorems for Certain Summability Methods in Non-archimedean Fields

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