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Graded
W
itt kernels of the compositum of multiquadratic extensions
w
ith the function fields of
Pfister
forms
详细信息
查看全文
作者:
Roberto Aravire
a
;
raravire@unap.cl" class="auth_mail" title="E-mail the corresponding author
;
Ahmed Laghribi
b
;
ahmed.laghribi@univ-artois.fr" class="auth_mail" title="E-mail the corresponding author
;
Manuel O'Ryan
c
;
moryan@inst-mat.utalca.cl" class="auth_mail" title="E-mail the corresponding author
关键词:
11E04
;
11E81
刊名:Journal of Algebra
出版年:2016
出版时间:1 March 2016
年:2016
卷:449
期:Complete
页码:635-659
全文大小:500 K
文摘
Let
F
be a field of characteristic 2 and
w the MathML source">W
q
(F)
w="scroll">
w>
W
w>
w>
q
w>
(
F
)
be the Witt group of nonsingular quadratic forms over
F
. Let
φ
be a bilinear
Pfister
form over
F
and
L
be a multiquadratic extension of
F
of separability degree less than of equal to 2. In this paper
w
e compute the kernel of the natural homomorphism
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w
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w="scroll">
w>
H
w>
w>
2
w>
w>
m
+
1
w>
(
F
)
⟶
w>
H
w>
w>
2
w>
w>
m
+
1
w>
(
L
(
φ
)
)
,
w
here
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width="69" alt="Vie
w
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w
the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869315005918-si3.gif">
width="69" alt="Vie
w
the MathML source" title="Vie
w
the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0021869315005918-si3.gif">
w="scroll">
w>
H
w>
w>
2
w>
w>
m
+
1
w>
(
F
)
is the cokernel of the Artin–Schreier operator
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w="scroll">
&
w
eierp;
:
w>
Ω
w>
w>
F
w>
w>
m
w>
⟶
w>
Ω
w>
w>
F
w>
w>
m
w>
/
d
w>
Ω
w>
w>
F
w>
w>
m
−
1
w>
given by
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w
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w
the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869315005918-si5.gif">
width="307" alt="Vie
w
the MathML source" title="Vie
w
the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0021869315005918-si5.gif">
w="scroll">
x
w>
d
w>
x
w>
w>
1
w>
w>
w>
x
w>
w>
1
w>
∧
⋯
∧
w>
d
w>
x
w>
w>
m
w>
w>
w>
x
w>
w>
m
w>
↦
(
w>
x
w>
w>
2
w>
−
x
)
w>
d
w>
x
w>
w>
1
w>
w>
w>
x
w>
w>
1
w>
∧
⋯
∧
w>
d
w>
x
w>
w>
m
w>
w>
w>
x
w>
w>
m
w>
,
w
here
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width="25" alt="Vie
w
the MathML source" title="Vie
w
the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869315005918-si6.gif">
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w
the MathML source" title="Vie
w
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w="scroll">
w>
Ω
w>
w>
F
w>
w>
m
w>
is the space of
m
-differential forms over
F
, and
w the MathML source">F(φ)
w="scroll">
F
(
φ
)
is the function field of the affine quadric given by the diagonal quadratic form associated to the bilinear form
φ
. As a consequence,
w
e deduce the kernel of the natural homomorphisms
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w="scroll">
w>
w>
I
w>
w>
q
w>
w>
m
+
1
w>
w>
‾
(
F
)
⟶
w>
w>
I
w>
w>
q
w>
w>
m
+
1
w>
w>
‾
(
L
(
φ
)
)
and
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w="scroll">
w>
I
w>
w>
q
w>
w>
m
+
1
w>
(
F
)
⟶
w>
I
w>
w>
q
w>
w>
m
+
1
w>
(
L
(
φ
)
)
,
w
here
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w="scroll">
w>
w>
I
w>
w>
q
w>
w>
m
+
1
w>
w>
‾
(
F
)
denotes the quotient
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w="scroll">
w>
I
w>
w>
q
w>
w>
m
+
1
w>
(
F
)
/
w>
I
w>
w>
q
w>
w>
m
+
2
w>
(
F
)
such that
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w>
I
w>
w>
q
w>
w>
m
+
1
w>
(
F
)
=
w>
I
w>
w>
m
w>
F
⊗
w>
W
w>
w>
q
w>
(
F
)
and
w the MathML source">I
m
F
w="scroll">
w>
I
w>
w>
m
w>
F
is the
m
-th po
w
er of the fundamental ideal
IF
of the Witt ring of
F
-bilinear forms. We also include some results concerning the case
w
here
φ
is replaced by a bilinear Pfister neighbor or a quadratic Pfister form.
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