Invariant Theory
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585 T. A. Springer
Invariant Theory
Springer-Verlag Berlin-Heidelberq • New York 1977
Author T. A, S p r i n g e r Mathematisch Instituut der Rijksuniversiteit Budapesttaan 6 Utrecht, N e d e r l a n d
Library of Congress Cataloging in Publication Data
Springer s TormyAlbert, Imv&riant theory.
1926-
(Lecture notes in mathematics ; 585) Includes bibliographies and index. i. Linear algebraic groups. 2. Invariants. I. Title. II. Series: I~c%]~-e notes in mathematics
(Berlin) ; 585.
QA3.~8
no. 585 [% N.
1.2.2.
Theorem
(Hilbert's
the p o l y n o m i a l
r i n g R[T].
For the proof
see
1.2.3.
Corollary.
1.2.4.
Zeros
1.2.5. ideal
theorem).
If R is n o e t h e r i a n
then
so is
[6, Ch.VI,§2] .
S is noetherian.
of ideals
Let I be an ideal for all
basis
of S.
of S.
Then v C V is called
a zero of I if f(v)
= 0
f E I.
Theorem
(Hilbert's
Nullstellensatz).
(i)
(first
form)
A proper
I o f S has a zero~
(ii)
(second
form)
f(v)
= 0 for ~II
See
[6,
Let I be an ideal of S and
zeros
Ch.X,§2] .
v of i. Then there
let f E S be such that
is n ~ I such that
fn E I.
1.3. The Zariski 1.3.1.
topology
If I is an ideal
have the f o l l o w i n g
on v.
of S, let ~(I)
: V, ~(S)
(b) I C J
~
(c) ~(ICU)
: ~(1) u ~(j);
: ~;
zr(1) D ~(J);
(d) If ( I ) a E A is a set of ideals f , with f
See
~
~ I
~z
a
(x
and f i (x ) =
from
(a),
~
(c) and
closed
sets are the ~ I ) ,
this).
This
1.3.2.
Exercises.
(i) The
and
~(I (x ). (d) that there
I running
topology
is TI~
i.e.
(2) If dim V = 1 the Z a r i s k i - c l o s e d (3) If V and k are both given f 6 S
are continuous.
1.3.3.
If X is a subset ~(X)
If I is an ideal which
lies
1.3.4.
Zf($(X)).
: {f e S
I f(X) by
~
points
of S
(check
(i) ~(J(X))
are closed.
sets are the finite topology,
the ideal Y(X)
ones.
the functions
of S by
= 0}. the ideal of the f E S a power of
is indeed an
ideal).
= X, the closure
and contains
of X;
X, by the definitions.
to show that any closed Now
the ideals
on V whose
= WY.
is closed
have only
of V, define
of S, denote
Proposition.
~(2(X))
through
the Zariski
in I (check that this
(ii) U(~(1))
is a t o p o l o g y
Zariski topology.
is the
Zariski
a
Z I the ideal of the sums aeA = 0 for all except f i n i t e l y many ~, then
aEA
[ 6, Ch.X,§3] .
It follows
We then
properties:
(a) ~({0})
aEA
be the set of its zeros.
if X C ~(I),
set ~(I)
then f(X)
containing
To prove X also
(i) we
contains
= 0 for all f E I, so I C J(X)
and
~(I) • ~(a(x)). (ii)
is a t r a n s l a t i o n
of the
second
form of H i l b e r t ' s
Nullstellensatz.
1.3.5. Exercises. (1) The map I ~
~(1) defines a bijection of the c o l l e c t i o n of ideals
I of S with I = W~ onto the collection of closed subsets of V. (2) Every n o n - e m p t y c o l l e c t i o n of closed
subsets of V contains a
minimal element. 1.3.6.
I r r e d u q i b ! l i t ¥.
A t o p o l o g i c a l space X is reducible sets X1,X 2
with X = X i U X2,
if there are n o n
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