INTRODUCTION TO MANIFOLDS — IV
Appendix: algebraic language in Geometry
1. Algebras.
♥
Definition.
A (
commutative associatetive
)
algebra
(
over reals
) is a lin
ear space
A
over
R
, endowed with two operations, + and
·
, satisfying the natural
axioms of arithmetics: (
A,
+) is an additive Abelian (=commutative) group with
the neutral element denoted by 0, while (
A,
·
) is a commutative semigroup. If there
is a
·
neutral element, then it is denoted by 1, though existence of such an element
is not usually assumed.
♦
Example.
The basic example is that of real numbers. Another elementary exam
ples: matrices Mat
n
(
R
). Other examples follow.
The Principal Example.
Let
M
be a smooth
n
dimensional
manifold, and
A
=
C
∞
(
M
)
the space of al smooth functions
on it. Then if one sets
(
f
+
g
)(
x
) =
f
(
x
) +
g
(
x
)
,
(
f
·
g
)(
x
) =
f
(
x
)
g
(
x
)
,
(
λf
)(
x
) =
λf
(
x
)
,
then al the axioms will be satisfied.
Now the principal wisdom comes.
The main idea of algebraic approach to geometry is to
study properties of the manifold via algebraic properties
of the algebra
(
ring
)
C
∞
(
M
)
.
2. Reconstruction of points.
♥
Definition.
An
ideal
I
of an algebra
A
is a subalgebra with the property
A
·
I
⊆
I,
which is to be understood as
∀
a
∈
A,
∀
u
∈
I
au
∈
I.
♣
Problem 1.
Prove that 1
∈
I
=
⇒
I
=
A
.
Typeset by
A
M
S
T
E
X
1
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APPENDIX: ALGEBRAIC LANGUAGE IN GEOMETRY
♣
Problem 2.
Prove that
{
0
}
is always an ideal in any algebra.
The principal example of an ideal is the following one.
♣
Problem 3.
If
Z
⊆
M
is a closed subset, then
I
Z
=
{
f
∈
C
∞
(
M
):
f

Z
≡
0
}
is an ideal in
A
=
C
∞
(
M
).
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 Spring '08
 PROTSAK
 Algebra, Geometry, Vector Space, Manifold

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