Real Algebraic Geometry II – Exercise Sheet 3

Universität Konstanz Tom-Lukas Kriel Department of Mathematics and Statistics María López Quijorna

Summer Term 2017 Markus Schweighofer

Real Algebraic Geometry II – Exercise Sheet 3

Exercise 1(4P) An subset of a topological space is calledclopenif it is closed and open.

Let Abe a commutative ring.

(a) Show that every constructible subset of sperAis quasicompact.

(b) Show that the constructible subsets of sperA are exactly the clopen subsets of sperAwith respect to the constructible topology.

Exercise 2 (4P) Let A be a commutative ring. Show that the set of maximal prime cones ofAis a compact subset of sperA.

Exercise 3(4P) LetAbe a commutative ring. Let S

⊆

sperAbe closed with respect to the constructible topology. Prove thatS

= {

P

∈

sper

(

A

) | ∃

Q

∈

S:Q

⊆

P

}

.

Exercise 4(6P) LetAbe a commutative ring. Show that the correspondence F

7→

^\_F,

{

B

∈

_C

|

A

⊆

B

} ←

_[A

defines a bijection between the set of filters in the Boolean algebraCAof constructible subsets of sperAand the set of nonempty closed subsets of sperAwith respect to the constructible topology.

Exercise 5(4P) Show the following:

(a) IfP

∈

sperR

[

_X

]

_{andU :}

=

_{UP, thenP}is Archimedean if and only if the ultrafilter U contains a bounded semialgebraic set.

(b) U :

= {

S

∈

_S2

| ∃

N

∈

_N :

{(

x²,x

) |

x

∈

_R,x

≥

N

} ⊆

S

}

is an ultrafilter on the Boolean algebra of semialgebraic subsets ofR² such that P :

=

P_U is a maximal element of sperR

[

X₁,X₂

]

but Pis not Archimedean.

Exercise 6 (2P) Let R be a real closed field,m

∈

_Nand P₁, . . . ,P_m

∈

sperR

[

X

]

. Show that there exists a real closed extension field H of R and x₁, . . . ,x_m

∈

Hⁿ such that P_i

= {

f

∈

H

[

X

] |

f

(

x_i

) ≥

0

}

for alli

∈ {

1, . . . ,m

}

.

Please submit until Tuesday, May 16, 2017, 11:44 in the box named RAG II near to the room F411.