• Keine Ergebnisse gefunden

Exercises for discussion in the exercise class on 06.12.2018

N/A
N/A
Info
Herunterladen
Protected

Academic year: 2021

Aktie "Exercises for discussion in the exercise class on 06.12.2018"

Copied!
1
0
0

Wird geladen.... (Jetzt Volltext ansehen)

Jetzt herunterladen ( 1 Seite )

Volltext

(1)

Singularities, Monodromy and Zeta Functions Blatt 7

Exercises for discussion in the exercise class on 06.12.2018

Aufgabe 1:

Let X = {(x

1

, ..., x

n

)|f

1

(¯ x) = ... = f

k

(¯ x) = 0} be an affine variety over Z and X(R) = {(x

1

, ..., x

n

) ∈ R

n

|f

1

(¯ x) = ... = f

k

(¯ x) = 0} the set of R-valued points of X .

Show that X : R 7→ X (R) induces a functor from the category of rings to the category of sets.

Hint: How should one write X(f ) for f a ring homomorphism?

Clarification: Here rings are (by definition) commutative and unital; morphisms in the category of rings preserve 1.

Aufgabe 2:

Verify that X := {x|v(x) = 1} is a uniformly definable set in terms of Definition 1.5.5.

That is, write out X in terms of affine algebraic varieties, ( Z

p

)

p∈P

and the operations of co-ordinate projection, cartesian product and boolean combination.

Aufgabe 3:

Let X := {x|v(x) ≥ 0 ∧ ∃y : y

2

= x}.

In the lecture we saw that µ

p

(X( Q

p

)) =

2(p+1)p

for all p ≥ 3.

(a) Calculate µ

2

(X( Q

2

)).

(b) Write down a single Z ∈ C

0

such that ev

p

(Z) = µ

p

(X ( Q

p

)) for every prime p (including 2).

If you’re too lazy for (a), you can denote the rational number µ

2

(X( Q

2

)) by r and solve (b) with that.

Course website: http://reh.math.uni-duesseldorf.de/~internet/Zeta_WS18/

Referenzen

Jetzt herunterladen ( PDF - 1 Seite - 142.24 KB )

ÄHNLICHE DOKUMENTE

Exercises for the Course Datawarehousing, SS 2002, Prof. R. Bayer, TUM Exercise Sheet 1

Exercise 3, Ch.4: How large is the hierarchically aggregated cube (without base table) for an aggregation up to the levels of the prefixes of calls, geographic 100*100 m squares

Exercises for the Course Datawarehousing, SS 2002, Prof. R. Bayer, TUM Exercise Sheet 2

Reduce the dimensionality of the fact table by exploiting the fact that the location of a mobile phone user usually can only be determined by the current cell.. Make sure that

Exercises for the Course Datawarehousing, SS 2002, Prof. R. Bayer, TUM Exercise Sheet 2

Reduce the dimensionality of the fact table by exploiting the fact that the location of a mobile phone user usually can only be determined by the current cell.. Make sure that

Exercise Class

For the following code fragment, count the number of times the function f is invoked and express this in Θ -notation as concisely as possible.. For the following code

SS 2012 18.07.2012 In-class Exercises to the Lecture Logics Sheet 7 Jun.-Prof. Dr. Roland Meyer Discussion on 19./20.07.2012

A variant of Herbrand’s Theorem reads: A formula in first order predicate logic without “=” is satisfiable if and only if it has a Herbrand model.. a) Present a satisfiable formula

•Discussion on Fri 29.5. 13:00-15:45 Exercises Praktikum

• Atmospheric neutrino oscillations confirmed by long-baseline accelerator neutrino experiments... 31: The zenith angle distribution for fully-contained 1-ring events,

In the future the exercises must be done at home and will be discussed during the exercise class

You have to ensure that you have copied the source code of your solved exercises to the server before the exercise class starts.. If necessary, ask your tutor how to copy files

Exercise 2 Show that the class

In this exercise we want to show that the model construction for FO 2 -formulae from the lecture is optimal in the following sense: in general it does not suffice to take only