Workshop on group schemes and p-divisible groups: Homework 3.
1. Let K be the fraction eld of a complete discrete valuation ring with mixed characteristic
(0, p) and perfect residue eld. Let K/K be an algebraic closure and CK its completion. Let
GK = Gal(
Mathematics Department Stanford University
Math 205a Lecture Correction
The argument (given on Tuesday Nov. 12) to prove the lemma needed for the Hahn decomposition
theorem (that (A) < 0 implies A contains a negative set E with (E ) (A) should indeed be
x
2
Stanford Mathematics Department
Math 205A Lecture Supplement #5
Riesz Representation for Lp ( )
Here (X ; A; ) is any measure space and 1 p 1, 1 q 1 are conjugate
exponents, meaning that
1
1
C D 1;
p
q
()
1
where of course we take 1 D 0. Lp ( ) will her
Workshop on group schemes and p-divisible groups: Homework 1.
1. Let S be a scheme, and G and G group schemes over S .
(i) Using Yonedas Lemma and group theory, show that the identity section and inversion morphism for G are uniquely determined by the mul
Workshop on group schemes and p-divisible groups: Homework 2.
1. (i) Using the structure theorem and Frobenius morphisms, prove that a nite group scheme over
a eld is killed by its order. (Exer. 3(ii) in HW1 gives a non-commutative group of order p2 .)
(i
2
1 Borel Regular Measures
We now state and prove an important regularity property of Borel regular
outer measures:
Stanford Mathematics Department
Math 205A Lecture Supplement #4
Borel Regular & Radon Measures
1.1 Theorem. Suppose X is a topological spa
Mathematics Department Stanford University
Math 205a Lecture Correction
In lecture on Tuesday Nov. 5 we were looking at a Borel regular outer measure in a topological
space X , we were given a subset A X , and we dened the restriction A of to A; this is
t
Mathematics Department Stanford University
Solutions to Math 205A Sample Mid-term Test (75 minute test)
1 (i) If A be a -algebra of subsets of a space X and if f : X R, prove that the collection E = cfw_E : E
R, f 1 (E ) A forms a -algebra of subsets of
Mathematics Department Stanford University
Solutions of Math 205A Mid-term Test
1. If f : R R is continuous and g : R R is Lebesgue measurable (i.e. M-measurable, where M is the
set of -measurable subsets of R), prove that the composite f g is Lebesgue me
Mathematics Department Stanford University
Math 205A Sample Final Examination Solutions
Unless otherwise indicated, you can use results
covered in lecture, provided they are clearly stated.
If necessary, continue solutions on backs of pages
Note: work she
Mathematics Department Stanford University
Math 205A 2013Lecture Supplement #3
Dierentiability Theory for Functions and Measures
As a preliminary to the discussion of dierentiation of functions and measures, we need the following
important covering lemma,