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
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 con
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;
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 inve
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)
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 & Rad
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
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 = c
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 sub
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 necessa
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 me