Beilinson-Bernstein over the base affine Gcfw_N
October 23, 2015
Our goal is to explain some of the background used to understand the main result of [BN] and compute some
examples. There is more general theory in [BN2]. Much of that theory a
MATH 104 - WEEKLY ASSIGNMENT 3
DUE 16 SEPTEMBER 2016, BY 16:00
(i) Let bn = n12 +1 + n12 +2 + . . . + n12 +n , for all n N. What is the limit of (bn )nN ?
(Hint: Be very careful here: each of the terms above converges to 0, but that
MATH 104 - WEEKLY ASSIGNMENT 6
DUE 14 OCTOBER 2016, BY 16:00
(1) The aim of this exercise is to clarify a quite delicate point when it comes to rearrangements of series, that may be causing a lot of confusion.
(i) In Weekly Assignment 4, we proved that th
MATH 104 - WEEKLY ASSIGNMENT 8
DUE 24 OCTOBER 2016, BY 16:00
(1) (Sandwich lemma for functions.) Let f, g, h : A R, where A R. Let x0 R be an
accummulation point. Suppose that
f (x) g(x) h(x),
for all x A. Show that, if limxx0 f (x) = limxx0 h(x) = l R, t
Group Homework Assignment #1
1. For this problem, the goal is to prove that real numbers correspond bijectively to decimal expansions not terminating in an infinite string of nines.
uppose x 2 R. The decimal expansion of x is
Chapter II: Sequences and functions
Denition 1. Let E be a set. A sequence with E -values is a function u : N E . We
note un := u(n), it is the n-th term of the sequence. We note the sequence (un )nN
or sometimes only (un ) and E N the set o
CHAPTER VI: COMPACTNESS
Denition 1. Let (E, d) metric. We say that (E, d) is compact if every sequence with
values E has a convergent subsequence. If X (E, d), we say that X is compact, if
(X, d|XX ) is compact.
Example : [a, b] is compact.
CHAPTER V: METRIC SPACES
Notion of metric
Denition 1. Let E be a set. We call d a distance (or a metric) on E a function
R+ such that:
x, y E, d(x, y) = 0 x = y (separation).
x, y E, d(x, y) = d(y, x) (symmetry)
x, y, z E, d(x, z) d(x,
Chapter IV : Series
1.1. Preliminaries. Before explaining what is a serie, we will start to make some
denitions similar to those we know for the functions. Let K = R, C. Let (un ) and
(vn ) KN , with (vn ) nonzero starting from a certain r
1.1 Why the rational numbers are not sucient?
Let Q be the set of rational numbers, Q+ the positive rational and Q the non-zero
rationals. The set of rational numbers has two disadvantages:
Lemma 1. In Q, 2 is not a square, i.e.
MATH 104 - WEEKLY ASSIGNMENT 5
DUE 30 SEPTEMBER 2016, BY 16:00
(1) True of False? Justify your answers. (Some may be misleading. Of course, so may be
(i) If |ak | 0, then +
k=1 ak converges absolutely.
(ii) If ak > 0 and 0 < k+1
MATH 104 - WEEKLY ASSIGNMENT 2
DUE 9 SEPTEMBER 2016, BY 16:00
(1) Show that, for any a, b R with a < b, there exist infinitely many irrational numbers x
with a < x < b.
(2) Show that, for any a, b R, it holds that
|a| |b| |a + b|
|a| |b| |a b|.
Group Homework Assignment #3
1. Prove that every function defined on a discrete metric space is uniformly continuous.
2. For all subsets E, F of a metric space (S, d) prove:
(a) E is clopen i @E = ;.
(b) @E = @E C
(c) @E @E
(d) @E = @E
(e) @(E [
Group Homework Assignment #2
1. Let b = sup S where S is a bounded non-empty subset of R.
(a) Given > 0, show that there exists an s 2 S with
(b) Can s 2 S always be found so that b
< s < b?
2. Given y 2 R, n 2 N, and
Group Homework Assignment #4
1. Definition: The diameter of a nonempty set E S is the supremum of all distances d(x, y) between points of E.
Let (An ) be a nested decreasing sequence of non-empty closed sets in the metric spa
MATH 104 - WEEKLY ASSIGNMENT 4
DUE 26 SEPTEMBER 2016, BY 16:00
(1) Find a sequence (an )nN in R, with an+1 an 0, but (an )nN not Cauchy.
(2) Let (an )nN be a sequence in R. Show that (an )nN converges if and only if: (a2n )nN
and (a2n1 )nN converge and ha
MATH 104 - WEEKLY ASSIGNMENT 9
DUE 28 OCTOBER 2016, BY 16:00
(1) (Very basic.) Let f : A R, where A R, and let x0 be an accumulation point of A.
lim f (x) = l
if and only if
lim f (x) = l and lim f (x) = l,
in the case where l R.
MATH 104 - WEEKLY ASSIGNMENT 7
DUE 17 OCTOBER 2016, BY 16:00
(1) The aim of this exercise is to show the necessity of certain assumptions in some of the
theorems regarding continuity. You can just draw the graph of your function if you want,
you dont have
MATH 104 - WEEKLY ASSIGNMENT 10
DUE 4 NOVEMBER 2016, BY 16:00
(1) Explain the following:
(i) The function f : R R, with f (x) = x2 for all x R, is not uniformly continuous.
(Note that this means that products of uniformly continuous functions are
MATH 104 - WEEKLY ASSIGNMENT 1
DUE 5 SEPTEMBER 2016, BY 16:00
(1) Let (F, +, ) be a field. Show the following:
(i) For a, b, c F, a + c = b + c a = b.
(ii) For a, b, c F and c 6= 0, ac = bc a = b.
(iii) For every a F, (a) = a.
(iv) For every a F with a 6=
Chapter III : Functions
In this whole chapter, I will be an interval.
Denition 1. Let f : I R be a function, we say that is the limit of f at x0 I
> 0, > 0, |x x0 | = |f (x) | .
We have a sequential criterion