F -SINGULARITIES AND FROBENIUS SPLITTING NOTES
9/2-2010
KARL SCHWEDE
1. Criteria for local Frobenius splitting I (Fedders criteria)
Now we need some notation.
e
Denition 1.1. Suppose that S is a ring and I is an ideal. If : F S S is an S -linear
e
map, we
FIELDS AND POLYNOMIAL RINGS
MATH 435 SPRING 2012
NOTES FROM APRIL 6TH, 2012
1. Irreducible polynomials
Throughout this section, k denotes a eld. Before really starting, Id like to point out a couple
lemmas. The rst ties together the notions of ideal conta
WORKSHEET # 4 SOLUTIONS
MATH 435 SPRING 2011
We rst recall some facts and denitions about cosets. For the following facts, G is a group and H
is a subgroup.
(i) For all g G, there exists a coset aH of H such that g aH . (One may take a = g ).
(ii) Cosets
QUIZ # 3
MATH 435 SPRING 2011
1. Suppose that R is a ring. Prove that 0x = 0 for all x R. (1 point)
Solution: We know 0 + 0 = 0 thus 0x = (0 + 0)x = 0x + 0x and now because we have an
Abelian group under addition, we can cancel and so 0 = 0x as desired.
2
QUIZ # 2
MATH 435 SPRING 2011
1. Compute all of the left cosets of the subgroup cfw_1, 10 inside the group U (11) (the group of integers between 1 and 11 relatively prime to 11, under multiplication). Is cfw_1, 10 a normal subgroup?
(2 points)
Solution :
INFO ON EXAM #1
MATH 435 SPRING 2011
There will be 4 pages of regular questions on the exam and one extra-credit question.
(1) There will be one page of short answer questions (for example, dene the term subgroup or
give an example of an non-Abelian group
QUIZ # 1
MATH 435 SPRING 2011
1. Consider the group Z mod 15 under addition. Write down all the elements of the group and
identify the order of each element. For which g Z mod 15 is it true that Z mod 15 = g ? (3 points)
Solution: The elements of Z
mod 15
HOMEWORK # 10
DUE WEDNESDAY APRIL 13TH
MATH 435 SPRING 2011
For this homework, we assume all rings are commutative, associative and with multiplicative
identity. We assume that all homomorphisms send 1 to 1.
1. Show that a ring R is Noetherian if and only
HOMEWORK # 2
DUE FRIDAY JAN. 21ST
MATH 435 SPRING 2011
1. Below are sets with binary operations. Determine if each set is (or is not) a group and prove
your answer.
(a) For a xed n, the numbers cfw_0, 1, 2, . . . , n 1. The binary operation is mutliplicat
HOMEWORK #0 (WARM UP) MATH 435
DUE FRIDAY JANUARY 13TH
(1) What is wrong with the following inductive proof that either all cats are the same color?
For example, will will prove they are all orange, or all black, or all grey, or etc. We will
ignore the po
F -SINGULARITIES AND FROBENIUS SPLITTING NOTES
11/11-2010
KARL SCHWEDE
1. Vanishing theorems via finite maps and direct summand conditions
Using the methods discussed previously, one can show the following.
Proposition 1.1. [HH92] [Also see [Smi97] and er
F -SINGULARITIES AND FROBENIUS SPLITTING NOTES
9/21-2010
KARL SCHWEDE
Remark 0.1. If is eective, we see that (X, , at ) is klt if and only if J (X, , at ) = OX .
Furthermore, if (X, , at ) is log canonical, then J (X, , at ) is a radical ideal. Furthermor
F -SINGULARITIES AND FROBENIUS SPLITTING NOTES
9/21-2010
KARL SCHWEDE
1. Deformations of F -split and rational singularities.
One very fundamental property of rational singularities is the fact that they behave well
in families. In fact, one also has the
F -SINGULARITIES AND FROBENIUS SPLITTING NOTES
9/21-2010
KARL SCHWEDE
1. F -rationality
Denition 1.1. Given (M, ) as above, the module (M, ) is called the test submodule of
(M, ). With R : F R R , the module (R , R ) = (R ) is called the simply the test
s
F -SINGULARITIES AND FROBENIUS SPLITTING NOTES
10/26-2010
KARL SCHWEDE
1. Characteristic p > 0 analogs of LC-centers and subadjunction
We recall the following denition (for now, we work in characteristic zero). Most of the
results found here (including mo
F -SINGULARITIES AND FROBENIUS SPLITTING NOTES
10/28-2010
KARL SCHWEDE
1. Characteristic p > 0 analogs of LC-centers and subadjunction continued
Using the same idea (Fedders lemma), we have the following method for checking whether
an ideal is a non-F -re
F -SINGULARITIES AND FROBENIUS SPLITTING NOTES
11/9-2010
KARL SCHWEDE
Before continuing on, we need a very brief introduction to Matlis/local-duality. Suppose
that (R, m) is a local ring. We know every R-module lives inside an injective R-module. In
parti
F -SINGULARITIES AND FROBENIUS SPLITTING NOTES
10/19-2010
KARL SCHWEDE
1. F -singularities and birational maps
Our goal in this section is to relate F -singularities and test ideals with log canonical and
log terminal singularities as well as multiplier i
F -SINGULARITIES AND FROBENIUS SPLITTING NOTES
9/21-2010
KARL SCHWEDE
1. Pairs in positive characteristic
Weve already studied pairs in a certain context. Consider pairs of the form (R, ) where
e
: F R R is an R-linear map. Our rst goal will be to see th
F -SINGULARITIES AND FROBENIUS SPLITTING NOTES
9/21-2010
KARL SCHWEDE
1. Reduction to characteristic p > 0
Note that if one also has the coordinates of a point x X (closed or not), one can reduce
that closed point to characteristic p as well. Let x R be a
HOMEWORK # 3
DUE FRIDAY JAN. 28TH
MATH 435 SPRING 2011
1. Prove that any group with 2, 3, 4 or 5 elements must be Abelian.
2. Give an example of a group with 6 elements which is not Abelian.
3. Consider the set G = cfw_a + b 2|a, b Qwhere either a = 0 or
HOMEWORK # 11
DUE WEDNESDAY APRIL 20TH
MATH 435 SPRING 2011
For this homework, we assume all rings are commutative, associative and with multiplicative
identity. We assume that all homomorphisms send 1 to 1.
1. Suppose k is a eld and f (x) and g (x) are e
HOMEWORK # 9
DUE WEDNESDAY MARCH 30TH
MATH 435 SPRING 2011
In this homework assignment, all rings with be commutative associative with unity (multiplicative identity).
Ring homomorphisms will always be assumed to send 1 to 1. The terms that appear on this
INFO ON THE FINAL EXAM
MATH 435 SPRING 2011
There will be 6 pages of regular questions on the exam and one extra-credit question.
(1) There will be two pages of short answer questions (for example, dene the term subgroup,
ideal, Noetherian ring, transcend
FIELD EXTENSION REVIEW SHEET
MATH 435 SPRING 2011
1. Polynomials and roots
Suppose that k is a eld. Then for any element x (possibly in some eld extension,
possibly an indeterminate), we use
k [x] to denote the smallest ring containing both k and x.
k (
WORKSHEET # 7
MATH 435 SPRING 2011
In this worksheet, well learn about factoring elements in abstract rings. For this worksheet, we
follow Rotmans denition of a ring. In particular, all rings are commutative, associative, and have
a multiplicative identit
WORKSHEET # 6
MATH 435 SPRING 2011
In this worksheet we will learn about subrings.
We will follow the books (Rotmans) notation. In particular, ALL rings will be commutative,
associative, and with multiplicative identity.
Denition 0.1. Given a ring R a sub
WORKSHEET # 2
MATH 435 SPRING 2011
Denition 0.1. A permutation Sn is called even if it can be written as a product of an even
number of transpositions (ie, cycles of the form (ij ). A permutation Sn is called odd if it isnt
even.
1. Set An to be the set o
SOLUTIONS TO FIELD EXTENSION REVIEW SHEET
MATH 435 SPRING 2011
1. Polynomials and roots
Exercise 1.1. Prove that Q[i] = Q(i).
Solution: We need to prove that Q[i] is a eld. So choose a + bi Q[i], with a + bi = 0.
a
Then 1/(a + bi) = (a bi)/(a2 + b2 ) = a2
MATH 435, EXAM #1
Your Name
You have 50 minutes to do this exam.
No calculators!
No notes!
For proofs/justications, please use complete sentences and make sure to explain any steps which
are questionable.
Good luck!
Problem
Total Points
1
30
2
25
3
20
4