### Create a StudySoup account

#### Be part of our community, it's free to join!

Already have a StudySoup account? Login here

# Special Topics Algebraic Combinatorics MATH 996

KU

GPA 3.6

### View Full Document

## 36

## 0

## Popular in Course

## Popular in Mathematics (M)

This 5 page Class Notes was uploaded by Edgar Jacobi on Monday September 7, 2015. The Class Notes belongs to MATH 996 at Kansas taught by Staff in Fall. Since its upload, it has received 36 views. For similar materials see /class/182393/math-996-kansas in Mathematics (M) at Kansas.

## Reviews for Special Topics Algebraic Combinatorics

### What is Karma?

#### Karma is the currency of StudySoup.

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 09/07/15

Chem Classes by Induction Last Lecture Satya Mandal May 2005 Suppose X is an algebraic Scheme over a eld With dimX n and 5 be a locally free sheaf of rank 7 We Will try to de ne all the chem classes of 5 Notation 01 1 ATX will denote the Chow group of codimension 7 cycles and 2 ATX will denote the Chow group of dimension 7 cycles 3 AX 0ATX ZOATX will denote the total chow group 1 Nonsingular Case 1 We Will assume X is smooth and hence the total Chow group AX i xAkX is a GRADED ring 2 We want the Chern classes to have the following properties a the k 7 th Chern class 0195 6 AWX So7 0195 0 for k gt dimX Also 005 1 b C d The total Chern class of 5 Will be denoted by 05 1Cl5 025 e So7 the total Chern class is an UNlT in AX f Given any exact sequence Gasasas a0 We must have the 05 C5 C5 g For a rank one locally free sheaf on X we have de ned the rst chern class Cquotl 7 as in the book of Fulton ln fact7 if is isomorphic to the invertible sheaf of ideals I Which is not alaways the case then 015 magi I The total Chern class of is 051 015 h Fullback Let us write X1 ProjSymm5 and p X1 a X be the projection map Then the pullback must commute With chern classes That means Cp f WOW 0R 0191 p0k5 i Let p X1 a X be as above Then the pullback map AX AX1 is an injective map of GRADED rings See page 15 of Mohan Kumar7s Note 3 Splitting Principle Let X1 ProjSymmS and p X1 a X be the projection map Then there is an exact sequence 0a gtpggt0lgt0 Where the kernel 5 is7 clearly7 a locally free sheaf of rank 7 7 1 2 4 Inductive De nition Use the injectivity of AX AX1 and de ne the total Chern class Here n 01ltOlt1gtgt CyczewI Where I 2 gtosymm 5w U Exercise Let f be a FREE sheaf of rank 7 over X Prove that the total Chern class Cf 1 G Exercise Let 5 be a locally free sheafofrank 7 Prove that 0195 0 for all k gt 7 1 Exercise Let 5 be a locally free sheaf of rank 7 over X It needs a proof that 0195 6 AkX 2 First and the Top Chern Class As above7 suppose X is an algebraic Scheme over a eld With dimX n and 5 be a locally free sheaf of rank 7 The 7 Chern class CNS of 5 Will be called the TOP Chern class of 5 1 Description of the rst Chern class is given by 015 01det5 For the right hand side7 we have to look at an invertible subsheaf of KX that is isomorphic of 16755 OR the Cartier divisor corresponding to 16755 2 OJ 3 For simplicity7 assume that X SpecA and dimA 71 Now let P be a projective Aimodule of rank 7 To describe the top Chern class of P we do the following Let A P 7 I Q A be surjective linear map7 where I is a locally complete intersection ideal of height 7 Such maps and ideals exist The CTP 71TCycleAI AND CTP CycleAI where P H07nP7 A Same can be done for non a ine schemes Let 5 be a locally free sheaf on a scheme X Let 3 E P57X be a global section7 such that Y m E X 8 0 is a locally complete intersection subscheme7 of codimension 7 7 of X Such sections may not exist Then the top Chern class of 5 is given by CT 5 cycleY The Singular Case Now we assume that X is not necessarily nonsingular H N3 03 So7 the total Chow group AX AWX does not have a ring struc ture De nitionA group homomorphism Lp AX 7 AX is said to be a graded homomorphism of degree d7 if gpATX Q ATdX for all 70172 For a cartier diVisor or a line bundle D7 intersection was de ne D AX 7 AX as a homomorphism of degree one see Section 23 of Fulton Let GrHomAX OHOWTAX denote the group of all graded homomorphisms7 where HomTAX is the group of homomorhisms of degree 7 4 U 53gt 1 00 H H Note that GrHomAX has a graded ring structure under composition also note that D f E Hom1AX de ne total Chern class of a line bundle L as CL101L1Dm This is an element in 1 Hom1AX Q GrHomAX For a locally free sheaf 5 of rank 7 total Chern class is de ned 051 015 CT5 where 0195 6 HankRX is a homomorphism of degree k The rest is using induction as above in the nonsingular case For our purpose7 GrHomAX behaves quite like the Chow group AX in nonsingular case For locally free sheaf 5 of rank 73 we will use the above exact sequence and de ne the total chern class C5 CO1 C5 1 77 where 77 AX1 a AX1 is the rst chern class of 91 12 It needs a proof to show that 0195 AX a AX I did not have chance to proof read Thanks you all

### BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.

### You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

#### "Knowing I can count on the Elite Notetaker in my class allows me to focus on what the professor is saying instead of just scribbling notes the whole time and falling behind."

#### "I signed up to be an Elite Notetaker with 2 of my sorority sisters this semester. We just posted our notes weekly and were each making over $600 per month. I LOVE StudySoup!"

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

#### "Their 'Elite Notetakers' are making over $1,200/month in sales by creating high quality content that helps their classmates in a time of need."

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.