Class Note for MATH 996 at KU
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Date Created: 02/06/15
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 lEXAkX 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 b So7 0195 0 for k gt dimX c Also 005 1 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 cycze 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 1400 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 5 Exercise Let f be a FREE sheaf of rank 7 over X Prove that the total Chern class Cf 1 6 Exercise Let 5 be a locally free sheafofrank 7 Prove that 0195 0 for all k gt 7 7 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 ofdet5 OR the Cartier divisor corresponding to 16755 2 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 1 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 10 11 12 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 05 009415 C5CO1 C51 77 where 77 AX1 a AX1 is the rst chern class of 91 It needs a proof to show that 0195 AX a AX I did not have chance to proof read Thanks you all
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