Some Results in Complete Intersections
964th AMS Meeting of AMS, March 30-31, 2001
University of Kansas, Lawrence, Kansas 66045

Theorem(Boratynski). Let A be a commutative ring and I be an ideal with

I=(f1, ¼, fk) + I2.

Let

J=(f1, ¼, fk-1) + I(k-1)!.

Then J is image of a projective module Q of rank k:

Q ® J

This theorem of Boratynski had a far reaching impact in the study of complete intersections in affine varieties. Mohan Kumar used this theorem of Boratynski to prove the following theorem.

Theorem(Mohan Kumar). Let A be a reduced affine ring with dim(A)= n over an algebraically closed field k, and let Q be a projective A-module of rank n. Suppose that Q maps onto a complete intersection ideal J = (f₁, ¼, f_n) of height n. Then Q = Q₀ ÅA.

This theorem of Mohan Kumar also had a far reaching consequence. Eventually, using Monhan Kumar's theorem, Murthy (Annals of Mathematics, 1994) proved :

Theorem(Murthy). Let A be a reduced affine ring with dim(A)= n over an algebraically closed field k, and let Q be a projective A-module of rank n.
Then Q = Q₀ ÅA if and only if the top Chern Class C_n(Q) = 0 in the Chow Group of zero cycles.

The theorem of the day that I want present is as follows:

Theorem. Let A be a noetherian commutative ring. Let r₁, ¼, r_k be nonnegative integers and k £ dim(A). Let f₁, ¼, f_k be a regular sequence in A. Suppose Q is a projective A-module of rank k. Suppose we have a surjective map

Q ® (f1r1, ¼, fkrk)

If (k-1)! divides r₁r₂¼ r_k, then [Q] = [Q₀Å A] in K₀(A), for some projecive A-module Q₀.

In fact, we give a more general theorem on such decomposition of projective modules as follows.

Theorem. Let A be a noetherian commutative ring. Let r₁, ¼, r_k be positive integers and k £ dim(A).
Let

J0 =(f1, ¼, fk) + J02.

be a locally complete intersection ideal of height k. Define

J1 =(f2, ¼, fk) + J0r1, J2 =(f1r1, f3, ¼, fk) + J1r2, ¼     ¼     ¼     J = Jk =(f1r1, f2r2, ¼, fk-1rk-1) + Jk-1rk .

( So, J = Jk =(f1r1, f2r2, ¼, fk-1rk-1 , fkrk) + J2).

Suppose that Q is a projective A-module of rank k and there is a surjective map :

f : Q ® J

Assume that (k-1)! divides r₁r₂¼ r_k. Then

[Q] = [Q0Å A] - (r1r2¼ rk)¤(k-1)!)[A/J0]

for some projective A-module Q₀ of rank k-1.

So, if (x, y, z) is a unimodular row then (x ², y, z) is the first row of an invertible matrix.

Suslin extended the above:

Theorem. Let A be a commutative ring. Let r₀, r₁, ¼, r_n be nonnegative integers. Suppose

(

x 0,

x 1,

x 2,

¼ ,

x n

)

is a unimodular row in A.

Assume that n! divides r₀r₁¼ r_n.

Then there is an invertible matirx a with it's forst row

(

x 0r0,

x 1r1,

x 2r2,

¼ ,

x nrn

)

Boratynski used this theorem of Suslin to prove his theorem. Another ingrediant is the Murthy's extension of Boratynski's theorem.

Theorem. Let A be a commutative ring and I be a locally complete intersection ideal of height k with

I=(f1, ¼, fk) + I2.

Let

J=(f1, ¼, fk-1) + I(k-1)!.

Then J is image of a projective module P of rank k with [P] - k = -[A/I] in K₀(A).

The proof of this theorem is done by looking at the "universal ring"

Ak = Z[X1, X2, ¼ ,Xk,Y1,Y2, ¼ ,Yk, Z] ¤( åXi Yi - Z(1+Z))

and it's Chow Ring.
I used the flowing "universal ring"

Bk = Z[X1, X2, ¼ ,Xk,Y1,Y2, ¼ ,Yk, S, T, U, V] ¤(SU - TV -1, åXi Yi - ST))

and it's Chow Ring.