Satyagopal Mandal |
Department of Mathematics |
Office: 624 Snow Hall Phone: 785-864-5180 |
Theorem(Boratynski). Let A be a commutative ring and I be an ideal with
I=(f_{1}, ¼, f_{k}) + I^{2}. |
J=(f_{1}, ¼, f_{k-1}) + I^{(k-1)!}. |
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 = (f_{1}, ¼, f_{n}) of height n. Then Q = Q_{0} 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 = Q_{0} 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
r_{1}, ¼,
r_{k}
be nonnegative integers and k £ dim(A).
Let f_{1}, 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 ® (f_{1}^{r1}, ¼, f_{k}^{rk}) |
If (k-1)! divides r_{1}r_{2}¼ r_{k}, then [Q] = [Q_{0}Å A] in K_{0}(A), for some projecive A-module Q_{0}.
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
r_{1}, ¼,
r_{k}
be positive integers and k £ dim(A).
Let
J_{0} =(f_{1}, ¼, f_{k}) + J_{0}^{2}. |
J_{1} =(f_{2}, ¼,
f_{k}) + J_{0}^{r1}, J_{2} =(f_{1}^{r1}, f_{3}, ¼, f_{k}) + J_{1}^{r2}, ¼ ¼ ¼ J = J_{k} =(f_{1}^{r1}, f_{2}^{r2}, ¼, f_{k-1}^{rk-1}) + J_{k-1}^{rk} . |
( So, J = J_{k} =(f_{1}^{r1}, f_{2}^{r2}, ¼, f_{k-1}^{rk-1} , f_{k}^{rk}) + J^{2).} |
Suppose
that Q is a projective A-module of rank k and there is a surjective
map :
f : Q ® J |
[Q] = [Q_{0}Å A] - (r_{1}r_{2}¼ r_{k})¤(k-1)!)[A/J_{0}] |
÷ |
| ÷ | = xw-yz |
| = (ax + by +cz) ^{2} |
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
r_{0}, r_{1}, ¼,
r_{n}
be nonnegative integers.
Suppose
( | x _{0}, | x _{1}, | x _{2}, | ¼ , | x _{n} | ) |
Assume that n! divides r_{0}r_{1}¼ r_{n}.
Then there is an invertible
matirx a with it's forst row
( | x _{0}^{r0}, | x _{1}^{r1}, | x _{2}^{r2}, | ¼ , | x _{n}^{rn} | ) |
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=(f_{1}, ¼, f_{k}) + I^{2}. |
J=(f_{1}, ¼, f_{k-1}) + I^{(k-1)!}. |
Then J is image of a projective module P of rank k with [P] - k = -[A/I] in K_{0}(A).
The proof of this theorem is done by looking at the "universal ring"
A_{k} = Z[X_{1}, X_{2}, ¼ ,X_{k},Y_{1},Y_{2}, ¼ ,Y_{k}, Z] ¤( åX_{i} Y_{i} - Z(1+Z)) |
B_{k} = Z[X_{1}, X_{2}, ¼ ,X_{k},Y_{1},Y_{2}, ¼ ,Y_{k}, S, T, U, V] ¤(SU - TV -1, åX_{i} Y_{i} - ST)) |