Complete Intersections K-Theory and Chern Classes

Abstract

Throughout this abstruct $A$ will denote a noetherian commutative ring of dimension $n$. The paper has two parts. Among the interesting results in Part-1 are the following: 1) {\it suppose that $f_1, f_2, ..., f_r$ (with $r \leq n$) is a regular sequence in $A$ and suppose $Q$ is a projective $A$-module of rank $r$ that maps onto the ideal $(f_1, f_2, ..., f_{r-1},f_r^{(r-1)!})$. Then $[Q]=[Q_0 \oplus A]$ in $K_0(A)$ for some projective $A-module~Q_0$ of rank $r-1$.} 2) The set $$F_0K_0(A) = \{[A/I] \in K_0(A): I~ is~ a~ locally~ complete ~intersection~ ideal~ in~ A~ of~ height~n \}$$ is a {\it subgroup} of $K_0(A)$. We also show that if $A$ is a reduced affine algebra over a field $k$ then $F_0K_0(A)$ {\it is indeed the Zero Cycle Subgroup of} $K_0(A)$ {\it that is generated by smooth maximal ideals} $\Cal M$ {\it of height} $n$. 3){\it let $A$ be such that whenever $I$ is a locally complete intersection ideal of height $n$ with $[A/I]=0$ then $I$ is the image of a projective $A-module$ of rank $n$. Then for any locally complete intersection ideal $J$ of height $n$ with $[A/J]$ divisible by $(n-1)!$ in $F_0K_0(A)$, there is a projective $A-module$ of rank $n$ that maps onto $J$}. The main result in Part-2 is the following construction: 1) {\it let $X=Spec A$ be a Cohen-Macaulay scheme of dimension $n$ and let $r_0,~r$ be two integers with $n/2 \leq r_0 \leq r \leq n$. Let i) $Q_0$ be a projective $A-module$ of rank $r_0-1$ such that the restriction $Q_0|Y$ is trivial for all locally complete intersection subvarieties $Y$ of codimension at least $r_0$. Also ii) for $k= r_0$ to $r$, let $I_k$ be locally complete intersection ideals of height $k$ so that $I_k/I_k^2$ has a generators of the type $f_1, ..., f_{k-1}, f_k^{(k-1)!}$. Then there is a projective $A-module~Q$ of rank $r$ such that