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$$K_ 0$$ of a union of planes through the origin. (English) Zbl 0537.13010
Let A be the coordinate ring of a union of 2-dimensional planes through the origin of $${\mathbb{A}}^ p_ k$$ where k is a field of characteristic 0. (Alternately, A is the homogeneous coordinate ring of a union X of straight lines in $${\mathbb{P}}_ k^{p-1}.)$$ This paper is largely devoted to the calculation $$K_ 0(A)={\mathbb{Z}}\oplus^+A/A\oplus SK_ 0(A)$$ where $${}^+A$$ is the seminormalization of A and $$SK_ 0(A)$$ is a quotient of $$H\otimes \Omega_ k$$ where $$\Omega_ k$$ is the module of absolute Kähler differentials and H is a k-module which can be described combinatorially. It is shown that A is $$K_ 0$$-regular if and only if A is seminormal and $$H=0$$. It is now known that when A is seminormal, $$H=0$$ if and only if the sheaf cohomology group $$H^ 1(X,{\mathcal O}_ X(1))=0.$$ This characterization of $$K_ 0$$-regularity will appear in a joint paper of L. G. Roberts and the author.

##### MSC:
 13D15 Grothendieck groups, $$K$$-theory and commutative rings 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry 14C22 Picard groups 13F20 Polynomial rings and ideals; rings of integer-valued polynomials
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