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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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