# Is the algebraic Grothendieck group of a weighted projective space finitely generated ?

This is to be confronted with Joseph Gubeladze' paper : "Toric varieties with huge Grothendieck group" !

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By "Algebraic Grothendieck group", do you mean the Grothendieck group of vector bundles? –  David Speyer Jun 11 '13 at 17:40
Yes , it is the Grothendieck group of vector bundles. The one of coherent sheaves is well-known. –  Al-Amrani Jun 11 '13 at 17:51
It is mentioned in the above paper that it was the initial idea to construct such examples with huge $K_0$ among weighted projective spaces, but so far without success. –  Dietrich Burde Jun 11 '13 at 18:53
Yes, that is right. But recently, Adam Massey ( KH-Theory of Complete Simplicial Toric Varieties and Algebraic K-theory of Weighted Projective Spaces ) obtained some progress in very particular case of weights, that is (1,...,1,q). –  Al-Amrani Jun 11 '13 at 19:25
In other words, the question is an open problem that looks well known among the people that know this sort of problems. Notice there is a minisection in the FAQ mathoverflow.net/faq about this. –  Mariano Suárez-Alvarez Jun 11 '13 at 19:40