On the \(K\)-groups of spherical varieties (Q1392653)

Let \(B\) be a split connected solvable group over a field \(k\). Let \(X\) be a \(B\) variety with finitely many orbits. For \(i\in \mathcal{N}\), \(K_{i}'(X)\) (resp., \(K_{i}'(X,B))\)) denotes the \(i\)-th \(K\)-group of the category of coherent sheaves (resp., \(B\)-equivariant coherent sheaves) on \(X\). Furthermore, \(K_{i}(X,B)\) is the \(i\)-th \(K\)-group of the category of \(B\)-equivariant locally free sheaves on \(X\) and \(K_{0}'(X,B)\) is a \(K_{0}(k,B)\)-module. Suppose that \[ I =\{rm-r\mid r\in K_{0}(k,B),\;m\in K_{0}'(X,B)\} \] and \(K_{0}'(X,B)_{M}= K_{0}'(X,B)/I\). Then the main result of the paper shows the theorem: There is a natural isomorphism \[ K_{0}'(X,B)_{M}\rightarrow K_{0}'(X). \]