Let $X$ be a smooth complete intersection in $\mathbb{P}^n$. I am searching for literature on the Ktheory for $X$? I guess the Ktheory is known...
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As suggested by Daniel Loughran, I will try to answer the question whether $K_0(X)$ is finitely generated. First, the Chern character from $K_0(X)$ to the Chow ring $CH(X)$ is an isomorphism modulo torsion: see for instance Fulton, Intersection Theory, Cor. 18.3.2. Thus we can transfer the question to $CH(X)$, and use this question. So, at least conjecturally, for a complete intersection $X$ the group $K_0(X)$ is finitely generated iff $h^{p,q}(X)=0$ for $p\neq q$. This holds if and only if $X$ is a quadric, a cubic surface, or an evendimensional intersection of 2 quadrics (see M. Rapoport, Complément à l'article de P. Deligne "La conjecture de Weil pour les surfaces K3", Inv. math. 15 (1972), 227236). 

