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I do not have the reference with me right now, but I think the localization sequence for K-theory over general base was handled in: R. W. Thomason, T. Trobaugh, Higher algebraic K-theory of schemes …

Let $X$ be a smooth projective variety over $\mathbb C$ with $d = \dim X \geq 1$. Let $CH(X)$ denotes the total Chow group of (cycles modulo rational equivalences of) $X$ and $CH(X)_{\mathbb Q} = CH( …

This question is motivated by some issue raised by David Speyer in this question. Let $R$ be a ring. Let $K_0(R)$ and $G_0(R)$ be the Grothendieck groups of f.g. projective modules and f.g. modules …

Let $f: R\to S$ be a map between two commutative Noetherian rings. Let $G_0(R)=K_0(mod R)$ be the Grothendieck group of finite generated modules over $R$. It means $G_0(R)$ is the quotient of the free …

A nice property of $\mathbb P^n$ is: Property 1: Two subvarieties $U,V$ such that $\operatorname{dim} U +\operatorname{dim} V \geq n$ always intersect. (for example, any 2 curves in $\mathbb P^2 …

Assume $k= \mathbf C$ and $W$ homogeneous. Let $X=Proj (k[x_1,\cdots,x_n]/(W))$. $X$ is then a smooth hypersuraface in $\mathbb P_{n-1}$. Assume $n=2d$ is even. Corollary 3.10 of the paper you quote …