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I only know that the functorial property of $G$-theory for proper morphisms of Noetherian schemes implies that the induced diagram of $G$-theory spaces for the blow-up diagram of $X$ is commutative. …

Suppose that $k$ is an algebraically closed field and $A$ is the ring $k[a,b,c,d]/(ac-b^2,bd-c^2,ad-bc)$. Let $X$ be Spec$A$, and $m$ be the maximal ideal of $A$ generated by the quotient images of $a …

$\newcommand{\red}{\mathrm{red}}$Let $k$ be an algebraically closed field of characteristic zero and $m$ be a positive integer. Let $R$ be the subring $k[x,xy,xy^2,…,xy^m]$ of the polynomial ring $k[x …

Let $X$ be the fiber product of two copies of $\mathbb{P}^1_k$ over the affine scheme $\operatorname{Spec}(k)$.I am trying to compute the $G$-theory groups of the noetherian scheme $X$. …

By setting $U=U_{\sigma_0},V=U_{\sigma_1}$, we have that $\{U,V\}$ is an affine open cover of the noetherian scheme $X$, and the intersection $U\cap V$ is the affine toric variety $U_{\sigma_0\cap\sigma …