Let $\textbf {X}$ be a noetherian scheme

$\textbf {M(X)}$ be the categroy of coherent sheaf over the scheme $\textbf {X}$

We denote $ \textbf {K$_0$(M(X))}$ to be $ \textbf {G$_0$(X)}$

Now I know that since $\textbf {X}$ is noetherian it has a finite cover by Spec(A$_i$) for some $i$ = 1,2,..n. and A$_i$ for each $i$ is noetherian.

and there is an equivalence between the categories of M(Spec(A$_i$)) and fintely generated A$_i$ modules.

I have proved that G$_0$ (A$_i$$_{red}$) $\cong $ G$_0$ (A$_i$) where G$_0$ (R) is K$_0$(M(R));

Notation M(R) = finitely generated R module (R -noetherian in this case).

Now Can I say this, that since every open set in the cover of X has G$_0$ (A$_i$$_{red}$) $\cong $ G$_0$ (A$_i$) then

G$_0$(X) $\cong$ G$_0$(X$_{red}$) .

So my query is to prove the above mentioned statement does it suffice to prove it for affine noetherian scheme? As I know that every open affine subset of a noetherian scheme is noetherian.

Let $\textbf {X}$ be a noetherian scheme,

$\textbf {M(X)}$ be the categroy of coherent sheaf over the scheme $\textbf {X}$.

We denote $ \textbf {K$_0$(M(X))}$ to be $ \textbf {G$_0$(X)}$.

Now I know that since $\textbf {X}$ is noetherian it has a finite cover by Spec(A$_i$) for some $i$ = 1,2,..n. and A$_i$ for each $i$ is noetherian.

and there is an equivalence between the categories of M(Spec(A$_i$)) and fintely generated A$_i$ modules.

I have proved that G$_0$ (A$_i$$_{red}$) $\cong $ G$_0$ (A$_i$) where G$_0$ (R) is K$_0$(M(R));

Notation M(R) = finitely generated R module (R -noetherian in this case).

Now Can I say this, that since every open set in the cover of X has G$_0$ (A$_i$$_{red}$) $\cong $ G$_0$ (A$_i$) then

G$_0$(X) $\cong$ G$_0$(X$_{red}$) .

So my query is to prove the above mentioned statement does it suffice to prove it for affine noetherian scheme? As I know that every open affine subset of a noetherian scheme is noetherian.