Let's assume we have a sheaf of spectra on some scheme. As an example I will assume that we are working with the $K$-theory sheaf. There are certain local to global spectral sequences, like descent spectral sequence. If one knows the algebraic $K$-groups on some open cover and the intersections of the opens, then it is possible to use the descent spectral sequence to recover the algebraic $K$-group globally. I was wondering whether it is possible to recover the algebraic $K$-groups by knowing its values on local rings?
Another way of looking at the problem is the following: Let's assume for two schemes $X$ and $Y$ we have a morphism $f:X\rightarrow Y$. Assume we have two open coverings of $X$ and $Y$ denoted by $U_{\alpha}^X$ and $U_{\alpha}^Y$ in a way that $f$ induces a map from $\bigcap_{\alpha\in I}U_{\alpha}^X$ to $\bigcap_{\alpha\in I}U_{\alpha}^Y$ ($I$ is a finite collection of indices) which that map induces isomorphism of algebraic $K$-groups then by descent spectral sequence this implies that $f$ induces isomorphisms $K_n(X)\cong K_n(Y)$.
Now instead of working with an open cover, assume $f$ induces isomorphism of algebraic $K$-groups on the local rings of $X$ and $Y$, is it necessarily true that $K_n(X)\cong K_n(Y)$.
Another version: Is it enough these isomorphisms to happen for the local rings of closed points?
