I was wondering whether in general it is known that for an invertible prime $l$, the mod-$l$ algebraic $K$-group of a regular Noetherian scheme $X$ injects into the mod-$l$ etale $K$-groups?
I just thought that the localization automatically means injection but I am not sure about that. Does mod $l$ $K$-theory inject into the Bott inverted mod $l$, $K$-groups? Is it possible inverting the Bott element send certain elements to zero?
There are some conditions on $X$ given at Theorem 2.45, which implies that mod $l$ etale $K$-groups coincide with a localization of mod $l$ algebraic $K$-groups, so the natural map from algebraic to etale is injective. The problem is that I cannot comprehend all those conditions completely, when are they satisfied and how mild are they. I am interested in a case that $X$ is a regular projective scheme over $\text{Spec}(\mathbb{F}_p[[t]])$.
