This is somehow related to (or maybe a simplified version of) an earlier question (see here) regarding Gersten complexes for singular varieties. The Gersten complexes arise from the coniveau spectral sequence, and at some crucial point, devissage is used: for a closed immersion $Z\hookrightarrow X$ of codimension $c$ between smooth schemes, we can identify $K_n(X {\rm on } Z)\cong K_{n-c}(Z)$. Essentially, this follows from the existence of a normal bundle, whose Thom space looks, to the eyes of an orientable theory like K-theory, just like a suspension of $Z$.

Now we all know that devissage fails when the smoothness condition above is not met. But do we know how badly it fails?

The topological picture I have in mind is that of an isolated singularity, algebraic, so that I still have local contractibility for the complex points. The cohomology of a contractible neighbourhood of the singularity with support in the singular point is the shifted cohomology of the link. (In the smooth case, the link is a sphere, tying up with the devissage picture above.) So I would expect that the failure of devissage in algebraic K-theory could somehow be related to the topology of links of singular points. But I don't even know how to precisely say what "topology of links" should be.

Does anyone know about work in this spirit, describing the failure of devissage in terms of some invariants of the singularities. More specifically, of course, we can ask about isolated hypersurface singularities. Are there results in the literature about K-theory of isolated hypersurface singularities with support in the singular point? (And potential relations with Milnor fiber?)

Any hints welcome. Of course, comments on the big picture (if applicable), how to formulate "topology of links" and which algebraic invariants could be related to this are also welcome.

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    $\begingroup$ A very simple remark: if $Z$ and $X\setminus Z$ are regular then the negative relative $K$-theory groups are isomorpic to that of $X$, and there is some literature on negative $K$-theory. $\endgroup$ – Mikhail Bondarko Aug 14 at 16:20

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