It is one of the important facts in K-theory/motivic cohomology that the Gersten-type complexes (for Quillen K-theory, Milnor K-theory or more generally Rost's cycle modules) are exact for smooth local schemes. (I'd like to think of this as an algebraic analogue of the local contractibility of manifolds in differential topology.) Mostly out of curiosity I wonder what is known about the failure of exactness for local schemes which are singular.

What aspects of singularity theory are detected by the non-exactness of the Gersten complex?

To convey a feeling what I mean by that: are there conditions on a local singularity that still imply exactness of the Gersten complex? Are there results relating properties of the singularities (like those relevant for the minimal model program) to properties of the Gersten complex for K-theory (like vanishing of specific cohomology groups)? References to instructive examples from the literature would also be very welcome.

For K-theory, I suppose the Brown-Gersten-Quillen spectral sequence (starting from Gersten complexes and converging to G-theory) indicates that the question above is also related to the difference between K-theory and G-theory. (That relates to this unanswered MO-question.) By Corollary V.9.6.4 of Weibel's K-Book, Gersten complexes for mod $\ell$ K-theory are exact even for singular local schemes (essentially of finite type), so the homology of the Gersten complex for singular local schemes should be rational vector spaces. This suggests speculating about relations between homology of the Gersten complexes for K-theory and some of the differential forms invariants arising in trace methods. (But I'm not aware of any explicit statements)

Can something general be said about the cohomology of the Gersten complex?

For instance: does the homology depend only on the homotopy type of the dual intersection complex of a resolution of singularities? If so, could the homology be described in terms of this? I would also be interested in any philosophical explanations what should/shouldn't be true in this situation.


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