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Gersten's conjecture for regular local rings

Let X=Spec⁡(R)X=\operatorname{Spec}(R) be the spectrum of a regular local ring RR. For each integer ll, let Ml(X)\mathrm{M}_l(X) be the category of coherent sheaves on XX whose supports have codimension at least ll, and let Kp(Ml(X))\mathrm{K}_p(\mathrm{M}_l(X)) be its Quillen K\mathrm{K}-group for p≥0p\geq 0. Gersten's conjecture. For every p≥0p\geq 0 and every ll, the homomorphism

Kp(Ml+1(X))⟶Kp(Ml(X))\mathrm{K}_p(\mathrm{M}_{l+1}(X))\longrightarrow \mathrm{K}_p(\mathrm{M}_l(X))

induced by the inclusion Ml+1(X)→Ml(X)\mathrm{M}_{l+1}(X)\to\mathrm{M}_l(X) is zero. This is the classical Gersten vanishing statement in Quillen algebraic K\mathrm{K}-theory for regular local rings; the candidate appears as the statement recalled in the discussion of the triangulated Gersten conjecture.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Gersten's conjecture for regular local rings

    Let RR be a commutative regular local ring. The conditions in the equivalent formulation above—equivalence of the relevant Quillen and Brown–Gersten–Thomason spectral-sequence terms, exactness of the Gersten sequence, and vanishing of the maps induced by MRp↪MRp−1\mathcal{M}_R^p\hookrightarrow\mathcal{M}_R^{p-1} on KK-theory—are the conditions under consideration.

    Gersten's conjecture. The conditions above are true for any commutative regular local ring.

    This is the central conjecture surveyed in the paper. The supplied text discusses implications from other conjectures and counterexamples when commutativity is dropped, but gives no resolution status for the commutative case.

    source: Satoshi Mochizuki, “A survey of Gersten's conjecture”, arXiv:1608.08114 (2016).

Sources & referencesView supporting material

Primary source

Sebastian Klein, “Intersection products for tensor triangular Chow groups”, arXiv:1505.07615 (2015).

algebraic-k-theorygersten-conjectureregular-rings

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