Gersten conjecture

Gersten conjecture
Name	Gersten conjecture
Field	Algebraic K-theory
Introduced	1973
Proposer	Stephen Gersten
Status	Partially resolved; counterexamples known
Keywords	K-theory, regular local rings, Gersten complex, Quillen, Bloch, Suslin

Gersten conjecture The Gersten conjecture is a statement in Algebraic K-theory about the exactness of a canonical complex of sheaves on the Zariski site of a regular scheme. It connects foundational work of Quillen on higher K-groups, constructions of Bloch and Beilinson, and localization techniques used by Thomason and Trobaugh. The conjecture influenced developments involving Milnor K-theory, Suslin, and computations in arithmetic geometry associated with Grothendieck.

Statement

The conjecture asserts that for a regular local ring R the Gersten complex, built from the K-groups K_n of residue fields along codimension filtrations, is exact; this statement references constructions by Quillen and localization sequences similar to those in work of Bass and Matsumura. In concrete terms the claim concerns the exactness of the sequence of abelian groups formed from K-theory of function fields, discrete valuation rings, and residue fields indexed by codimension, echoing techniques of Gabber and compatibility with purity principles studied by Deligne and Hartshorne. Variants consider étale sheaves and coefficients relating to results of Suslin and formulations paralleling conjectures of Beilinson and Bloch on motivic cohomology.

Historical background and motivation

The conjecture originated in surveys and unpublished notes of Stephen Gersten motivated by computations in algebraic K-theory and the localization theorems of Quillen. It drew upon earlier work by Bass on K_0 and K_1, the homotopy-theoretic methods of Atiyah and Hirzebruch, and the use of spectral sequences as in contributions by Adams and Brown. Interest accelerated after foundational advances by Thomason and Trobaugh on K-theory of schemes and after techniques from Gabber on absolute purity and from Geisser and Levine on motivic cohomology began to clarify links with Bloch's higher Chow groups.

Known cases and partial results

Exactness was established in many cases: for regular local rings containing a field by work of Quillen, Suslin, and Gabber; for equicharacteristic regular schemes via methods of Morrow and Kerz building on advances by Panin and Fasel; and for certain mixed characteristic cases after results by Kerz and Saito informed by techniques from Kato and Niziol. Further progress used comparison theorems of Geisser and Levine relating K-theory to motivic cohomology, input from Voevodsky on homotopy invariance, and theorems by Cortinas and Weibel on negative K-theory. Additional verifications cover low-degree K-groups following classical analyses by Bass and computations by Milnor.

Counterexamples and limitations

Counterexamples to naive generalizations appear in pathological situations inspired by constructions in Nagata-type examples, observation by Roberts in mixed characteristic, and subtleties identified by Hesselholt and Madsen when topological cyclic homology comparisons fail. Limitations are evident for singular schemes or non-noetherian rings as highlighted in works of Weibel, Thomason, and Rosenschon, and certain torsion phenomena were revealed through computations influenced by Suslin and by obstruction-theoretic analyses related to Bloch and Kato. These results show the necessity of hypotheses such as regularity, finite Krull dimension, or coefficient restrictions used in proofs by Gabber and Kerz.

Methods and techniques

Proof techniques draw on localization sequences of Quillen, devissage arguments originating with Bass and Matsumura, and purity results proven by Gabber and refined by Quillen in influential work relying on homotopical methods of Adams and Boardman. Étale descent and comparison with motivic cohomology use tools from Voevodsky's theory, cycles and higher Chow groups developed by Bloch and computations by Levine and Geisser. Modern approaches exploit topological cyclic homology and trace methods introduced by Bökstedt and developed by Hesselholt and Madsen, together with alterations techniques due to de Jong and categorical methods building on Thomason and Trobaugh.

Applications and related conjectures

The Gersten conjecture underpins computations in arithmetic algebraic geometry connected to conjectures of Beilinson on regulators, influences formulations of the Bloch–Kato conjecture later proven by work of Voevodsky and Rost, and informs structural results for motivic complexes used by Levine and Suslin. It plays a role in Riemann–Roch type theorems originating from Grothendieck and in purity conjectures related to Gabber and Deligne. Related open problems include comparisons between algebraic K-theory and topological cyclic homology studied by Hesselholt and Nikolaus, and refinements in the direction of Bass negativity conjectures and questions arising in the program of Bloch and Beilinson on motivic filtrations.

Category:Algebraic K-theory