K-theory K-theory in geometry Conjecture

Gersten conjecture

Builds on Gersten complex Regular local ring Algebraic K-theory

Statement

Let $R$ be a regular local ring with field of fractions $F$ , and let $n \ge 0$ . For a prime ideal $\mathfrak{p}$ of $R$ let $k(\mathfrak{p})$ be its residue field, and let $K_m$ denote Quillen's algebraic K-groups, with $K_m = 0$ for $m < 0$ . Then the Gersten complex $0 \to K_n(R) \to K_n(F) \to \bigoplus_{\operatorname{ht} \mathfrak{p} = 1} K_{n-1}(k(\mathfrak{p})) \to \bigoplus_{\operatorname{ht} \mathfrak{p} = 2} K_{n-2}(k(\mathfrak{p})) \to \cdots$ is exact.

Remarks

In particular the conjecture says that $K_n(R) \to K_n(F)$ is injective. Quillen proved the conjecture when $R$ is a local ring of a smooth variety over a field, or more generally essentially smooth over a field. Panin extended it to all regular local rings that contain a field. For regular local rings of mixed characteristic it is open in general.

For $n = 1$ the conjecture says that a unit of $F$ with valuation $0$ at every height one prime lies in $R^\times$ . This holds because a regular local ring is a unique factorization domain.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Tag	`N0ZTHA8YBB`
URI	`m2://N0ZTHA8YBB`
Kind	conjecture
Writing	W0 stub
MSC	19E08, 19D45, 13D15
Status	This copy of the site has no build data. Run `m2 status N0ZTHA8YBB`.
About	Gersten complex, Algebraic K-theory
Uses	Gersten complex, Regular local ring, Algebraic K-theory
Changed	3 November 2026

Gersten conjecture

Statement

Remarks

Reading path

Further reading

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