A regular local ring can lose integral K-theory at its fraction field

← All digests

Family 209 · Algebra
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-21

1. The question

A local ring describes geometry near one point. Its fraction field describes the generic point, where all nonzero denominators have been allowed. Passing to that field discards special loci, so some information can disappear. Gersten’s conjecture predicts a particularly strong form of control over this process for algebraic K-theory of regular local rings.

The first requirement is injectivity: a K-theory class that becomes zero over the fraction field should already have been zero in the local ring. Family 209 contains two manuscripts claiming explicit failures of that requirement in ramified mixed characteristic.

M1, dated 25 September 2026, constructs a kernel in integral $K_5$. M2, dated 26 September, constructs one in integral $K_3$ at a different ring. The examples concern regular total rings with singular special fibers. Neither manuscript determines the smallest degree of possible failure or settles the rational-coefficient version.

2. Background and essential definitions

Regularity and ramification are different conditions

A local ring $A$ has a unique maximal ideal $\mathfrak m$. Its residue field is $A/\mathfrak m$. A Noetherian local ring is regular when the minimum number of generators of $\mathfrak m$ equals its Krull dimension, the length of the longest chain of prime ideals.

Regularity is the algebraic counterpart of nonsingularity. In the examples, the dimension is two and the maximal ideal has two generators.

Mixed characteristic $(0,p)$ means that the ring has characteristic zero but its residue field has characteristic $p$. In this setting, the standard unramified condition is $p\notin\mathfrak m^2$. Ramification allows $p$ to lie deeper in the maximal ideal.

A regular ring can be ramified. It can also fail to be smooth over a chosen discrete valuation ring. The special fiber, obtained by setting the base uniformizer to zero, can be singular even when the total local ring is regular. These distinctions explain why the examples do not contradict smooth or unramified positive results.

What the K-groups measure

Algebraic K-theory assigns abelian groups $K_n(A)$ to a ring. The lowest group $K_0$ records finitely generated projective modules and their direct-sum relations. Over a field $F$, $K_1(F)=F^\times$, its multiplicative group. Higher groups encode more elaborate compatibility information among matrices and algebraic relations.

The digests need no construction of the full K-theory spectrum. The useful properties are functoriality, localization and products: ring maps induce maps on K-groups, removing a closed set fits into exact sequences, and classes can be multiplied.

For a regular local domain with fraction field $F$, the beginning of the Gersten complex is

$$0\longrightarrow K_n(A)\longrightarrow K_n(F) \longrightarrow\bigoplus_{\operatorname{ht}\mathfrak p=1} K_{n-1}(\kappa(\mathfrak p))\longrightarrow\cdots.$$

The fields $\kappa(\mathfrak p)$ belong to codimension-one points, and the arrows after the generic term are residue maps. Exactness at the initial term requires $K_n(A)\to K_n(F)$ to be injective.

Integral classes versus finite coefficients

“Integral K-theory” means the original abelian groups, before reducing coefficients or tensoring with $\mathbb Q$. Finite-coefficient K-theory is not simply the quotient group in every degree. There is an exact sequence

$$0\longrightarrow K_n(A)/5K_n(A) \longrightarrow K_n(A;\mathbb Z/5) \longrightarrow K_{n-1}(A)[5]\longrightarrow0,$$

where $K_{n-1}(A)[5]$ denotes elements killed by five. A coefficient class can therefore fail to lift to an integral class. Even a lift may have a nonzero but five-divisible generic image. An integral counterexample must address both issues.

3. Development history and previous partial results

Gersten’s questions about transfer and higher K-theory led to the localization problem. Quillen formulated the regular-local conjecture and proved it for regular semilocalizations of finite-type algebras over a field. Sherman treated the equicharacteristic discrete valuation ring case, and Panin established the full equicharacteristic theorem.

Mixed characteristic requires additional care. Gillet and Levine developed a relative theorem reducing smooth local situations over a discrete valuation ring to the associated valuation-ring problem. Geisser and Levine proved finite-coefficient exactness for semilocal rings essentially smooth over such a base, including coefficients of residue-characteristic order.

Skalit proved finite-coefficient exactness for arbitrary unramified regular local rings. The condition excluding deep ramification is therefore mathematically consequential, not an incidental technicality. The new examples deliberately lie outside it.

The manuscripts also discuss Druzhinin’s stated smooth Henselian result. Smoothness over the base excludes the structure maps in the new examples, so that statement has a different scope.

A particularly close predecessor is Feld’s finite-coefficient example in ramified mixed characteristic. It gives a degree-two kernel modulo three at a ring whose integral $K_2$ map is injective. This demonstrates why a coefficient counterexample alone does not disprove integral Gersten injectivity. An integral lift there can retain a nonzero generic image divisible by three.

The new papers adapt the strategy of pushing a class from a divisor and detecting it after a suitable base change or localization. Their claimed advance is to arrange an integral class whose generic restriction is already zero, and only then use finite coefficients to detect that the class is nonzero.

4. What the new manuscripts claim

M1: a quartic example in degree five

Let $V_0$ be the ring of integers in the unramified degree-eight extension of $\mathbb Q_5$. Define

$$A_5=\left(V_0[x,y]/(5+x^4+y^4)\right)_{(5,x,y)}.$$

The subscript means localization at the displayed maximal ideal. M1 asserts that $A_5$ is a two-dimensional Noetherian regular local domain and

$$\ker\bigl(K_5(A_5)\longrightarrow K_5(\operatorname{Frac}A_5)\bigr)\ne0.$$

Its maximal ideal is generated by $x,y$, and $5\in(x,y)^4$. Thus the ring is ramified and is not smooth over $V_0$.

M2: a cyclotomic quintic example in degree three

Fix a primitive fifth root of unity $\zeta$, put $L=\mathbb Q_5(\zeta)$, let $V$ be its ring of integers, and write $\pi=1-\zeta$. Consider the projective plane curve model over $V$ defined by

$$(X-Z)^5-XY^4+Y^5+\zeta Z^5=0.$$

Let $A_3$ be its local ring at the special-fiber point $[0:0:1]$. M2 asserts that $A_3$ is a two-dimensional regular local domain, with $\pi\in\mathfrak m^2$, and constructs a class $c\in K_3(A_3)$ satisfying

$$c|_{\operatorname{Frac}A_3}=0, \qquad c|_{A_3[1/\pi]}\bmod5\ne0.$$

The latter nonvanishing is stated in coherent K-theory with coefficients, denoted $G_3(-;\mathbb Z/5)$. Coherent K-theory permits pushforwards even when auxiliary curves are singular; on the regular rings at issue it agrees with ordinary K-theory.

The two constructions are independent and concern different rings. A degree-three example is a lower-degree failure than a degree-five example, but it does not establish that degree three is the first possible failure in every regular local setting.

5. Why the advance matters

Importance: 5/5, provisional. The family claims counterexamples to the unrestricted integral form of a foundational conjecture connecting local algebra, K-theory and residue geometry.

Failure occurs at the very first injection, before the more elaborate residue terms become relevant. It would show that regularity alone does not prevent integral K-theory information from disappearing at the generic point in ramified mixed characteristic.

The examples also clarify the role of coefficients. The detecting calculation is performed modulo five, but the objects detected are integral classes already known to vanish generically. That logical order is what separates these claims from earlier finite-coefficient failures.

The rating concerns the unrestricted integral conjecture. The examples leave important positive theories intact and do not justify describing every form of Gersten exactness as false.

6. The main idea

Support guarantees generic vanishing

A class pushed forward from a closed divisor is supported on that divisor. After passing to the fraction field, the divisor disappears, so the pushed-forward class becomes zero.

This supplies one half of the desired counterexample automatically. The hard tasks are to produce the divisor class integrally and to show that its pushforward is not already zero on the local ring.

M2 arranges an integral lift through a Steinberg relation

On the affine chart $Z\ne0$, M2 uses the divisor $y=0$. Its local ring $S=A_3/yA_3$ is a discrete valuation ring with parameter $t$. Its fraction field $D$ is a totally ramified degree-five extension of $L$, with

$$N_{D/L}(t)=\pi,\qquad (1-t)^5=\zeta.$$

A Bott-type class $\beta\in K_2(L;\mathbb Z/5)$ has coefficient boundary $[\zeta]$. Multiplying its restriction to $D$ by the unit class $[t]$ produces a degree-three coefficient class.

The obstruction to lifting this class integrally is killed by the displayed fifth-power relation and the Steinberg identity

$$\{1-t,t\}=0$$

in $K_2(D)$. The braces denote the standard multiplicative symbol in degree two. Quillen’s finite-field vanishing $K_2(\mathbb F_5)=0$ then allows the integral class to extend from $D$ to $S$. Pushing it into $A_3$ produces the candidate $c$ with zero generic restriction.

A norm valuation detects nonvanishing

The special fiber is designed so that every generic-curve point specializing away from $[0:0:1]$ has residue degree divisible by five. The distinguished point on $y=0$ has residue degree one.

After identifying degree-three coefficient classes over the relevant local fields with units modulo fifth powers, pushforward corresponds to field norm. Normalized valuation gives

$$v_L(N_{E/L}(u))=[k_E:\mathbb F_5],v_E(u).$$

Thus points specializing elsewhere contribute zero modulo five. The distinguished class contributes one because its norm is $\pi$.

If the supported class vanished on $A_3[1/\pi]$, localization would express it using points omitted from a suitable affine open. Those points all specialize elsewhere. Proper pushforward and the norm valuation would then equate one with zero modulo five. This is the manuscript’s nonvanishing mechanism.

M1 uses an arithmetic lift and a Picard test

M1 follows a different route. It starts with a cyclotomic coefficient class involving two Bott factors and proves that its obstruction to lifting into integral $K_5$ vanishes. Localization extends the class to a valuation ring, which appears as a divisor of $\operatorname{Spec}A_5$.

After a flat cyclotomic extension, that divisor splits into four points on a Fermat quartic. A specialization criterion translates any hypothetical relation among their supported coefficient classes into a relation in a special-fiber Picard group, the group of line bundles or divisor classes.

A degree-two map from the quartic to a genus-one curve supplies the detector. The four specialized points map to $Q$ or $-Q$, with signs matching the squared cyclotomic character contributed by the two Bott factors. Their weighted sum is $4Q$, nonzero modulo five. This detects the already integral, generically vanishing class.

7. Scope, limitations, and open questions

Both examples use ramified regular local rings in mixed characteristic $(0,5)$, and both fail generic injectivity. They do not contradict the equicharacteristic theorem or the cited smooth and unramified finite-coefficient results.

The nonzero coefficient reductions show that the constructed integral classes are not five-divisible in the relevant K-groups. They do not determine the classes’ orders. In particular, they do not establish that the classes remain nonzero after tensoring with $\mathbb Q$. The rational-coefficient conjecture is therefore not settled by these examples.

Neither manuscript determines the smallest degree in which integral injectivity can fail. M2’s auxiliary projective generic curve need not be smooth, which is why coherent K-theory is used for its pushforward argument.

M1 includes a separate discussion of a proposed proof in earlier literature. That appendix is not an input to its kernel construction. The mathematical content summarized here is the stated counterexample and its detection, without relying on an independent assessment of that earlier proof.

8. Sources and further reading

Gersten, Problems about higher K-functors, and Quillen, Higher algebraic K-theory: I, appear in Lecture Notes in Mathematics 341 (1973). Panin, The equicharacteristic case of the Gersten conjecture, Proceedings of the Steklov Institute of Mathematics 241 (2003), 154–163, gives the equicharacteristic theorem.

Gillet and Levine, The relative form of Gersten’s conjecture over a discrete valuation ring: The smooth case, Journal of Pure and Applied Algebra 46 (1987), 59–71, supplies the relative framework. Skalit, Regular morphisms and Gersten’s conjecture, states the unramified finite-coefficient result discussed above.

Feld, Finite-coefficient Gersten injectivity fails in ramified mixed characteristic, is the close coefficient-level predecessor. Weibel, The K-book: An introduction to algebraic K-theory (2013), provides background on localization, symbols and coefficient sequences.

Source manuscripts

• M1. OpenAI, An integral counterexample to Gersten’s conjecture (September 25, 2026).
• M2. OpenAI, An integral degree-three Gersten counterexample (September 26, 2026).

Sources consulted on 21 October 2026 at repository revision 1581d4c. Download the manuscript citations.