Gersten's conjecture, stated in 1973 in the language of higher algebraic
K-theory, is a prediction about residue complexes on regular local rings. For a
regular local ring R with fraction field F and residue fields kappa(p),
kappa(q), ... at its points of increasing codimension, it asserts that the
localization sequence
0 -> K_n(R) -> K_n(F) -> sum_{p codim 1} K_{n-1}(kappa(p))
-> sum_{q codim 2} K_{n-2}(kappa(q)) -> ...
is exact in every degree n >= 0. The first and most basic clause is injectivity
of K_n(R) -> K_n(F): a K-class that dies on passage to the fraction field should
already have been zero on R. That clause is what fails below.
Gersten, Problems about higher K-functors, LNM 341 (1973) 43-56, doi:10.1007/BFb0067050.
The conjecture is known in many regimes. Quillen proved it for semilocalizations
of finite-type algebras over a field at regular primes, Panin proved the full
equicharacteristic case, and in mixed characteristic the smooth local case is
Gillet-Levine's relative theorem. Finite-coefficient refinements are due to
Geisser-Levine, and Skalit obtained them for arbitrary unramified regular local
rings.
Quillen, Higher algebraic K-theory I, LNM 341 (1973) 85-147, doi:10.1007/BFb0067053.
local pdf: refs/weibel-kbook-VI.pdf (Weibel, The K-book, ch. VI, restates Quillen localization.)
Every positive result in mixed characteristic assumes, in one form or another,
that the local ring is smooth or unramified at p: that p does not sit deep in
the square of the maximal ideal. The example constructed here is the opposite.
Fix the unramified extension of the 5-adic field of degree eight, let V be its
ring of integers, let k = F_{5^8} be the residue field, and form
A = ( V[x,y] / ( 5 + x^4 + y^4 ) )_(5,x,y).
The maximal ideal of A is (5,x,y) modulo the equation, and the relation
5 = -(x^4 + y^4) puts 5 in the fourth power of that ideal, so 5 is maximally
ramified. The quotient A/5A is the plane curve x^4 + y^4 = 0, a curve with an
ordinary singular point at the origin; A itself is nonetheless a regular local
domain of dimension two, because 5 + x^4 + y^4 reduces to 5 modulo the square of
the maximal ideal of V[x,y]_(5,x,y) and so is part of a regular system of
parameters. The structure map V -> A is therefore not smooth, and A lies
outside every hypothesis listed above.
The result is this.
The map K_5(A) -> K_5(Frac A) has nonzero kernel. Consequently Gersten's
conjecture, in its unrestricted integral form, is false.
The proof breaks into two movements that are independent in spirit. The first
produces a nonzero integral class in K_5(A) that is automatically killed on the
fraction field, by pushing a well-chosen K_5-class forward from a divisor inside
Spec A. The second proves that pushed-forward class is genuinely nonzero, by
reducing it modulo 5 and testing it against a homomorphism built from an
elliptic curve. Neither movement may assume Gersten injectivity, since that is
exactly what is being disproved; both therefore rest only on localization,
devissage, and the geometry of a specific Fermat quartic.
Throughout, K-theory is taken with mod-5 coefficients. Let M = S/5 be the
cofiber of multiplication by 5 on the sphere spectrum S. For a Noetherian
scheme X, write K(X) for the algebraic K-theory spectrum of perfect complexes
and G(X) for the K-theory spectrum of coherent modules, and set
K_i(X;5) = pi_i( K(X) (wedge) M ), G_i(X;5) = pi_i( G(X) (wedge) M ).
For X the spectrum of a regular ring with finite Krull dimension (an affine or
quasiprojective scheme over an affine base, which covers every case below), the
canonical resolution identifies K(X) with G(X), so the two groups coincide.
The cofiber sequence defining M gives, for i >= 1, a natural exact sequence
0 -> K_i(X)/5 -> K_i(X;5) -> K_{i-1}(X)[5] -> 0, (*)
and the same with G in place of K. Here A/5 = A/5A and A[5] = ker(5 : A -> A);
the last map is the coefficient boundary d_coef. For a field H, K_1(H) = H^x
and K_0(H) = Z give K_1(H;5) = H^x/H^{x5} and K_0(H;5) = Z/5. The class of a
unit a in K_1(-;5) is written [a]_5, and K-groups are written additively even
when the elements are unit classes.
Quillen, Higher algebraic K-theory I, LNM 341 (1973); localization and the
devissage residue maps are recorded in Weibel, The K-book, ch. VI.
local pdf: refs/weibel-kbook-VI.pdf
The coefficient groups are F_5-vector spaces, and there is a canonical product
on them. The next statement is the one ingredient that lets a single
coefficient class act as a degree-two operator.
Lemma. The spectrum M carries a unital pairing mu : M (wedge) M -> M, unique in
the stable homotopy category, compatible with the usual tensor products of
K-theory spectra. Multiplication by 5 on M is null-homotopic, so every
coefficient group above is an F_5-vector space; tensor product followed by
mu gives natural bilinear products in coefficient K-theory and module
actions on coefficient G-theory.
The proof uses only the low homotopy groups of the sphere. Since
pi_1 S = pi_2 S = Z/2, the cofiber sequence S -> S -> M gives pi_1 M = pi_2 M =
0. Applying maps into M to the same cofiber sequence makes restriction along
the unit eta : S -> M injective on [M,M] -> [S,M] = Z/5; hence 5 id_M = 0. The
cofiber sequence M -> M -> M (wedge) M -> Sigma M then lets id_M extend to a pairing
mu, and [Sigma M, M] = 0 (the same two vanishing homotopy groups applied to
Sigma M) makes the extension unique. Restriction of mu along 1 (wedge) eta agrees
with restriction along eta (wedge) 1, so mu is unital. Smashing 5 id_M = 0 with any
spectrum kills 5 in every coefficient group. The products are the usual tensor
pairings of K-theory spectra composed with this fixed mu.
The mod-5 multiplicative structure on the coefficient theory, in the form used
later for the motivic spectral sequence, is Friedlander-Suslin, The spectral
sequence relating algebraic K-theory to motivic cohomology, Ann. Sci. ENS 35
(2002), Propositions 16.1 and 16.2.
local pdf: refs/friedlander-suslin-2002.pdf
Fix a base ring R and a class beta in K_2(R;5). For an R-scheme X define
b = b_beta : G_i(X;5) -> G_{i+2}(X;5), x |-> beta . x,
left multiplication by the restriction of beta, and write b^2 = b (compose) b,
so b^2(x) = beta . (beta . x) with the parentheses as displayed. No
associativity of the coefficient theory is assumed; only this fixed two-fold
iteration is used.
Lemma. The operation b commutes with every arrow and boundary map of the
localization sequences of G-theory over R, including localization in a
multiplicative set, localization at a generic point, and the devissage
residue maps. Under devissage its action on a residue-field term is
multiplication by the restricted class beta. It is compatible with finite
coherent pushforward and with flat pullback over R.
The proof is a functoriality statement for tensor products. Tensoring with a
finitely generated projective R-module is exact on coherent modules and
preserves the supports that occur, so the action of K(R) commutes with the
localization data. Concretely, for a term B of the localization triangle, the
pairing is the composite
( K(R) (wedge) M ) (wedge) ( B (wedge) M ) -> K(R) (wedge) B (wedge) M (wedge) M -> B (wedge) M,
whose last arrow is the integral tensor action followed by mu. These maps
commute with every arrow of the triangle by functoriality of tensor product.
The representative beta : S^2 -> K(R) (wedge) M has even degree 2, so moving it past
a localization boundary produces no sign and the diagrams for b commute. The
residue-field assertion is the same computation after tensoring over the residue
field, which is exactly the restriction of beta to that field; compatibility
with pushforward and flat pullback is the projection formula and flat base
change.
Weibel, The K-book, ch. VI, for the localization triangles and residue maps.
local pdf: refs/weibel-kbook-VI.pdf
The next result is the precise form of Bott periodicity in low degree that the
argument consumes. It fixes the multiplying class rather than asserting an
abstract isomorphism, and it covers only the field types that occur.
Lemma. Let H be a perfect field of characteristic different from 5 that
contains a primitive fifth root zeta_H, and let beta_H in K_2(H;5) satisfy
d_coef(beta_H) = [zeta_H]. Let b be left multiplication by beta_H.
(1) if cd_5(H) <= 3, then b : K_{2r}(H;5) -> K_{2r+2}(H;5) is an
isomorphism for every r >= 1;
(2) if cd_5(H) <= 2, then b : K_{2r+1}(H;5) -> K_{2r+3}(H;5) is an
isomorphism for every r >= 0;
(3) if H is finite, then b : K_i(H;5) -> K_{i+2}(H;5) is an isomorphism
for every i >= 0.
Over an algebraically closed field of characteristic different from 5, the
group K_{2j}(H;5) is generated by the iterated product b^j(1), and the
odd-degree groups vanish, with 1 in K_0(H;5) the unit class.
The instances needed later are these. For a number field or a finite extension
of Q_5 the 5-cohomological dimension is 2, so b^2 : K_1(H;5) -> K_5(H;5) is an
isomorphism. For the function field J of a curve over a finite extension of
Q_5, the bound cd_5(J) <= 2 + 1 = 3 holds, so b^2 : K_2(J;5) -> K_6(J;5) is an
isomorphism. For a finite residue field k_v, clause (3) gives b^2 : K_0(k_v;5)
-> K_4(k_v;5) an isomorphism, which raises a valuation target from degree zero
to degree four. The cohomological-dimension bounds are in Weibel, The K-book,
VI.7 and VI.8.2; the function-field bound cd_5 <= 3 is Stacks Project, Tag
0F0T.
local pdf: refs/weibel-kbook-VI.pdf
The proof runs through the motivic spectral sequence, which is strongly
convergent and multiplicative for mod-5 coefficients:
E_2^{p,q} = H^{p-q}( H, Z/5(-q) ) ==> K_{-p-q}( H; 5 ).
Reparametrizing with s = p - q and j = -q makes the total degree 2j - s and the
term H^s(H, Z/5(j)). The mod-5 motivic complex in weight j vanishes in
cohomological degree above j, so the only nonzero terms satisfy 0 <= s <= j.
The differential d_rho goes from (s,j) to (s + 2 rho - 1, j + rho - 1), hence
raises the total degree by one and lowers j - s by rho - 1. The spectral
sequence and its multiplicative structure are Weibel, The K-book, VI.4.2 and
Addendum VI.4.2.1; the corrected construction is Suslin, On the Grayson
spectral sequence; the product is the motivic cup product of
Friedlander-Suslin, Propositions 16.1 and 16.2, which by uniqueness agrees with
the coefficient product above.
local pdf: refs/friedlander-suslin-2002.pdf
The algebraically closed case comes first and without extra input. If H'-bar is
an algebraic closure of H, only the terms with s = 0 survive. In even degree 2j
there is exactly one term, H^0(H'-bar, mu_5^{(x)j}) = F_5, and in odd degree
none, and no differential can occur. So the edge map is an isomorphism in every
even degree and K_{2j}(H'-bar;5) = F_5, K_{2j+1}(H'-bar;5) = 0. The coefficient
sequence (*) maps K_2(H'-bar;5) onto H'-bar^x[5] = mu_5(H'-bar); both groups have
dimension one, so this boundary is an isomorphism, the restriction of beta_H is
nonzero, and its edge image is nonzero. Multiplicativity then shows each b^j(1),
computed by iterated left multiplication, has nonzero edge image and generates
the corresponding even group. This gives the last clause without any separate
rigidity theorem.
For the field H itself, naturality makes the edge image alpha in H^0(H, mu_5) of
beta_H nonzero: its geometric restriction is the nonzero class just found, and
restriction on H^0 is injective. The edge image of b^r(1) is the r-fold tensor
power of alpha, which generates H^0(H, mu_5^{(x)r}) because H contains mu_5. So
the whole line F_5 b^r(1) consists of permanent cycles for every r >= 1.
In total degree 2r the only possible terms are H^0(H, mu_5^{(x)r}) and
H^2(H, mu_5^{(x)(r+1)}), since s <= j forbids others. Neither has an incoming
differential; from the first the only outgoing differential is d_2 into an H^3
term, killed by permanence of F_5 b^r(1); from the second every outgoing
differential lands in cohomological degree at least 5 and is zero. This yields a
two-step filtration
0 -> H^2(H, mu_5^{(x)(r+1)}) -> K_{2r}(H;5) -> H^0(H, mu_5^{(x)r}) -> 0.
The multiplier b acts on each graded piece by tensoring with the nonzero class
alpha. On the H^0 piece this is an isomorphism between one-dimensional spaces;
together with the collapse of the spectral sequence on these two columns this
makes b : K_{2r}(H;5) -> K_{2r+2}(H;5) an isomorphism for r >= 1, which is
clause (1). When cd_5(H) <= 2 the sole term in total degree 2r + 1 is
H^1(H, mu_5^{(x)(r+1)}), with no possible nonzero differential, and b again acts
by tensoring with alpha; this is clause (2). A finite field has 5-cohomological
dimension one, so the same computation applies in every positive degree, and in
degree zero the class 1 generates K_0(H;5) while b(1) = beta_H generates
K_2(H;5) = H^0(H, mu_5); this is clause (3).
The arithmetic input lives in the cyclotomic ring at 5. Fix a primitive fifth
root of unity zeta, put N = Q(zeta), S = Z[zeta, 1/5], u = zeta - 1, and
Delta = Gal(N/Q), cyclic of order four. The cyclotomic character is
chi : Delta -> F_5^x, defined by delta(zeta) = zeta^{chi(delta)}. Substituting
zeta = 1 + u into Phi_5(X) = X^4 + X^3 + X^2 + X + 1 gives
Phi_5(1+U) = U^4 + 5 U^3 + 10 U^2 + 10 U + 5.
This polynomial is Eisenstein at 5, hence irreducible over Q_5, so N has a
single place above 5. The ring Z_5[u] is finite free over Z_5, a domain of
dimension one; modulo 5 the polynomial becomes U^4, so its maximal ideal is
(5,u) = (u) and its residue field is F_5. A one-dimensional Noetherian local
domain with principal maximal ideal is a discrete valuation ring, integrally
closed in its fraction field, with residue field F_5 and ramification index 4.
The relation behind the principality is u^4 = -5(1 + 2u + 2u^2 + u^3), whose
parenthesis is a unit.
The ring of integers of N is Z[zeta] and its discriminant is 125. The norm of u
is Phi_5(1) = 5, and Phi_5'(zeta) = 5 zeta^4 / u, so the discriminant of the
power basis 1, zeta, zeta^2, zeta^3 is N_{N/Q}(Phi_5'(zeta)) = 5^4 / 5 = 125.
The discriminant-index formula then allows no prime other than 5 in the index
[O_N : Z[zeta]], while the local computation above already makes Z[zeta]
maximal at 5; hence O_N = Z[zeta]. N is totally imaginary of degree four, so
Minkowski's bound reads (4!/4^4)(4/pi)^2 sqrt(125) < 2, every ideal class has a
representative of norm one, and the class number is one; equivalently, the
valuation map from N^x onto the direct sum of Z over the closed points of
Spec S is surjective and Pic(S) = 0. The discriminant, the ring of integers and
the Minkowski bound are Milne, Algebraic Number Theory, ch. 4.
local pdf: refs/milne-ant.pdf
Every closed point v of Spec S has residue field k_v of characteristic different
from 5, and the image of zeta in k_v has order exactly five: the order divides
five, and it is not one because u is a unit at v. These residue fields are
exactly the finite fields to which clause (3) of the Bott lemma is later
applied.
In K_1(S) = S^x the class [zeta] again has order five, since the determinant
S^x -> K_1(S) splits and detects it. The coefficient sequence (*) in degree two,
0 -> K_2(S)/5 -> K_2(S;5) -> K_1(S)[5] -> 0,
supplies a class beta' with d_coef(beta') = [zeta]. The group K_2(S;5) is an
F_5-vector space and Delta acts on it by functoriality. With 1/4 denoting the
inverse of 4 in F_5, set
e_chi = (1/4) sum_{delta in Delta} chi(delta)^{-1} delta in F_5[Delta],
beta = e_chi beta'.
The coefficient boundary is Delta-equivariant and delta([zeta]) =
chi(delta)[zeta], so d_coef(beta) = e_chi[zeta] = [zeta]. Relabeling the sum
gives tau e_chi = chi(tau) e_chi for every tau in Delta, hence
d_coef(beta) = [zeta], delta(beta) = chi(delta) beta (delta in Delta).
This single computation is what produces the square of the cyclotomic character
in the detection step. The cyclotomic-unit and Bott pattern follows Soule,
Elements cyclotomiques en K-theorie, Asterisque 147-148 (1987), sections
4.2-4.4.
local pdf: refs/soule-cyclotomiques.pdf
One lemma isolates the only arithmetic fact that the vanishing of the integral
obstruction needs. It concerns reductions of integral K_2 classes, not the whole
of K_2(N;5).
Lemma. If a class in K_2(N)/5 has zero tame residue in k_v^x/k_v^{x5} at
every closed point v of Spec S, then the class is zero.
By Matsumoto's theorem K_2 of a field is generated by Steinberg symbols, and the
classical-to-Quillen comparison identifies this with K_2(N). The degree-two
norm-residue theorem identifies K_2(N)/5 with the 5-torsion of the Brauer group,
Br(N)[5], and carries the tame residue at v to the Brauer residue
Br(N_v)[5] -> H^1(k_v, mu_5) = k_v^x/k_v^{x5}. So a class in K_2(N)/5 with all
tame residues zero is an element a of Br(N)[5] whose residue vanishes at every
finite place. For the completion N_v at such a place, the kernel of the Brauer
residue map is Br(k_v), which vanishes because k_v is finite; hence a restricts
to zero in Br(N_v). The infinite completions of N are complex, and Br(C)
vanishes, so no invariant survives at infinity. Only the single place above 5
remains. In the global Brauer invariant sequence the local invariants sum to
zero; since every other invariant has been shown to vanish, the invariant at the
place above 5 is zero as well, and a restricts to zero there too. The map
Br(N) -> product over all places of Br(N_v) is injective, so all local
restrictions being zero forces a = 0. No residue calculation at the place above
5 is used. The presentation of K_2 by symbols is Weibel, The K-book, III.6.1;
the norm-residue theorem is Voevodsky, On motivic cohomology with Z/l
coefficients, Ann. of Math. 174 (2011); the invariant sequence and injectivity
are Milne, Class Field Theory, IV.4.
local pdf: refs/weibel-kbook-VI.pdf
local pdf: refs/milne-cft.pdf
local pdf: refs/harari-brauer.pdf
With this, the integral obstruction disappears.
Lemma. For S = Z[zeta, 1/5], the group K_4(S)[5] is zero.
Since S is a Dedekind ring with Pic(S) = 0, the normalized valuation map from N^x
onto the direct sum of Z over the closed points of Spec S is onto. Quillen
localization and devissage for S identify the boundary K_1(N;5) -> sum_v
K_0(k_v;5) with this valuation map modulo 5, so that boundary is onto. Passing
to degree 5 uses the Bott maps. The field N has 5-cohomological dimension 2, so
b^2 : K_1(N;5) -> K_5(N;5) is an isomorphism; each residue field k_v is finite
and contains the primitive image of zeta, so b^2 : K_0(k_v;5) -> K_4(k_v;5) is an
isomorphism. The same restricted class beta acts on every term, and
compatibility with localization carries the surjection in degree one to a
surjection
K_5(N;5) -> sum_v K_4(k_v;5).
In the exact localization segment
K_5(N;5) -> sum_v K_4(k_v;5) -> K_4(S;5) -> K_4(N;5),
the middle arrow is therefore zero, and the restriction K_4(S;5) -> K_4(N;5) is
injective.
It remains to see where that injection lands. By the Bott clause b : K_2(N;5)
-> K_4(N;5) is an isomorphism, so the image x_N of a class x in K_4(S;5) is
b(z) for a unique z in K_2(N;5). Because the coefficient boundary of x_N lies
in K_1(N)[5] = mu_5(N) = <zeta>, there is m in F_5 with d_coef(x_N) = m[zeta],
and hence d_coef(z - m beta_N) = 0 for beta_N the restriction of beta to N. The
coefficient exact sequence now identifies z - m beta_N with an element of
K_2(N)/5, and naturality of coefficient reduction with localization identifies
its coefficient residues with the tame residues of that element. These residues
vanish at every closed point of Spec S: the residues of beta_N do, since beta
extends over S, and those of z do because x_N, being the image of a class on S,
is unramified. The preceding lemma then forces z - m beta_N = 0, so z = m
beta_N and x_N = m b(beta_N) = m b^2(1). The image of K_4(S;5) in K_4(N;5) is
therefore contained in the F_5-line spanned by b^2(1).
That line survives passage to an algebraic closure. Over an algebraic closure
N'-bar of N, the last clause of the Bott lemma makes b^2(1) the nonzero
generator of K_4(N'-bar;5), so restriction is injective on F_5 b^2(1), and
together with the containment just proved this makes
K_4(S;5) -> K_4(N'-bar;5)
injective. The same clause gives K_5(N'-bar;5) = 0, and the coefficient sequence
in degree 5 then shows K_4(N'-bar)[5] = 0; indeed K_4(N'-bar) has no nonzero
element of any finite five-power order, since if x has order 5^k then 5^{k-1}x
has order 5 and is zero. It follows that K_4(S;5) injects into K_4(S)[5]. The
two groups have the same order, so K_4(S;5) and K_4(S)[5] coincide as finite
five-primary groups.
A finitely generated group with this property has no five-primary part. By
Quillen's finite-generation theorem K_4(O_N) is finitely generated, so the
group G in the localization segment 0 = K_4(F_5) -> K_4(O_N) -> G -> K_3(F_5) =
Z/24 is finitely generated, the two finite-field values being Quillen's integral
computation. Write G = Z^r + P + T with P its finite five-primary subgroup and T
finite of order prime to 5. The coincidence just proved gives P contained in
5G. If p = 5g lies in P, the free part of g is zero because 5g is torsion, and
the T-part of g is zero because multiplication by 5 is an automorphism of T;
hence g lies in P, and P is contained in 5P. The reverse inclusion is
automatic, so P = 5P, and iterating to the exponent of the finite group P gives
P = 0. In particular K_4(S)[5] = 0. Finite generation is Quillen, Finite
generation of the groups K_i of rings of algebraic integers, LNM 341 (1973),
Theorem 1; the finite-field values are Quillen, On the cohomology and K-theory
of the general linear groups over a finite field, Ann. of Math. 96 (1972).
local pdf: refs/weibel-kbook-VI.pdf
The obstruction is now gone, so the chosen coefficient class lifts to an honest
integral class, and the four cyclotomic automorphisms act on its reduction in a
controlled way.
Proposition. Let u = zeta - 1 in S^x, and let b be multiplication by beta.
There exists c in K_5(S) with
c mod 5 = b^2[u]_5 in K_5(S;5).
For every delta in Delta the reduced conjugate satisfies
(delta c) mod 5 = chi(delta)^2 b^2[delta u]_5 in K_5(S;5).
Restriction along any fixed map iota : S -> H to a field gives the
corresponding identities in K_5(H;5); in the second, with b_iota the
multiplication by the restriction of beta,
res_iota(delta c) mod 5 = chi(delta)^2 b_iota^2[ iota(delta u) ]_5.
No action of delta on H is required.
The vanishing of K_4(S)[5] and the coefficient sequence (*) with i = 5 give
0 -> K_5(S)/5 -> K_5(S;5) -> K_4(S)[5] = 0,
so K_5(S) -> K_5(S;5) is surjective and the class b^2[u]_5 has an integral
preimage c, which is the first assertion. For the second, apply delta after
reduction and use naturality and bilinearity of the parenthesized product:
(delta c) mod 5 = delta( beta . (beta . [u]_5) )
= (delta beta) . ( (delta beta) . [delta u]_5 )
= chi(delta) beta . ( chi(delta) beta . [delta u]_5 )
= chi(delta)^2 b^2[delta u]_5.
The last line is the displayed identity, and the computation makes no equivariant
choice of the lift: any integral lift of b^2[u]_5 satisfies the first identity and
hence the second. Restriction along iota is functorial, and it introduces no
action of delta on H. The output is an integral class whose mod-5 reduction knows
the square of the cyclotomic character.
The geometric half rests on a single reusable criterion about sums of
coefficient classes pushed forward from points of a curve. A vanishing such sum
would be a localization boundary; two Bott operations turn that boundary into
ordinary tame symbols, and the divisor boundary on a smooth model then forces a
relation on the special fiber that a Picard-group homomorphism can refuse.
Proposition. Let L be a finite extension of Q_5 containing mu_5, with
valuation ring R and residue field k. Let X -> Spec R be a smooth
projective morphism of relative dimension one with geometrically integral
fibers C = X_L and D = X_k. For a closed point q of C let r(q) be its
specialization on D, the image of the closed point under the unique
extension of Spec kappa(q) -> X to the valuation ring of kappa(q). For a
closed point r of D write [r] in Pic(D) for the class of the divisor (r).
Assume:
(H1) U is the spectrum of a localization of the coordinate ring of a
nonempty affine open of C, and p_1, ..., p_m are distinct L-rational
closed points of U;
(H2) Z is a finite set of closed points of D, and every closed point q of
C with r(q) not in Z belongs to U;
(H3) B is an abelian group killed by 5, and psi : Pic(D) -> B is a
homomorphism with psi([r]) = 0 for every r in Z.
Put r_i = r(p_i) in D(k), choose beta_L in K_2(L;5) with
d_coef(beta_L) = [zeta], let b be left multiplication by the restrictions of
beta_L, and let v_L be the normalized valuation of L. For a_i in L^x and
e_i in F_5, the condition
sum_{i=1}^m e_i v_L(a_i) psi([r_i]) != 0 in B
implies
sum_{i=1}^m (p_i)_* ( e_i b^2[a_i]_5 ) != 0 in G_5(U;5).
The proof has three stages.
Stage one untwists. The sum over the empty set is zero, so the hypothesis forces
m > 0. Suppose the second sum vanishes. Let J = L(C) be the function field of
the generic fiber and A_U = Gamma(U, O_U) the coordinate ring of U, a localization
of the coordinate ring of a smooth affine curve, hence a Noetherian regular
domain of dimension one with fraction field J (the dimension is one because some
p_i is retained). Every finitely generated torsion A_U-module is killed by a
nonzero element, its quotient of A_U has dimension at most zero and is Artinian,
and its simple factors are the residue fields of the retained closed points; this
does not need U to be semilocal or the multiplicative set to be finite. Quillen
localization for the torsion category, followed by devissage, gives the exact
sequence
K_6(J;5) -> sum_{q in U^(1)} K_5(kappa(q);5) -> G_5(U;5),
the first arrow being the localization residues and the second the sum of
coherent pushforwards. The index set may be infinite but each class has finite
support, and every q in U^(1) is a closed point of C with kappa(q) a finite
extension of L. Since the second sum is assumed zero, exactness supplies xi in
K_6(J;5) whose residue at p_i is e_i b^2[a_i]_5 and whose other residues vanish.
The field J has cd_5(J) <= 3 and each kappa(q) has cd_5 = 2; all are perfect and
contain the same primitive root, by restriction from L. The Bott lemma therefore
makes b^2 : K_2(J;5) -> K_6(J;5) surjective and b^2 : K_1(kappa(q);5) ->
K_5(kappa(q);5) injective. Choose xi_0 in K_2(J;5) with b^2 xi_0 = xi. Because
b is compatible with localization and has even degree, d_q(b^2 xi_0) = b^2(d_q
xi_0), and injectivity on residue fields gives
d_q(xi_0) = e_i[a_i]_5 if q = p_i,
d_q(xi_0) = 0 for all other q in U^(1).
One coefficient obstruction to representing xi_0 integrally remains. Its
coefficient boundary lies in K_1(J)[5] = mu_5(J) = <zeta> (the last equality
because L already contains mu_5), so d_coef(xi_0) = lambda[zeta] for some lambda
in F_5. Let beta_U and beta_J be the restrictions of beta_L; naturality gives
d_coef(beta_J) = [zeta], while every localization residue of beta_J on U vanishes
because beta_J extends to the coefficient class beta_U over U. Hence
eta = xi_0 - lambda beta_J in K_2(J;5)
has zero coefficient boundary and the same residues as xi_0. The coefficient
exact sequence then supplies an integral representative z in K_2(J) of eta, whose
ordinary tame symbols d_q^int z in K_1(kappa(q)) = kappa(q)^x satisfy
v_q(d_q^int z) = e_i v_L(a_i) at q = p_i, (mod 5)
v_q(d_q^int z) = 0 at every other q in U^(1),
where v_q is the normalized valuation of kappa(q) and v_L = v_{p_i} because p_i
is L-rational. The assumed vanishing has been converted into a statement about
ordinary tame symbols of the single integral class z. The exactness of the
localization sequence is Quillen, Higher algebraic K-theory I, section 7,
Proposition 3.2, as presented in Weibel, The K-book, chapter VI.
local pdf: refs/weibel-kbook-VI.pdf
Stage two aggregates the tame symbols on the special fiber. The model X is
smooth over the regular DVR R, hence a regular surface with function field J.
The codimension filtration of coherent K-theory gives the row of the coniveau
complex
K_2(J) -> sum_{s in X^(1)} kappa(s)^x -> sum_{r in X^(2)} Z.
Only the vanishing of the composite is needed, and that is the formal identity
d_1^2 = 0 for the first differential of the coniveau spectral sequence. Exactly
one vertical point lies in X^(1), the generic point of the integral special curve
D; its residue d_D z in k(D)^x contributes the principal divisor div_D(d_D z).
The other codimension-one points are the closed points q of C.
The contribution of a closed point q of C is computed without any normal-flatness
assumption. On Spec R_q the localization boundary K_1(kappa(q)) -> K_0(k_q) = Z
is the normalized valuation v_q: a unit has boundary zero, and a uniformizer
pi_q has boundary [k_q]. Functoriality of coherent localization for the finite
map j_q : Spec R_q -> X identifies the horizontal boundary on X with the
pushforward of this valuation-ring boundary. At the generic point the pushforward
is the identity on kappa(q); at the closed point it sends [k_q] to
[k_q : kappa(r(q))][kappa(r(q))]. So the contribution at r(q) is exactly
f_q v_q(d_q^int z), f_q = [k_q : kappa(r(q))],
with no extra ramification factor: the boundary already uses the normalized
valuation on kappa(q), and the closed-point pushforward contributes only the
residue degree f_q. Define the horizontal divisor
H(z) = sum_{q in C^(1)} f_q v_q(d_q^int z) (r(q)).
The sum is finite because the first arrow lands in a direct sum, and all
codimension-two points of the proper surface lie on D. Applying d_1^2 = 0 to z
gives, as an equality of divisors on D,
H(z) + div_D(d_D z) = 0.
A principal divisor is linearly trivial, so
[H(z)] = 0 in Pic(D).
The tame symbols of z, aggregated on the special fiber, are a specific principal
divisor class.
Stage three tests that class. Applying psi to the vanishing of [H(z)] gives a
relation in B. If r(q) lies in Z, the q-th summand vanishes because psi([r(q)]) = 0. If r(q) does not lie in Z, hypothesis (H2) puts q in U; for such a q other
than the p_i, the congruence v_q(d_q^int z) = 0 mod 5 makes the q-th summand zero
in B, which is killed by 5. Only q = p_i remains. There kappa(p_i) = L, k_{p_i} = k = kappa(r_i), so f_{p_i} = 1 and v_{p_i} = v_L, and the remaining congruence
reads v_{p_i}(d^int z) = e_i v_L(a_i) mod 5. The identity psi([H(z)]) = 0 is
therefore
sum_{i=1}^m e_i v_L(a_i) psi([r_i]) = 0 in B,
contrary to the hypothesis. This proves the implication.
The exactness used in stage one comes from torsion-category localization and holds
for the arbitrary localization U. On the mixed-characteristic surface only
d_1^2 = 0 is used, never Gersten injectivity or exactness: the criterion must be
independent of the very statement being disproved. The coniveau identity and the
localization sequence are Quillen, Higher algebraic K-theory I, section 7,
Theorem 5.4 and equation (5.5), in the form given by Weibel, The K-book, chapter
VI.
local pdf: refs/weibel-kbook-VI.pdf
The geometry is a single degree-two quotient of the Fermat quartic onto an
elliptic curve, and it lives over two finite fields. Put k_0 = F_{5^4} and
k = F_{5^8}. Over F_5 let D be the projective plane curve X^4 + Y^4 = Z^4 and
let E be the smooth projective model of w^2 = s^4 + 1.
The curve D is smooth and geometrically integral. The three partial derivatives
4X^3, 4Y^3, -4Z^3 cannot vanish together at a projective point, so D is smooth
over F_5 and remains so after any base change; it is geometrically integral
because two distinct projective plane components would meet and create a
singular point. The curve E is smooth of genus one. The polynomial s^4 + 1 has
four simple roots; at infinity the coordinates q = 1/s and r = w/s^2 give r^2 =
1 + q^4 with two unramified points r = +1 and r = -1 above q = 0, so the double
cover E -> P^1_s has exactly four branch points and the Hurwitz formula gives
genus (4/2) - 1 = 1.
There is a finite degree-two morphism
h : D -> E, s = X/Y, w = Z^2/Y^2.
On the chart Y != 0, with v = Z/Y, the equation becomes v^4 = s^4 + 1; the
extension of function fields is obtained by adjoining v with v^2 = w, and the
involution v |-> -v shows the degree is two. The rational map extends to the
smooth projective models and is finite, and it carries the four points of D with
Z = 0 (where v = 0 and w = 0) to the four branch points of E -> P^1, the points
with s^4 = -1 and w = 0. The construction is the passage from a diagonal quartic
to a genus-one quotient by a sign involution, as in Flynn-Wetherell, Covering
collections and a challenge problem of Serre, Acta Arith. 98 (2001), section 1.
doi:10.4064/aa98-2-9
The eighth roots of unity lie in k_0 because 8 divides 5^4 - 1 = 624, so all
four branch points are k_0-rational. Choose one, written o, as the origin of
E_{k_0}. Every fiber of E -> P^1 is linearly equivalent to 2(o), so the
involution w |-> -w is negation for this group law, using the identification of a
degree-zero divisor class with its point on a genus-one curve with chosen origin.
In particular every branch point is a two-torsion point of E(k_0). These facts
are Silverman, The Arithmetic of Elliptic Curves, GTM 106, 2nd edition,
Proposition III.3.4.
doi:10.1007/978-0-387-09494-6
Base change to k and continue to write h : D_k -> E_k, with o in E(k_0) defining
the group law on E_k. Put Z_inf = D_k intersect {Z = 0}, the four points above.
Define a map on divisor classes by
psi : Pic(D_k) -> E(k)/5E(k),
[D'] |-> [ h_*D' - deg(D')(o) ] mod 5E(k).
This is well defined on the whole Picard group. Every line bundle on the smooth
curve D_k has a nonzero rational section and so a representing divisor D'. On a
closed point r the finite divisor pushforward is h_*(r) =
[kappa(r):kappa(h(r))](h(r)), of degree [kappa(r):kappa(h(r))][kappa(h(r)):k] =
[kappa(r):k], so it preserves degree over k; and it takes principal divisors to
principal divisors by the norm formula. The identification Pic^0(E_k) = E(k)
sends [(R) - (o)] to R, so psi is a homomorphism, and for a k-rational point r of
D_k it gives psi([r]) = h(r) modulo 5E(k). The norm formula for divisors is the
Stacks Project, Tag 02RT.
https://stacks.math.columbia.edu/tag/02RT
Each point of Z_inf is k-rational and maps under h to a branch point of E_k ->
P^1_k, hence to a two-torsion point of E_k. A two-torsion class is zero modulo 5,
because 5 is invertible on E[2]. Therefore
psi([r]) = 0 for every r in Z_inf.
This is the point-killing property (H3) of the specialization criterion, with Z =
Z_inf. It remains to find an orbit whose weighted image under psi is nonzero.
For a in F_5^x define an automorphism of D by g_a(X:Y:Z) = (aX : aY : Z). On the
dense chart Y != 0 it fixes s = X/Y and multiplies w = Z^2/Y^2 by a^{-2}; on the
branch-point fiber w = 0 this stays true, and since D is integral and E separated
the identity of morphisms extends. Hence
h(g_a P) = ( s(P), a^{-2} w(P) ),
so h(g_a P) = Q when a^2 = 1 and h(g_a P) = -Q when a^2 = -1.
Lemma. There are points Q in E(k_0) and P in D(k) such that Q is not in
5E(k), h(P) = Q, Z(P) != 0, and
sum_{a in F_5^x} a^2 psi([g_a P]) = 4Q != 0 in E(k)/5E(k).
The curve E has four points over F_5: at s = 0 there are two affine points w =
+1 and w = -1; for each nonzero s the value s^4 + 1 = 2 is a nonsquare in F_5, so
none; and there are two rational points at infinity because the leading
coefficient of the quartic is a square. With any rational point as origin the
Frobenius polynomial is T^2 - 2T + 5, whose roots compute the extension-field
point counts. If a_m is the sum of the m-th powers of the roots, then a_0 = a_1 = 2 and a_m = 2 a_{m-1} - 5 a_{m-2}; this gives a_4 = -14 and
#E(k_0) = 5^4 + 1 - a_4 = 640.
The finite group E(k_0) therefore has order 640 = 2^7 . 5, and its quotient by 5
is Z/5, so there is a point Q not in 5E(k_0). Such a point stays nondivisible by
5 over k: if Q = 5R in E(k), the trace for the quadratic extension k/k_0, namely
2Q = 5(R + Frob_{k_0}(R)) in E(k_0), would put Q in 5E(k_0), since
multiplication by 2 is invertible on E(k_0)/5E(k_0). Every branch point is
two-torsion and hence divisible by 5, so Q is not a branch point. A fiber of the
degree-two morphism h has a point over an extension of degree at most two, and k
is the quadratic extension of k_0, so some P in D(k) satisfies h(P) = Q; because
Q is not a branch point, P is not over Z = 0 and Z(P) != 0.
For the weighted sum, each summand is a^2 h(g_a P): for a^2 = 1 (the two values
a = +1 and a = -1) it is +Q, contributing 2Q, and for a^2 = -1 (the two values a = +2 and a = -2) it is (-1)(-Q) = Q, contributing 2Q. The total is 4Q, nonzero
because 4 is invertible in F_5 and Q is nonzero in E(k)/5E(k). The point count
is Silverman, Theorem V.2.3.1.
doi:10.1007/978-0-387-09494-6
The divisor inside Spec A is built from a cyclotomic uniformizer. Let H_0 be
the unramified extension of Q_5 of degree eight, with ring of integers V and
residue field identified with k = F_{5^8}. Retain zeta, u = zeta - 1,
Delta = Gal(Q(zeta)/Q) and chi : Delta -> F_5^x. Embed Q(zeta) in an algebraic
closure of H_0 and put
L = H_0(zeta), OO = the ring of integers of L.
The translated cyclotomic equation u^4 + 5u^3 + 10u^2 + 10u + 5 = 0 is Eisenstein
over V, so L/H_0 is totally ramified of degree four, u is a uniformizer of OO,
the residue field is k, and restriction of automorphisms identifies Gal(L/H_0)
with Delta.
There is a uniformizer t in OO with
t^4 = -5, t/u = 1 (mod u), delta(t)/t in mu_4(V),
residue of delta(t)/t equal to chi(delta).
To build it, the cyclotomic equation gives u^4/(-5) = 1 + 2u + 2u^2 + u^3, a
unit congruent to 1 modulo u; by Hensel's lemma in the complete valuation ring V
(since 4 is invertible in V) the inverse of this unit has a fourth root congruent
to 1 modulo u, and multiplying u by that root gives t. Then delta(t)^4 = -5 =
t^4, so delta(t)/t is a fourth root of unity. Its residue equals that of
delta(u)/u, because t and u differ by a unit congruent to 1 modulo u; and
delta(u)/u = (zeta^{chi(delta)} - 1)/(zeta - 1) = zeta^{(chi(delta)-1)} + ... + 1
has residue chi(delta) modulo (zeta - 1). All fourth roots of unity lie in V,
since 4 divides 5^4 - 1 and V is unramified with residue field F_{5^8}, again by
Hensel.
The whole ring of integers is generated by t: V[t] = OO. The element t
satisfies the Eisenstein polynomial T^4 + 5 over V, so V[t] is finite free over V
with fraction field L; reducing modulo 5 gives T^4, so V[t] is local with maximal
ideal (5,t) = (t) and dimension one, hence a discrete valuation ring, integrally
closed in L. Every element of OO is integral over V and lies in L, so it lies in
V[t]. These local-field facts are Milne, Algebraic Number Theory, chapter 1.
local pdf: refs/milne-ant.pdf
The chosen point of D(k) lifts to V. Write P = (X_0bar : Y_0bar : 1), at least
one of X_0bar, Y_0bar nonzero. If X_0bar is nonzero the partial derivative
4X_0bar^3 is a unit of V, so by Hensel's lemma V lifts (X_0bar, Y_0bar) to
(X_0, Y_0) in V^2 with X_0^4 + Y_0^4 = 1; symmetrically if Y_0bar is nonzero.
The equation is smooth in the corresponding direction, and at least one of X_0,
Y_0 is a unit of V.
Lemma. The ring A = ( V[x,y] / ( 5 + x^4 + y^4 ) )_(5,x,y) is a regular
local domain of dimension two. The assignment x |-> tX_0, y |-> tY_0 is a
surjective local homomorphism A -> OO with nonzero kernel of height one.
Regularity: the local ring V[x,y]_(5,x,y) is regular of dimension three, and
5 + x^4 + y^4 is congruent to 5 modulo the square of its maximal ideal, hence is
part of a regular system of parameters; the quotient A is a regular local ring of
dimension two and a domain. The map is defined because (tX_0)^4 + (tY_0)^4 =
t^4(X_0^4 + Y_0^4) = (-5)(1) = -5, and every inverted denominator has nonzero
constant residue at x = y = 0, hence maps to a unit of OO; so the map is local. It
is surjective because at least one of X_0, Y_0 is a unit, so the image contains t,
and then contains V[t] = OO. The element Y_0 x - X_0 y lies in the kernel and has
nonzero image in m_A/m_A^2; the kernel is nonzero, and it is not maximal because
the quotient OO is a discrete valuation ring, so it has height one in the
two-dimensional local domain A. Since Y_0 x - X_0 y is a regular parameter it
generates a height-one prime and hence generates the kernel. The regularity
criterion is Stacks Project, Tags 00NQ and 00NP.
https://stacks.math.columbia.edu/tag/00NQ
The arithmetic surface used for detection is the Fermat quartic over OO. Let
Dcal = Proj( OO[X,Y,Z] / ( X^4 + Y^4 - Z^4 ) ),
C = Dcal_L (generic fiber), D_k = Dcal_k (special fiber),
T = Spec( A (x)_V L ).
Over an algebraic closure of L or of k the three partial derivatives 4X^3, 4Y^3,
-4Z^3 cannot vanish at a projective point, so each geometric fiber of Dcal over
OO is smooth of dimension one, and each is integral since two distinct plane
components would meet in a singularity. The same Jacobian calculation on the
relative affine charts, where 4 is invertible, shows Dcal is smooth over the
discrete valuation ring OO, hence regular; flatness over the valuation ring
excludes a component supported in the special fiber, and the generic fiber is
integral, so Dcal is an integral surface with special fiber D_k. Smoothness over
a valuation ring by the Jacobian criterion is the Stacks Project, Tag 00TE.
https://stacks.math.columbia.edu/tag/00TE
Lemma. The scheme T is regular, integral, Noetherian and one-dimensional,
and flat over Spec A. The substitutions X/Z = x/t, Y/Z = y/t identify T
with the spectrum of a localization of the coordinate ring of the chart
Z != 0 in C. The base change to T of the closed immersion Spec OO ->
Spec A is the disjoint union of four L-rational closed points p_delta,
indexed by Delta, with
p_delta : ( X/Z, Y/Z ) = ( (delta(t)/t) X_0, (delta(t)/t) Y_0 ).
The specializations of the p_delta on Dcal are g_{chi(delta)} P, and the
delta-factor is obtained from OO by the ring map a |-> delta(a).
The V-module L is flat, since V -> H_0 inverts 5 and L is a vector space over
H_0, so A (x)_V L is flat over A. The substitution x = tX, y = tY turns
5 + x^4 + y^4 = 0 into X^4 + Y^4 = 1 (as t^4 = -5), so after base change and
localization T is the spectrum of a localization of the coordinate ring of the
chart Z != 0 of C; that coordinate ring is a localization of a smooth
geometrically integral affine curve over L, hence regular, integral, Noetherian
and of dimension at most one.
Since 5 is invertible in the second tensor factor, the coordinate ring of the
base-changed divisor splits,
OO (x)_V L = L (x)_{H_0} L -> prod_{delta in Delta} L,
a (x) b |-> ( delta(a) b )_delta,
the usual splitting of a finite Galois extension. The image of x under A -> OO
and then in the delta-component is delta(t) X_0, and the image of y is delta(t)
Y_0; dividing by t gives the displayed coordinates. These directions are
nonzero, so each delta-component is an actual point; the pairs are four distinct
points because the ratios delta(t)/t are the four distinct fourth roots of unity
of V and at least one of X_0, Y_0 is a unit; and every inverted denominator
evaluates to a unit of OO, so T has closed points and dimension exactly one. The
coordinates are integral over OO and satisfy the Fermat equation, so they define
sections of Dcal; reducing modulo 5, where delta(t)/t becomes chi(delta) and X_0,
Y_0 become X_0bar, Y_0bar, gives g_{chi(delta)} P on D_k. The last assertion is
the definition of the splitting map. The isomorphism OO (x)_V L = prod L is the
standard normal basis decomposition, Milne, Algebraic Number Theory, chapter 1.
local pdf: refs/milne-ant.pdf
The two halves now join. Let c in K_5(S) be the integral lift of the arithmetic
half and c_L in K_5(L) its restriction along the fixed embedding iota : S -> L.
There is no ring map S -> OO, since OO has characteristic zero and 5 is not
invertible there, so c is not restricted directly to OO. Instead integral
localization for the discrete valuation ring OO, with residue field k = F_{5^8},
gives the segment
K_5(OO) -> K_5(L) -> K_4(k) -> K_4(OO).
By Quillen's computation for finite fields K_4(k) = 0, so K_5(OO) -> K_5(L) is
surjective and c_L has a lift c-tilde in K_5(OO). The closed immersion
i : Spec OO -> Spec A of the local-ring lemma then defines
gamma = i_*( c-tilde ) in G_5(A).
Because A is regular, the canonical resolution identifies G_5(A) = K_5(A), so
gamma lives in K_5(A). It is killed by generic restriction: restriction to
Frac A is the direct image along the open immersion Spec(Frac A) -> Spec A, and
every coherent module supported on the proper closed subscheme Spec OO restricts
to zero there, so every class in the image of i_* dies:
gamma in ker( K_5(A) -> K_5(Frac A) ).
What remains is to show gamma is nonzero.
Flat base change V -> L splits the divisor. Put B_T = A (x)_V L and T = Spec
B_T. The square
Coprod_delta Spec L --(Coprod p_delta)--> T
| |
v v
Spec OO ---------- i ----------> Spec A
is Cartesian, and for every finitely generated OO-module M scalar extension gives
natural isomorphisms of B_T-modules
B_T (x)_A i_* M = ( OO (x)_V L ) (x)_OO M
= (direct sum over delta in Delta) L (x)_{OO,delta} M.
The delta-summand is a B_T-module through the quotient defining p_delta, and the
functors are exact because B_T is flat over A and each OO -> L followed by an
automorphism is a localization. So the identity holds on integral coherent
K-theory.
On the delta-factor, since c-tilde restricts to c_L = res_iota(c), its image is
res_delta(c-tilde) = res_iota(delta c): here delta is an automorphism of OO
extending to L that commutes with the inclusion OO -> L, so pushing c-tilde along
delta and then the identity factor equals restricting the delta-image of c along
iota. This uses no equivariance of the chosen lift. Hence, integrally,
gamma|_T = sum_{delta in Delta} (p_delta)_* res_iota( delta c ).
Reducing modulo 5 uses naturality of coefficient reduction and the conjugacy of
the lift,
(gamma|_T) mod 5
= sum_{delta in Delta} (p_delta)_* ( chi(delta)^2 b^2[ delta u ]_5 ),
a class in G_5(T;5), with b the multiplication by the restriction of beta to L and
the character factors chi(delta)^2 from the integral-lift proposition. No choice
of the lift c-tilde affects this formula.
The specialization criterion is now applied with
R = OO, X = Dcal -> Spec OO, C = Dcal_L, D = D_k,
U = T, Z = Z_inf, B = E(k)/5E(k), psi the elliptic detector,
the field L being the fraction field of OO. Three hypotheses must be checked.
(H1): by the splitting lemma T is a localization of the coordinate ring of the
affine chart Z != 0 of C, and the points p_delta are distinct L-rational, so U = T
and p_1, ..., p_4 = p_delta satisfy (H1) with m = 4.
(H2): let q be a closed point of C whose specialization r(q) is not in Z_inf, and
let OO_q be the valuation ring of the finite field extension kappa(q)/L. Because
r(q) is in the chart Z != 0, the proper extension Spec OO_q -> Dcal lies in that
chart, since an open set containing the closed point of a valuation ring contains
its generic point; hence X/Z and Y/Z belong to OO_q, and the coordinates x = tX/Z
and y = tY/Z of A have positive valuation there. Every denominator inverted in
A outside (5,x,y) has nonzero constant residue and so is a unit at q. Thus q
lies in T = U.
(H3): psi([r]) = 0 for every r in Z_inf, which is the point-killing property
proved in the construction of the detector.
The test sum of the criterion is
sum_{delta in Delta} chi(delta)^2 v_L( delta u ) psi([ g_{chi(delta)} P ]).
Each delta u is a conjugate of the uniformizer u, so v_L(delta u) = 1, and the
specializations are r(p_delta) = g_{chi(delta)} P. The sum therefore equals
sum_{delta in Delta} chi(delta)^2 psi([ g_{chi(delta)} P ]),
which is the weighted orbit computed in the geometry, equal to 4Q in E(k)/5E(k)
and nonzero because Q is nonzero there. All hypotheses of the criterion hold, so
sum_{delta in Delta} (p_delta)_* ( chi(delta)^2 b^2[ delta u ]_5 ) != 0
in G_5(T;5).
By the reduction formula this left-hand side is (gamma|_T) mod 5, hence
(gamma|_T) mod 5 != 0 in G_5(T;5).
Since its reduction modulo 5 is nonzero, gamma|_T itself is nonzero in G_5(T). But
gamma|_T is the flat pullback of gamma along the flat map T -> Spec A, and a flat
pullback of zero is zero. Therefore gamma != 0 in G_5(A) = K_5(A). Together
with the containment in the kernel of K_5(A) -> K_5(Frac A), this gives a nonzero
element of that kernel, which is the theorem. The finiteness of K_4(k) used above
is Quillen, On the cohomology and K-theory of the general linear groups over a
finite field, Ann. of Math. 96 (1972), read through Weibel, The K-book, IV.1.13.
local pdf: refs/weibel-kbook-VI.pdf
Two features of the argument deserve emphasis, since both look like errors at
first sight. First, the divisor Spec OO is cut out by the single regular
parameter Y_0 x - X_0 y, so it is principal, and one might expect its pushforward
to vanish; it does not. The self-intersection formula gives i^* i_* = 0 on
assembled classes and the projection formula gives i_* i^* = 0, but neither
asserts i_* = 0, and the pushforward of an arbitrary class on the divisor need
not vanish in higher K-theory. In particular c-tilde need not be the
restriction of a class on A. Second, the position of the four points matters: if
one forgets the signed positions and retains only the four normalized valuations,
all equal to 1, the scalar sum 1 + 4 + 4 + 1 = 10 = 0 in F_5, and a purely scalar
functional would see nothing. The elliptic detector keeps the actual points
g_{chi(delta)} P and computes 4Q, which is nonzero. The detector distinguishes
the points where the scalar reduction cannot.
A separate question rises from the principal characteristic of the divisor. The
claim that a pushforward from a principal divisor vanishes would annihilate the
class gamma, and such a claim appears in a proof of local Gersten for regular
systems of parameters. Version 8 of that preprint states, for a commutative
Noetherian local ring B and a nonzerodivisor g, that the inclusion from finitely
generated projective B/gB-modules into finitely generated B-modules of projective
dimension at most one induces zero on K-theory. With B = A and g = Y_0 x - X_0 y,
so that A/gA = OO, composing the claimed zero map with inclusion into coherent
A-modules would indeed kill i_*(c-tilde). The main argument above never invokes
that theorem, and gamma is detected by the elliptic quotient, not by any
vanishing statement; but the proof construction supplied for the theorem can be
checked directly, and it fails on an explicit morphism.
The following calculation does not refute the theorem's statement; it exhibits a
failure in the functor assertion of Lemma 13(III) of the preprint, already with
both B and B/gB regular. Mochizuki, Local Gersten's conjecture for regular system
of parameters, arXiv:1503.07966v8, Lemma 13(III) on page 10 and the additivity
argument on page 11.
local pdf: refs/mochizuki-gersten.pdf
Let B be a regular local ring and let g be a nonzero regular parameter, so that
1 - g is a unit. The exact category consists of two-term complexes of finite free
B-modules in homological degrees one and zero whose differential is injective and
whose cokernel is killed by g; exact sequences are degreewise exact. Only the
objects
(n,m)_B = [ B^n + B^m --diag(gI_n, I_m)--> B^n + B^m ]
and split exact sequences between them are used, so no classification of the
other objects is needed.
For X = (1,1)_B put D = diag(g,1). An encoded endomorphism M = [[a,b],[c,d]] has
actual chain matrices
M_1 = [[a,b],[gc,d]], M_0 = [[a,gb],[c,d]],
which satisfy D M_1 = M_0 D; conversely the chain-map condition together with g a
nonzerodivisor gives this presentation. Multiplying the actual matrices shows
that the encoded composition is
[[a',b'],[c',d']] star [[a,b],[c,d]]
= [[a'a+gb'c, a'b+b'd],[c'a+d'c, gc'b+d'd]],
so the star of the encoded matrices represents the composite. "Upper triangular"
means c = 0 and "lower triangular" means b = 0, referring to the encoded matrices.
Writing Typ(h) = [B --h--> B], the assignments mu'_1, mu'_2 that precede
Definition 12 replace (n,m)_B by Typ(g)^{+n} or Typ(1)^{+n} and on a morphism
retain its upper-left block in both chain degrees. These assignments are not a
functor on all morphisms, because the upper-left entry of the star contains the
term g b' c.
An S_2-object is an admissible short exact sequence X_01 >-> X_02 ->> X_12 with the
zero objects on its diagonal, and morphisms are commuting maps of such sequences;
Definition 12 requires upper triangular structure arrows and allows lower
triangular componentwise isomorphisms. The claim of Lemma 13(III) is that mu'_1,
mu'_2 induce functors on this construction. The failure occurs already in degree
two.
Lemma. For every regular local ring B with a nonzero regular parameter g, the
upper-left-block assignments fail to send all lower triangular isomorphisms
between upper triangular S_2-objects to morphisms of the target diagrams.
Take the encoded endomorphisms of X = (1,1)_B,
U = [[1,1],[0,1]], U' = [[1-g,1],[0,1]],
L = [[1,0],[1,1]], L' = [[1,0],[1,1-g]],
with actual chain matrices
U_1 = [[1,1],[0,1]], U_0 = [[1,g],[0,1]],
U'_1 = [[1-g,1],[0,1]], U'_0 = [[1-g,g],[0,1]],
L_1 = [[1,0],[g,1]], L_0 = [[1,0],[1,1]],
L'_1 = [[1,0],[g,1-g]], L'_0 = [[1,0],[1,1-g]].
In each degree the determinants are 1, 1-g, 1, 1-g, all units, so all four are
chain isomorphisms; the first two are upper triangular and the last two lower
triangular. Computing the actual composites by matrix multiplication,
L'_1 U_1 = [[1,1],[g,1]] = U'_1 L_1,
L'_0 U_0 = [[1,g],[1,1]] = U'_0 L_0,
so the square
X --U--> X --> 0
| |
L L'
v v
X --U'-> X --> 0
commutes as a chain map. Each row is split exact, the structure arrows are zero
or identity and hence upper triangular, and the vertical maps (L, L', identity)
are lower triangular isomorphisms; so the diagram is a morphism in the source
category.
Under either assignment, the targets retain the upper-left blocks. The upper-left
blocks of U, U', L, L' are 1, 1-g, 1, 1, so the left square is sent to
Typ(h) --1--> Typ(h)
| |
1 1
v v
Typ(h) -1-g-> Typ(h), h = g or h = 1.
Its two composites are 1 and 1-g in both degrees, unequal because g != 0. So the
proposed image is not a morphism of S_2-diagrams. The calculation isolates the
precise condition missing from the proof: preservation of the upper-upper and
lower-lower compositions does not imply preservation of mixed upper-lower
commuting squares. The two projected composites can be chain homotopic, but the
target category requires strictly commuting natural transformations, and the later
homotopy comparison does not define replacement functors on this square. A
construction supplying the required higher compatibilities and additivity would be
an additional argument. None of it bears on the main result, whose class is
detected by the elliptic quotient of the Fermat quartic.