---
title: 'Kato Homology Groups: Cohomology in Characteristic p'
url: https://www.emergentmind.com/topics/kato-homology-groups
type: topic
---

# Kato Homology Groups: Cohomology in Characteristic p

Searching arXiv for the primary paper and closely related work to support the article.
Kato homology groups, in the sense developed in "Extended differential symbol and the Kato homology groups" [2501.11224], arise from Bloch–Ogus type complexes built from the $p$-primary Galois-cohomological groups attached to residue fields of points on a variety in characteristic $p>0$. The paper studies these groups through an extended differential symbol map that serves as a one-degree-higher analogue of the differential symbol in the Bloch–Gabber–Kato theorem. Within this framework, for suitable varieties, the $0$-th Kato homology group $KH_0^{(s)}(X,\mathbf Z/p^r)$ is treated as the cohomological analogue of the higher Chow group $CH^{d+s}(X,s)$, and its structure is determined in several arithmetic settings, including finite fields, local fields, and global fields of positive characteristic [2501.11224].

## 1. Definition and basic framework

Let $F$ be a field of characteristic $p>0$ such that
$$
[F:F^p]\le p^s
$$
for some integer $s\ge 0$, and let $X$ be a projective smooth geometrically irreducible $F$-scheme of dimension $d$. For each $j\ge 0$, let $X_j$ denote the set of points $x\in X$ with $\dim \overline{\{x\}}=j$, and let $F(x)$ be the residue field at $x$ [2501.11224].

The relevant Kato complex is the Bloch–Ogus type homological complex
$$
KC^{(s)}(X,\mathbf Z/p^r): \quad \cdots \xrightarrow{\partial} \bigoplus_{x\in X_j} H_p^{s+j+1}(F(x)) \xrightarrow{\partial} \cdots \xrightarrow{\partial} \bigoplus_{x\in X_1} H_p^{s+2}(F(x)) \xrightarrow{\partial} \bigoplus_{x\in X_0} H_p^{s+1}(F(x)).
$$
The associated Kato homology groups are defined by
$$
KH_j^{(s)}(X,\mathbf Z/p^r) := H_j\bigl(KC^{(s)}(X,\mathbf Z/p^r)\bigr).
$$
In particular,
$$
KH_0^{(s)}(X,\mathbf Z/p^r)=\operatorname{Coker}\!\left(\partial:\bigoplus_{x\in X_1}H_p^{s+2}(F(x))\to\bigoplus_{x\in X_0}H_p^{s+1}(F(x))\right).
$$
Here the boundary maps are those of the Bloch–Ogus/Kato complex, described as the standard residue maps of Kato’s Gersten-type complexes [2501.11224].

The structure morphism $f:X\to\operatorname{Spec}F$ induces
$$
f_*:KH_0^{(s)}(X,\mathbf Z/p^r)\to KH_0^{(s)}(\operatorname{Spec}F,\mathbf Z/p^r)=H_p^{s+1}(F).
$$
At the complex level, the paper considers
$$
\bigoplus_{x\in X_1} H_p^{s+2}(F(x)) \xrightarrow{\partial} \bigoplus_{x\in X_0} H_p^{s+1}(F(x)) \xrightarrow{\operatorname{Cor}} H_p^{s+1}(F),
$$
where $\operatorname{Cor}$ is the sum of the corestrictions $\operatorname{Cor}_{F(x)/F}$, and the kernel of $f_*^{KH}$ is the homology of this complex [2501.11224].

## 2. Extended differential symbol and the cohomological shift

The construction is motivated by the Bloch–Gabber–Kato theorem, which identifies Milnor $K$-theory mod $p^r$ with logarithmic Hodge–Witt cohomology:
$$
K_n^M(F)/p^r \xrightarrow{\sim} H^n(F,\mathbf Z/p^r\mathbf Z(n)).
$$
In the formulation used in the paper, for a field $F$ of characteristic $p>0$, Bloch–Gabber–Kato and Kahn’s theorem imply
$$
S T_{F,p^r}^n : \bigl(G_m^{\otimes_M n}(F)\bigr)/p^r \xrightarrow{\sim} K_n^M(F)/p^r = H^n(F,\mathbf Z/p^r\mathbf Z(n)),
$$
and equivalently
$$
S T_{F,p^r}^n : K_n^M(F)/p^r \xrightarrow{\sim} H^n(F,\mathbf Z/p^r\mathbf Z(n))=W_r\Omega^n_{F,\log},
\qquad
\{b_1,\dots,b_n\}\mapsto d\log[b_1]\cdots d\log[b_n].
$$
This is Theorem 3.2 of the paper [2501.11224].

The paper’s central new input is the extended differential symbol map, imported from the authors’ previous work and used here as an analogue of Bloch–Gabber–Kato one degree higher. For $r\ge 1$, with $\wp=Frob-id:W_r(E)\to W_r(E)$ the Artin–Schreier–Witt map, and for a Mackey functor $M$,
$$
(W_r\otimes_M M)(F)/\wp := \operatorname{Coker}(\wp).
$$
The groups
$$
H_p^{n+1}(F):=H^{n+1}(F,\mathbf Z/p^r\mathbf Z(n))=H^1(F,W_r\Omega^n_{F,\log})
$$
admit Kato’s explicit presentation, and the extended differential symbol is the isomorphism
$$
\widetilde S_{F,p^r}^n : \bigl(W_r\otimes_M G_m^{\otimes_M n}\bigr)(F)/\wp \xrightarrow{\sim} H_p^{n+1}(F),
$$
given on symbols by
$$
\{a,b_1,\dots,b_n\}_{E/F}\longmapsto \operatorname{Tr}_{E/F}\bigl([a,b_1,\dots,b_n)_E\bigr).
$$
This is Theorem 3.4 [2501.11224].

The paper emphasizes the compatibility between the classical differential symbol and the extended symbol through a commutative diagram in which the right vertical map sends
$$
d\log[b_1]\cdots d\log[b_n]\longmapsto [1,b_1,\dots,b_n)_F.
$$
Accordingly, Bloch–Gabber–Kato identifies $K_n^M/p^r$ with $H^n(-,\mathbf Z/p^r(n))$, whereas the extended differential symbol identifies the Witt/Mackey-theoretic enlargement
$$
(W_r\otimes_M G_m^{\otimes_M n})/\wp
$$
with
$$
H^{n+1}(-,\mathbf Z/p^r(n)).
$$
This one-degree shift is the mechanism by which Kato homology becomes a cohomological counterpart of the Milnor $K$-theoretic complexes underlying higher Chow groups [2501.11224].

## 3. Relation to higher Chow groups

The basic comparison starts from the Milnor $K$-complex
$$
\cdots \xrightarrow{\partial} \bigoplus_{x\in X_j} K_{s+j}^M(F(x)) \xrightarrow{\partial} \cdots \xrightarrow{\partial} \bigoplus_{x\in X_1} K_{s+1}^M(F(x)) \xrightarrow{\partial} \bigoplus_{x\in X_0} K_s^M(F(x)),
$$
whose $0$-th homology is canonically isomorphic to the higher Chow group
$$
\operatorname{Coker}\!\left(\partial:\bigoplus_{x\in X_1}K_{s+1}^M(F(x))\to\bigoplus_{x\in X_0}K_s^M(F(x))\right)\cong CH^{d+s}(X,s).
$$
The paper attributes this to Kato and Akhtar [2501.11224].

Because Bloch–Gabber–Kato gives the corresponding mod $p^r$ identification with Galois cohomology at the relevant degrees, the Kato complex $KC^{(s)}(X,\mathbf Z/p^r)$ is obtained from the Milnor complex after mod $p^r$ and a degree shift. In this precise sense, $KH_0^{(s)}$ is presented as the cohomological analogue of $CH^{d+s}(X,s)$ [2501.11224].

For any $r\ge 1$, Proposition 4.1 produces a canonical surjective homomorphism
$$
\psi:\bigl(W_r\otimes_M CH^{d+s}(X,s)\bigr)(F)/\wp \longrightarrow KH_0^{(s)}(X,\mathbf Z/p^r).
$$
The construction is fieldwise: it uses the canonical presentation of $CH^{d+s}(X_E,s)$ as the $0$-th homology of the Milnor complex over finite extensions $E/F$, then applies the extended differential symbol, and finally descends through compatibility with corestriction, transfer, and the projection formula. On a cycle
$$
Q=y\times b_1\times\cdots\times b_s
$$
the explicit formula is
$$
\psi_E(a\otimes[Q])=\operatorname{Cor}_{E(Q)/F}\bigl([a,b_1,\dots,b_s)_{E(Q)}\bigr).
$$
This furnishes the canonical bridge from the Witt/Mackey expression to the cohomological Kato complex [2501.11224].

The paper then isolates the kernel of the structure map on higher Chow groups. If
$$
f^{CH}:CH^{d+s}(X,s)\to K_s^M
$$
is induced by the structure morphism, define
$$
A^{d+s}(X,s):=\ker(f^{CH}).
$$
Under either condition (Pt) or condition (Cor), Proposition 4.3 yields the right exact sequence
$$
\bigl(W_r\otimes_M A^{d+s}(X,s)\bigr)(F)/\wp
\to
\bigl(W_r\otimes_M CH^{d+s}(X,s)\bigr)(F)/\wp
\to
\bigl(W_r\otimes_M K_s^M\bigr)(F)/\wp
\to 0.
$$
If one additionally assumes condition (Van), namely
$$
(G_a\otimes_M A^{d+s}(X,s))(F)/\wp=0,
$$
then $\psi$ becomes an isomorphism and so does the structure map
$$
f_*^{KH}:KH_0^{(s)}(X,\mathbf Z/p^r)\xrightarrow{\sim}H_p^{s+1}(F).
$$
Equivalently, the complex
$$
\bigoplus_{x\in X_1} H_p^{s+2}(F(x)) \xrightarrow{\partial} \bigoplus_{x\in X_0} H_p^{s+1}(F(x)) \xrightarrow{\operatorname{Cor}} H_p^{s+1}(F)
$$
is exact [2501.11224].

The paper explicitly frames these results as an analogue of Akhtar’s theorem on higher Chow groups. This suggests that the organization of the theory is governed by a transfer principle: higher Chow groups are computed by a Milnor $K$-complex, while Kato homology arises from the corresponding cohomological complex after application of the extended differential symbol [2501.11224].

## 4. Arithmetic behavior over finite, local, and global fields

A principal theme of the paper is that $KH_0^{(s)}$ can often be computed explicitly over arithmetic fields [2501.11224].

### Finite fields

If $F$ is finite of characteristic $p>0$ and $X$ is projective smooth geometrically irreducible over $F$, Proposition 4.4 states that if $F$ is perfect, then for $s>0$,
$$
(W_r\otimes_M CH_0(X)\otimes_M G_m^{\otimes_M s})(F)/\wp
=
KH_0^{(s)}(X,\mathbf Z/p^r)
=
0.
$$
If $F$ is algebraically closed, the same holds for $s\ge 0$ [2501.11224].

Theorem 4.6 proves that over a finite field the conditions (Van) and (Cor) hold. Therefore there is a canonical isomorphism
$$
\psi:\bigl(W_r\otimes_M CH_0(X)\bigr)(F)/\wp\xrightarrow{\sim}KH_0^{(0)}(X,\mathbf Z/p^r),
$$
and the structure map is an isomorphism
$$
f_*^{KH}:KH_0^{(0)}(X,\mathbf Z/p^r)\xrightarrow{\sim}H_p^1(F).
$$
Since
$$
H_p^1(F)\cong \mathbf Z/p^r\mathbf Z,
$$
one obtains
$$
KH_0^{(0)}(X,\mathbf Z/p^r)\cong \mathbf Z/p^r\mathbf Z.
$$
In addition, Theorem 1.1(1) gives
$$
KH_0^{(s)}(X,\mathbf Z/p^r)=0\qquad (s\ge 1).
$$
The paper notes that this is part of Kato’s conjecture and aligns with Jannsen–Saito’s theorem [2501.11224].

### Local fields

Let $F$ be a local field of characteristic $p>0$, that is, a complete discrete valuation field with finite residue field $k$. The paper assumes
$$
[F:F^p]=p,
$$
which holds in this setting [2501.11224].

Proposition 4.7 shows that for every $s\ge 2$,
$$
(W_r\otimes_M CH_0(X)\otimes_M G_m^{\otimes_M s})(F)/\wp
=
KH_0^{(s)}(X,\mathbf Z/p^r)
=
0.
$$
The paper attributes this to the $p$-divisibility of $K_s^M(F)$ for $s\ge 2$ [2501.11224].

For the case $s=1$, the strongest statement is proved when
$$
X=X_1\times\cdots\times X_d
$$
is a product of projective smooth geometrically irreducible curves over $F$. Writing
$$
SK_1(X)=CH^{d+1}(X,1),
\qquad
V(X):=A^{d+1}(X,1)=\ker\bigl(f^{CH}:SK_1(X)\to F^\times\bigr),
$$
Lemma 4.10 establishes
$$
(G_a\otimes_M V(X))(F)/\wp=0
$$
when $\dim(X)=1$ and $F$ is local, and Lemma 4.8 extends this vanishing from factors to products of curves [2501.11224].

Theorem 4.11(1) then yields, for any $r\ge 1$,
$$
\bigl(W_r\otimes_M CH^{d+1}(X,1)\bigr)(F)/\wp \xrightarrow{\sim} KH_0^{(1)}(X,\mathbf Z/p^r),
$$
and
$$
f_*^{KH}:KH_0^{(1)}(X,\mathbf Z/p^r)\xrightarrow{\sim}H_p^2(F).
$$
Since
$$
H_p^2(F)=Br(F)[p^r],
$$
this identifies
$$
KH_0^{(1)}(X,\mathbf Z/p^r)\cong Br(F)[p^r].
$$
If each factor $X_i$ has good reduction, Theorem 4.11(2) proves the residue isomorphism
$$
\partial_0:KH_0^{(1)}(X,\mathbf Z/p^r)\xrightarrow{\sim}KH_0^{(0)}(X_k,\mathbf Z/p^r),
$$
where $X_k$ is the special fiber [2501.11224].

### Global fields of positive characteristic

If $F$ is a global field of characteristic $p>0$, namely a function field of one variable over a finite field $k$, the paper again has
$$
[F:F^p]=p.
$$
By Proposition 4.7,
$$
KH_0^{(s)}(X,\mathbf Z/p^r)=0\qquad (s\ge 2)
$$
also in this case [2501.11224].

For $s=1$, let again
$$
X=X_1\times\cdots\times X_d
$$
be a product of projective smooth geometrically irreducible curves over $F$. Under condition (Pt) for each factor $X_i$, Lemma 4.13 proves, for a curve $X$,
$$
(G_a\otimes_M V(X))(F)/\wp=0,
$$
and Lemma 4.8 propagates this to products [2501.11224].

Theorem 4.14(1) therefore gives
$$
\bigl(W_r\otimes_M CH^{d+1}(X,1)\bigr)(F)/\wp \xrightarrow{\sim} KH_0^{(1)}(X,\mathbf Z/p^r),
$$
and
$$
f_*^{KH}:KH_0^{(1)}(X,\mathbf Z/p^r)\xrightarrow{\sim}H_p^2(F).
$$
Since
$$
H_p^2(F)\cong Br(F)[p^r],
$$
one gets
$$
KH_0^{(1)}(X,\mathbf Z/p^r)\cong Br(F)[p^r].
$$
Theorem 4.14(2) further establishes the short exact sequence
$$
0\to KH_0^{(1)}(X,\mathbf Z/p^r)\to \bigoplus_v KH_0^{(1)}(X_v,\mathbf Z/p^r)\to \mathbf Z/p^r\mathbf Z\to 0,
$$
deduced from the corresponding Hasse–Brauer–Noether exact sequence for Brauer groups [2501.11224].

## 5. Structural hypotheses and proof strategy

The paper’s comparison theorems are formulated for projective smooth geometrically irreducible schemes over $F$, and for the strongest arithmetic results in the $s=1$ case, more specifically for products of projective smooth geometrically irreducible curves [2501.11224].

Three auxiliary conditions are used systematically. The first is (Van),
$$
(G_a\otimes_M A^{d+s}(X,s))(F)/\wp=0.
$$
The second is (Pt): there is a finite separable extension $F'/F$ such that $X(F')\neq\varnothing$ and $[F':F]$ is prime to $p$. The third is (Cor): for every finite separable extension $F'/F$, the corestriction
$$
\operatorname{Cor}_{F'/F}:H_p^{s+1}(F')\to H_p^{s+1}(F)
$$
is injective [2501.11224].

The principal reduction step is to compare $CH^{d+s}(X,s)$ with $K_s^M(F)$ via the structure map, isolate the kernel
$$
A^{d+s}(X,s)=\ker(f^{CH}),
$$
and prove the vanishing of
$$
(G_a\otimes_M A^{d+s}(X,s))(F)/\wp.
$$
Once this is done, Proposition 4.3 yields the desired isomorphism with $H_p^{s+1}(F)$ [2501.11224].

In the finite field case, the paper states that (Cor) is easy because $H_p^1(F)\cong\mathbf Z/p^r\mathbf Z$ and corestriction is bijective. The proof of (Van) uses higher-dimensional class field theory and the exact sequence
$$
0\to T(X)\to A_0(X)\to Alb_X(F)\to 0,
$$
together with the facts that $T(X)$ is finite and eventually killed by passage to finite extension and that
$$
(G_a\otimes_M Alb_X)(F)=0.
$$
Norm arguments then imply
$$
(G_a\otimes_M T(X))(F)/\wp=0
$$
[2501.11224].

In the local field case, the vanishing
$$
(G_a\otimes_M V(X))(F)/\wp=0
$$
for curves uses semistable reduction of the Jacobian after finite extension, class field theory for curves over local fields, the exact sequence
$$
0\to V(X)^{fin}\to \pi_1^{ab}(X)^{geo}\to \widehat{\mathbf Z}^{\,r}\to 0,
$$
and surjectivity of the norm on $V(X)/p$ after passing to the Artin–Schreier extension $E=F(\wp^{-1}(a))$ [2501.11224].

In the global field case, the curve argument is more involved. The paper uses a Hochschild–Serre type spectral sequence
$$
E_2^{i,j}=H^i\bigl(F,H^j(X_{F^{sep}},\mathbf Z/p_{\log}(2))\bigr)\Rightarrow H^{i+j}(X,\mathbf Z/p_{\log}(2)),
$$
together with exact sequences connecting $V(X)/p$, $H^1(F,M)$, and $KH_0^{(1)}(X,\mathbf Z/p)$, plus local-global control at places $v$, Kato’s conjectures and local results for almost all places, and bijectivity of local corestriction maps under assumptions on reduction of Jacobians [2501.11224].

## 6. Reciprocity sheaves interpretation

The final section reformulates the comparison theory using reciprocity sheaves and modulus presheaves with transfers. The point of this reformulation is not that $KH_0$ itself is directly defined as the value of a reciprocity sheaf. Rather, the paper interprets the comparison maps and their kernels through tensor products in the reciprocity-sheaf setting, so that the obstructions become reciprocity-law-type relations [2501.11224].

The category $RSC_k$ is described as containing $A^1$-invariant presheaves, smooth commutative algebraic groups, and de Rham–Witt sheaves. The relevant tensor comparison gives canonical surjections
$$
(ho(F_1)\otimes_M ho(F_2))(F)\to h_0(F_1\otimes F_2)(F),
$$
and the kernel is interpreted as the part arising from Weil reciprocity or reciprocity with modulus [2501.11224].

Corollary 5.15 is the main reciprocity-sheaf reformulation. It states:

- if $F/k$ is a field extension of characteristic $p\ge 3$, the reciprocity for
  $$
  W_r\otimes_M G_m^{\otimes_M n}
  $$
  is killed by $\wp\otimes id$ over $F$;

- if $k$ is finite and $F/k$ is finite, and $X$ is projective smooth geometrically irreducible over $k$, then the reciprocity for
  $$
  W_r\otimes_M CH_0(X)
  $$
  is killed by $\wp\otimes id$ over $F$;

- if $k$ is finite and $F$ is either a local field with residue field $k$ or a global field over $k$, and
  $$
  X=X_k\otimes_k F
  $$
  for a projective smooth geometrically irreducible curve $X_k/k$, then the reciprocity for
  $$
  W_r\otimes_M CH^2(X,1)
  $$
  is killed by $\wp\otimes id$ over $F$ [2501.11224].

This suggests a conceptual interpretation of the comparison isomorphisms: the passage from Witt/Mackey tensor constructions to Kato homology can be understood as quotienting by reciprocity relations that become trivial modulo the Artin–Schreier–Witt operator [2501.11224].

## 7. Significance and scope

The paper’s main contribution is twofold. First, it places $KH_0^{(s)}(X,\mathbf Z/p^r)$ in a systematic cohomological framework parallel to the Milnor $K$-theoretic description of higher Chow groups. Second, it computes these groups in several arithmetic situations, frequently reducing them to classical invariants such as $\mathbf Z/p^r\mathbf Z$ or $Br(F)[p^r]$ [2501.11224].

The following summary collects the principal identifications established in the paper.

| Base field and hypotheses | Result |
|---|---|
| $F$ finite, $X$ projective smooth geometrically irreducible | $KH_0^{(0)}(X,\mathbf Z/p^r)\cong \mathbf Z/p^r\mathbf Z$ |
| $F$ finite, same hypotheses, $s\ge 1$ | $KH_0^{(s)}(X,\mathbf Z/p^r)=0$ |
| $F$ local of characteristic $p>0$, $X$ arbitrary as above, $s\ge 2$ | $KH_0^{(s)}(X,\mathbf Z/p^r)=0$ |
| $F$ local, $X$ a product of projective smooth geometrically irreducible curves | $KH_0^{(1)}(X,\mathbf Z/p^r)\cong Br(F)[p^r]$ |
| $F$ global of characteristic $p>0$, $X$ arbitrary as above, $s\ge 2$ | $KH_0^{(s)}(X,\mathbf Z/p^r)=0$ |
| $F$ global of characteristic $p>0$, $X$ a product of curves satisfying (Pt) | $KH_0^{(1)}(X,\mathbf Z/p^r)\cong Br(F)[p^r]$ |

In this presentation, Kato homology groups are neither treated as isolated invariants nor as formal analogues alone. They are integrated with Bloch–Gabber–Kato theory, higher Chow groups, Gersten-type residue complexes, class field theory, Brauer groups, and reciprocity sheaves. A plausible implication is that the paper identifies $KH_0$ as the natural one-degree-up cohomological companion to the higher Chow groups in characteristic $p>0$, with arithmetic consequences that are explicit and computable in the cases considered [2501.11224].

Source: https://www.emergentmind.com/topics/kato-homology-groups