Kato Homology Groups: Cohomology in Characteristic p

• Kato homology groups are cohomological invariants defined via Bloch–Ogus complexes using p-primary Galois cohomology on varieties in characteristic p > 0.
• They connect with higher Chow groups through an extended differential symbol, serving as a one-degree higher analogue of the classical differential symbol in the Bloch–Gabber–Kato theorem.
• Computed over finite, local, and global fields, these groups reveal arithmetic behavior by linking to classical invariants such as Brauer groups under various structural hypotheses.

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" (Hiranouchi et al., 20 Jan 2025), 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 (Hiranouchi et al., 20 Jan 2025).

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 $p>0$0-scheme of dimension $p>0$1. For each $p>0$2, let $p>0$3 denote the set of points $p>0$4 with $p>0$5, and let $p>0$6 be the residue field at $p>0$7 (Hiranouchi et al., 20 Jan 2025).

The relevant Kato complex is the Bloch–Ogus type homological complex

$p>0$8

The associated Kato homology groups are defined by

$p>0$9

In particular,

$0$0

Here the boundary maps are those of the Bloch–Ogus/Kato complex, described as the standard residue maps of Kato’s Gersten-type complexes (Hiranouchi et al., 20 Jan 2025).

The structure morphism $0$1 induces

$0$2

At the complex level, the paper considers

$0$3

where $0$4 is the sum of the corestrictions $0$5, and the kernel of $0$6 is the homology of this complex (Hiranouchi et al., 20 Jan 2025).

2. Extended differential symbol and the cohomological shift

The construction is motivated by the Bloch–Gabber–Kato theorem, which identifies Milnor $0$7-theory mod $0$8 with logarithmic Hodge–Witt cohomology:

$0$9

In the formulation used in the paper, for a field $KH_0^{(s)}(X,\mathbf Z/p^r)$0 of characteristic $KH_0^{(s)}(X,\mathbf Z/p^r)$1, Bloch–Gabber–Kato and Kahn’s theorem imply

$KH_0^{(s)}(X,\mathbf Z/p^r)$2

and equivalently

$KH_0^{(s)}(X,\mathbf Z/p^r)$3

This is Theorem 3.2 of the paper (Hiranouchi et al., 20 Jan 2025).

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 $KH_0^{(s)}(X,\mathbf Z/p^r)$4, with $KH_0^{(s)}(X,\mathbf Z/p^r)$5 the Artin–Schreier–Witt map, and for a Mackey functor $KH_0^{(s)}(X,\mathbf Z/p^r)$6,

$KH_0^{(s)}(X,\mathbf Z/p^r)$7

The groups

$KH_0^{(s)}(X,\mathbf Z/p^r)$8

admit Kato’s explicit presentation, and the extended differential symbol is the isomorphism

$KH_0^{(s)}(X,\mathbf Z/p^r)$9

given on symbols by

$CH^{d+s}(X,s)$0

This is Theorem 3.4 (Hiranouchi et al., 20 Jan 2025).

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

$CH^{d+s}(X,s)$1

Accordingly, Bloch–Gabber–Kato identifies $CH^{d+s}(X,s)$2 with $CH^{d+s}(X,s)$3, whereas the extended differential symbol identifies the Witt/Mackey-theoretic enlargement

$CH^{d+s}(X,s)$4

with

$CH^{d+s}(X,s)$5

This one-degree shift is the mechanism by which Kato homology becomes a cohomological counterpart of the Milnor $CH^{d+s}(X,s)$6-theoretic complexes underlying higher Chow groups (Hiranouchi et al., 20 Jan 2025).

3. Relation to higher Chow groups

The basic comparison starts from the Milnor $CH^{d+s}(X,s)$7-complex

$CH^{d+s}(X,s)$8

whose $CH^{d+s}(X,s)$9-th homology is canonically isomorphic to the higher Chow group

$F$0

The paper attributes this to Kato and Akhtar (Hiranouchi et al., 20 Jan 2025).

Because Bloch–Gabber–Kato gives the corresponding mod $F$1 identification with Galois cohomology at the relevant degrees, the Kato complex $F$2 is obtained from the Milnor complex after mod $F$3 and a degree shift. In this precise sense, $F$4 is presented as the cohomological analogue of $F$5 (Hiranouchi et al., 20 Jan 2025).

For any $F$6, Proposition 4.1 produces a canonical surjective homomorphism

$F$7

The construction is fieldwise: it uses the canonical presentation of $F$8 as the $F$9-th homology of the Milnor complex over finite extensions $p>0$0, then applies the extended differential symbol, and finally descends through compatibility with corestriction, transfer, and the projection formula. On a cycle

$p>0$1

the explicit formula is

$p>0$2

This furnishes the canonical bridge from the Witt/Mackey expression to the cohomological Kato complex (Hiranouchi et al., 20 Jan 2025).

The paper then isolates the kernel of the structure map on higher Chow groups. If

$p>0$3

is induced by the structure morphism, define

$p>0$4

Under either condition (Pt) or condition (Cor), Proposition 4.3 yields the right exact sequence

$p>0$5

If one additionally assumes condition (Van), namely

$p>0$6

then $p>0$7 becomes an isomorphism and so does the structure map

$p>0$8

Equivalently, the complex

$p>0$9

is exact (Hiranouchi et al., 20 Jan 2025).

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 $[F:F^p]\le p^s$0-complex, while Kato homology arises from the corresponding cohomological complex after application of the extended differential symbol (Hiranouchi et al., 20 Jan 2025).

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

A principal theme of the paper is that $[F:F^p]\le p^s$1 can often be computed explicitly over arithmetic fields (Hiranouchi et al., 20 Jan 2025).

Finite fields

If $[F:F^p]\le p^s$2 is finite of characteristic $[F:F^p]\le p^s$3 and $[F:F^p]\le p^s$4 is projective smooth geometrically irreducible over $[F:F^p]\le p^s$5, Proposition 4.4 states that if $[F:F^p]\le p^s$6 is perfect, then for $[F:F^p]\le p^s$7,

$[F:F^p]\le p^s$8

If $[F:F^p]\le p^s$9 is algebraically closed, the same holds for $s\ge 0$0 (Hiranouchi et al., 20 Jan 2025).

Theorem 4.6 proves that over a finite field the conditions (Van) and (Cor) hold. Therefore there is a canonical isomorphism

$s\ge 0$1

and the structure map is an isomorphism

$s\ge 0$2

Since

$s\ge 0$3

one obtains

$s\ge 0$4

In addition, Theorem 1.1(1) gives

$s\ge 0$5

The paper notes that this is part of Kato’s conjecture and aligns with Jannsen–Saito’s theorem (Hiranouchi et al., 20 Jan 2025).

Local fields

Let $s\ge 0$6 be a local field of characteristic $s\ge 0$7, that is, a complete discrete valuation field with finite residue field $s\ge 0$8. The paper assumes

$s\ge 0$9

which holds in this setting (Hiranouchi et al., 20 Jan 2025).

Proposition 4.7 shows that for every $X$0,

$X$1

The paper attributes this to the $X$2-divisibility of $X$3 for $X$4 (Hiranouchi et al., 20 Jan 2025).

For the case $X$5, the strongest statement is proved when

$X$6

is a product of projective smooth geometrically irreducible curves over $X$7. Writing

$X$8

Lemma 4.10 establishes

$X$9

when $p>0$00 and $p>0$01 is local, and Lemma 4.8 extends this vanishing from factors to products of curves (Hiranouchi et al., 20 Jan 2025).

Theorem 4.11(1) then yields, for any $p>0$02,

$p>0$03

and

$p>0$04

Since

$p>0$05

this identifies

$p>0$06

If each factor $p>0$07 has good reduction, Theorem 4.11(2) proves the residue isomorphism

$p>0$08

where $p>0$09 is the special fiber (Hiranouchi et al., 20 Jan 2025).

Global fields of positive characteristic

If $p>0$10 is a global field of characteristic $p>0$11, namely a function field of one variable over a finite field $p>0$12, the paper again has

$p>0$13

By Proposition 4.7,

$p>0$14

also in this case (Hiranouchi et al., 20 Jan 2025).

For $p>0$15, let again

$p>0$16

be a product of projective smooth geometrically irreducible curves over $p>0$17. Under condition (Pt) for each factor $p>0$18, Lemma 4.13 proves, for a curve $p>0$19,

$p>0$20

and Lemma 4.8 propagates this to products (Hiranouchi et al., 20 Jan 2025).

Theorem 4.14(1) therefore gives

$p>0$21

and

$p>0$22

Since

$p>0$23

one gets

$p>0$24

Theorem 4.14(2) further establishes the short exact sequence

$p>0$25

deduced from the corresponding Hasse–Brauer–Noether exact sequence for Brauer groups (Hiranouchi et al., 20 Jan 2025).

5. Structural hypotheses and proof strategy

The paper’s comparison theorems are formulated for projective smooth geometrically irreducible schemes over $p>0$26, and for the strongest arithmetic results in the $p>0$27 case, more specifically for products of projective smooth geometrically irreducible curves (Hiranouchi et al., 20 Jan 2025).

Three auxiliary conditions are used systematically. The first is (Van),

$p>0$28

The second is (Pt): there is a finite separable extension $p>0$29 such that $p>0$30 and $p>0$31 is prime to $p>0$32. The third is (Cor): for every finite separable extension $p>0$33, the corestriction

$p>0$34

is injective (Hiranouchi et al., 20 Jan 2025).

The principal reduction step is to compare $p>0$35 with $p>0$36 via the structure map, isolate the kernel

$p>0$37

and prove the vanishing of

$p>0$38

Once this is done, Proposition 4.3 yields the desired isomorphism with $p>0$39 (Hiranouchi et al., 20 Jan 2025).

In the finite field case, the paper states that (Cor) is easy because $p>0$40 and corestriction is bijective. The proof of (Van) uses higher-dimensional class field theory and the exact sequence

$p>0$41

together with the facts that $p>0$42 is finite and eventually killed by passage to finite extension and that

$p>0$43

Norm arguments then imply

$p>0$44

(Hiranouchi et al., 20 Jan 2025).

In the local field case, the vanishing

$p>0$45

for curves uses semistable reduction of the Jacobian after finite extension, class field theory for curves over local fields, the exact sequence

$p>0$46

and surjectivity of the norm on $p>0$47 after passing to the Artin–Schreier extension $p>0$48 (Hiranouchi et al., 20 Jan 2025).

In the global field case, the curve argument is more involved. The paper uses a Hochschild–Serre type spectral sequence

$p>0$49

together with exact sequences connecting $p>0$50, $p>0$51, and $p>0$52, plus local-global control at places $p>0$53, Kato’s conjectures and local results for almost all places, and bijectivity of local corestriction maps under assumptions on reduction of Jacobians (Hiranouchi et al., 20 Jan 2025).

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 $p>0$54 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 (Hiranouchi et al., 20 Jan 2025).

The category $p>0$55 is described as containing $p>0$56-invariant presheaves, smooth commutative algebraic groups, and de Rham–Witt sheaves. The relevant tensor comparison gives canonical surjections

$p>0$57

and the kernel is interpreted as the part arising from Weil reciprocity or reciprocity with modulus (Hiranouchi et al., 20 Jan 2025).

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

• if $p>0$58 is a field extension of characteristic $p>0$59, the reciprocity for

$p>0$60

is killed by $p>0$61 over $p>0$62;

• if $p>0$63 is finite and $p>0$64 is finite, and $p>0$65 is projective smooth geometrically irreducible over $p>0$66, then the reciprocity for

$p>0$67

is killed by $p>0$68 over $p>0$69;

• if $p>0$70 is finite and $p>0$71 is either a local field with residue field $p>0$72 or a global field over $p>0$73, and

$p>0$74

for a projective smooth geometrically irreducible curve $p>0$75, then the reciprocity for

$p>0$76

is killed by $p>0$77 over $p>0$78 (Hiranouchi et al., 20 Jan 2025).

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 (Hiranouchi et al., 20 Jan 2025).

7. Significance and scope

The paper’s main contribution is twofold. First, it places $p>0$79 in a systematic cohomological framework parallel to the Milnor $p>0$80-theoretic description of higher Chow groups. Second, it computes these groups in several arithmetic situations, frequently reducing them to classical invariants such as $p>0$81 or $p>0$82 (Hiranouchi et al., 20 Jan 2025).

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

Base field and hypotheses	Result
$p>0$83 finite, $p>0$84 projective smooth geometrically irreducible	$p>0$85
$p>0$86 finite, same hypotheses, $p>0$87	$p>0$88
$p>0$89 local of characteristic $p>0$90, $p>0$91 arbitrary as above, $p>0$92	$p>0$93
$p>0$94 local, $p>0$95 a product of projective smooth geometrically irreducible curves	$p>0$96
$p>0$97 global of characteristic $p>0$98, $p>0$99 arbitrary as above, $0$00	$0$01
$0$02 global of characteristic $0$03, $0$04 a product of curves satisfying (Pt)	$0$05

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 $0$06 as the natural one-degree-up cohomological companion to the higher Chow groups in characteristic $0$07, with arithmetic consequences that are explicit and computable in the cases considered (Hiranouchi et al., 20 Jan 2025).