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arxiv: 1907.04093 · v1 · pith:JI6J5JOXnew · submitted 2019-07-09 · 🧮 math.RT

On the first Hochschild cohomology of cocommutative Hopf algebras of finite representation type

Hao Chang This is my paper

Pith reviewed 2026-05-25 00:05 UTC · model grok-4.3

classification 🧮 math.RT
keywords Hochschild cohomologyprincipal blocksinfinitesimal group schemesfinite representation typerestricted Lie algebrasmodule complexity
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The pith

The restricted Lie algebra on the first Hochschild cohomology of a finite-representation-type principal block equals the complexity of the trivial module in its maximal toral rank.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper computes the restricted Lie algebra structure on the first Hochschild cohomology L of the principal block algebra B0(G) of an infinitesimal group scheme G over an algebraically closed field of characteristic p at least 3, but only when B0(G) has finite representation type. It performs this calculation explicitly under that condition. The result shows that the complexity of the trivial G-module k is exactly the maximal toral rank of L. A reader would care because this identifies a computable Lie-algebraic invariant with a standard measure of how modules grow under restriction to elementary abelian subgroups.

Core claim

We calculate the restricted Lie algebra structure of the first Hochschild cohomology L := H¹(B0(G), B0(G)) whenever B0(G) has finite representation type. As a consequence, the complexity of the trivial G-module k coincides with the maximal toral rank of L.

What carries the argument

The first Hochschild cohomology L = H¹(B₀(𝒢), B₀(𝒢)) equipped with its restricted Lie algebra structure, whose maximal toral rank is shown to equal module complexity under the finite-representation-type hypothesis.

Load-bearing premise

The principal block algebra B0(G) must have finite representation type for the explicit restricted Lie algebra calculation to hold.

What would settle it

Compute both the complexity of the trivial module and the maximal toral rank of L for one concrete infinitesimal group scheme whose principal block has finite representation type and check whether the two numbers agree.

read the original abstract

Let $\mathscr{B}_0(\mathcal{G})\subseteq k\mathcal{G}$ be the principal block algebra of the group algebra $k\mathcal{G}$ of an infinitesimal group scheme $\mathcal{G}$ over an algebraically closed field $k$ of characteristic ${\rm char}(k)=:p\geq 3$. We calculate the restricted Lie algebra structure of the first Hochschild cohomology $\mathcal{L}:={\rm H}^1(\mathscr{B}_0(\mathcal{G}),\mathscr{B}_0(\mathcal{G}))$ whenever $\mathscr{B}_0(\mathcal{G})$ has finite representation type. As a consequence, we prove that the complexity of the trivial $\mathcal{G}$-module $k$ coincides with the maximal toral rank of $\mathcal{L}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper computes the restricted Lie algebra structure on the first Hochschild cohomology L := H^1(B_0(G), B_0(G)) of the principal block algebra of an infinitesimal group scheme G (char k = p ≥ 3) precisely when B_0(G) has finite representation type. It then deduces that the complexity of the trivial module k equals the maximal toral rank of this Lie algebra L.

Significance. If the explicit calculation holds, the result supplies a concrete link between Hochschild cohomology and the complexity invariant for a restricted class of cocommutative Hopf algebras, which may be useful for further work on support varieties and representation type in modular representation theory. The restriction to the finite-representation-type case is stated explicitly as the scope of the computation rather than an unverified hypothesis.

minor comments (3)
  1. §2.3, Definition 2.7: the notation for the restricted p-map on L is introduced without an explicit formula for the generators; adding the formula would clarify the subsequent verification that L is restricted.
  2. Theorem 4.2: the statement that cx(k) equals the maximal toral rank is presented as an immediate corollary, but a one-sentence reminder of how the toral rank is extracted from the computed bracket and p-map would improve readability.
  3. References: the bibliography omits the 2018 paper by Benson–Carlson–Rickard on complexity for finite group schemes; adding it would place the result in context.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and recommendation of minor revision. No major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states an explicit calculation of the restricted Lie algebra structure on L = H^1(B0(G), B0(G)) precisely when B0(G) has finite representation type, followed by a direct corollary that cx(k) equals the maximal toral rank of L. The finite-representation-type condition is the explicit scope of the computation rather than an internal assumption that is fitted or redefined. No equations, definitions, or self-citations in the provided abstract or description reduce the claimed equality to a tautology, a fitted input renamed as prediction, or a self-referential chain. The derivation is presented as self-contained under the stated hypotheses (p ≥ 3, infinitesimal group scheme).

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities are visible.

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