The Serre--Hochschild plane of a finite-dimensional algebra
Abstract
For a finite-dimensional algebra A over a field we organize the Hochschild cohomology of A with coefficients in all derived tensor powers of the Serre bimodule ω=D A into a single bigraded derived invariant, the \emph{Serre--Hochschild plane} Tp,m(A), defined over all of ℤ2 without inverting ω. The column m=0 is the Tamarkin--Tsygan calculus, m=1 is dual Hochschild homology (the column governing Han's conjecture), m=2 recovers the τ-Hochschild shadow, and m=−1 Keller's Calabi--Yau completions. We prove a ladder theorem making every column a symmetric module over the calculus, a coefficient spectral sequence, a Serre reflection fixing the Han column, a dictionary identifying the plane of a geometric algebra with twisted polyvector cohomology of a smooth projective variety, finite generation of the homology column under the condition Fg with a growth trichotomy for dimHHn(A), and a complete graded Tate criterion for Han's conjecture on periodic algebras. Han's conjecture follows for symmetric periodic algebras in every characteristic and for local periodic algebras in characteristic zero. We exhibit the first example of a stably traceless periodic algebra, showing that the graded criterion is sharp.
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BibTeX
@misc{armenta2026serrehochschild,
title = {{The Serre--Hochschild plane of a finite-dimensional algebra}},
author = {Marco Armenta},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20220},
eprint = {2609.20220},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20220},
}
RIS
TY - UNPB
TI - The Serre--Hochschild plane of a finite-dimensional algebra
AU - Marco Armenta
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.20220
UR - https://arxiv.org/abs/2609.20220
AB - For a finite-dimensional algebra A over a field we organize the Hochschild cohomology of A with coefficients in all derived tensor powers of the Serre bimodule ω=D A into a single bigraded derived invariant, the \emph{Serre--Hochschild plane} T^p,m(A), defined over all of ℤ^2 without inverting ω. The column m=0 is the Tamarkin--Tsygan calculus, m=1 is dual Hochschild homology (the column governing Han's conjecture), m=2 recovers the τ-Hochschild shadow, and m=−1 Keller's Calabi--Yau completions. We prove a ladder theorem making every column a symmetric module over the calculus, a coefficient spectral sequence, a Serre reflection fixing the Han column, a dictionary identifying the plane of a geometric algebra with twisted polyvector cohomology of a smooth projective variety, finite generation of the homology column under the condition Fg with a growth trichotomy for dimHH_n(A), and a complete graded Tate criterion for Han's conjecture on periodic algebras. Han's conjecture follows for symmetric periodic algebras in every characteristic and for local periodic algebras in characteristic zero. We exhibit the first example of a stably traceless periodic algebra, showing that the graded criterion is sharp.
ER -