Fractional Calabi–Yau Properties of Higher Representation-Finite Algebras

Definitions and mathematical statement

The supplied question concerns higher representation-finite algebras and twisted fractionally Calabi--Yau algebras. The excerpt does not specify the ground field or the ambient finiteness hypotheses. The following gives a precise formulation under the usual finite-dimensional, finite-global-dimension interpretation; confirmation of the intended ambient class remains necessary.

Let $k$ be a field and let $A$ be a finite-dimensional $k$-algebra of finite global dimension. Write $\operatorname{mod}A$ for the category of finite-dimensional right $A$-modules and $\mathcal D_A=D^b(\operatorname{mod}A)$. Its Serre functor is $S_A=-\otimes_A^{\mathbf L}DA$, where $DA=\operatorname{Hom}_k(A,k)$. Write $[m]$ for the cohomological shift by $m\in{\mathbb{Z}}$. For a $k$-algebra automorphism $\phi$ of $A$, let $T_\phi$ be the induced twisting functor, obtained on right modules by replacing the action with $x\cdot a=x\phi(a)$.

In this formulation, $A$ is fractionally Calabi--Yau if there exist integers $\ell>0$ and $m\in{\mathbb{Z}}$ and a natural isomorphism of triangle functors \[ S_A^{\ell}\cong[m]. \] It is twisted fractionally Calabi--Yau if there additionally may be an automorphism $\phi$, so that \[ S_A^{\ell}\cong[m]\circ T_\phi \qquad\text{for some }\ell>0,\ m\in{\mathbb{Z}},\ \phi\in\operatorname{Aut}_{k\text{-alg}}(A). \] The fractional Calabi--Yau condition sought below may use different integers from those witnessing the twisted condition.

For an integer $n\geq1$, use the convention that $A$ is $n$-representation-finite if $\operatorname{gl.dim}A\leq n$ and there is a module $M\in\operatorname{mod}A$ such that \[ \begin{aligned} \operatorname{add}M &=\{X\in\operatorname{mod}A:\operatorname{Ext}_A^i(M,X)=0\text{ for every }1\leq i<n\}\\ &=\{X\in\operatorname{mod}A:\operatorname{Ext}_A^i(X,M)=0\text{ for every }1\leq i<n\}. \end{aligned} \] Here $\operatorname{add}M$ consists of direct summands of finite direct sums of copies of $M$; when $n=1$, the displayed vanishing conditions are empty. Interpret higher representation-finite as $n$-representation-finite for some integer $n\geq1$, subject to confirmation of the source's convention.

Question under this interpretation. For every algebra $A$ in the intended ambient class, does being higher representation-finite imply that \[ \exists\,p\in{\mathbb{Z}}_{>0}\ \exists\,q\in{\mathbb{Z}}:\quad S_A^p\cong[q]? \] More generally, does being twisted fractionally Calabi--Yau imply this same conclusion for every such $A$? The supplied curation reports that higher representation-finite algebras are twisted fractionally Calabi--Yau; that background assertion is not proved or independently verified here.

The surrounding excerpt discusses entropy vanishing results and imposes the additional hypothesis that $A$ is elementary for its Hochschild (co)homology entropy assertion. The supplied question does not explicitly impose that additional hypothesis. The excerpt's entropy notation is incomplete, so those surrounding results are not reconstructed as mathematical assertions here.

Short English statement

Must every higher representation-finite algebra have a positive power of its Serre functor naturally isomorphic to a shift? More generally, does the same conclusion follow whenever a positive Serre power is a shift composed with a twist by an algebra automorphism? The ground field, ambient hypotheses, and source conventions need confirmation. The supplied open status is dated curation, not an independently verified assessment.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005906$, source problem code catalogue entry, title: Fractional Calabi--Yau Properties of Higher Representation-Finite Algebras. No individual proposer is supplied; the recorded proposed year is 2024.
• Hochschild (Co)Homology and Applications (2024), Oberwolfach Reports, pp. 1097--1162, DOI 10.4171/owr/2024/20. Supplied report link: https://doi.org/10.4171/owr/2024/20. The supplied passage attributes the question to its internal reference [1, Question 1.6]; the record does not explicitly identify that reference's full bibliographic entry.
• Herschend and Iyama, supplied literature-source label and DOI link: https://doi.org/10.1112/blms/bdq101.
• Han, entropies of Serre functors, supplied literature-source label and DOI link: https://doi.org/10.1016/j.jalgebra.2024.03.031.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and reports that higher representation-finite algebras are known to be twisted fractionally Calabi--Yau, while their fractional Calabi--Yau property remains unknown. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its status and cited claims require independent checking.
• The registry labels the statement corrected_verified and says that only formula delimiters, notation, grammar, and typography were repaired while preserving mathematical content and scope. These supplied labels do not establish mathematical correctness or independent source verification.