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Abstract We investigate maximal tori in the Hochschild cohomology Lie algebra $${\operatorname {HH}}^1(A)$$ of a finite dimensional algebra $$A$$, and their connection with the fundamental groups associated to presentations of $$A$$. We prove that every maximal torus in $${\operatorname {HH}}^1(A)$$ arises as the dual of some fundamental group of $$A$$, extending the work by Farkas, Green, and Marcos; de la Peña and Saorín; and Le Meur. Combining this with known invariance results for Hochschild cohomology, we deduce that (in rough terms) the largest rank of a fundamental group of $$A$$ is a derived invariant quantity, and among self-injective algebras, an invariant under stable equivalences of Morita type. Using this we prove that there are only finitely many monomial algebras in any derived equivalence class of finite dimensional algebras; hitherto this was known only for very restricted classes of monomial algebras.
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The finite dual coalgebra as a quantization of the maximal spectrum
In pursuit of a noncommutative spectrum functor, we argue that the Heyneman-Sweedler finite dual coalgebra can be viewed as a quantization of the maximal spectrum of a commutative affine algebra, integrating prior perspectives of Takeuchi, Batchelor, Kontsevich-Soibelman, and Le Bruyn. We introduce fully residually finite-dimensional algebras A as those with enough finite-dimensional representations to let A^o act as an appropriate depiction of the noncommutative maximal spectrum of A; importantly, this class includes affine noetherian PI algebras. In the case of prime affine algebras that are module-finite over their center, we describe how the Azumaya locus is represented in the finite dual. This is used to describe the finite dual of quantum planes at roots of unity as an endeavor to visualize the noncommutative space on which these algebras act as functions. Finally, we discuss how a similar analysis can be carried out for other maximal orders over surfaces.
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- Award ID(s):
- 2201273
- PAR ID:
- 10505104
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Journal of Algebra
- Volume:
- 644
- Issue:
- C
- ISSN:
- 0021-8693
- Page Range / eLocation ID:
- 287 to 328
- Subject(s) / Keyword(s):
- noncommutative spectrum, quantum set, dual coalgebra, Azumaya locus
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1016/j.jalgebra.2023.12.035
