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PreprintarXiv licencearXiv

Additive invariants of Finite-dimensional DG-algebras

  • Isambard Goodbody
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Abstract

Orlov has shown that additive invariants such as K-theory and Hochschild homology of smooth finite-dimensional DG-algebras behave similarly to those of finite-dimensional algebras. The similarities only go so far: there are examples of finite-dimensional DG-algebras with vanishing and uncountable Grothendieck groups. We show that for DG-quivers with relations, the Grothendieck groups are free of the expected rank and that, in the smooth case, all additive invariants are determined by the base field. We use our results to rule out certain finite-dimensional DG-algebras from being phantoms.

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Cite

BibTeXRIS
BibTeX
@misc{goodbody2026additive,
  title = {{Additive invariants of Finite-dimensional DG-algebras}},
  author = {Isambard Goodbody},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.26302},
  eprint = {2610.26302},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.26302},
}
RIS
TY  - UNPB
TI  - Additive invariants of Finite-dimensional DG-algebras
AU  - Isambard Goodbody
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.26302
UR  - https://arxiv.org/abs/2610.26302
AB  - Orlov has shown that additive invariants such as K-theory and Hochschild homology of smooth finite-dimensional DG-algebras behave similarly to those of finite-dimensional algebras. The similarities only go so far: there are examples of finite-dimensional DG-algebras with vanishing and uncountable Grothendieck groups. We show that for DG-quivers with relations, the Grothendieck groups are free of the expected rank and that, in the smooth case, all additive invariants are determined by the base field. We use our results to rule out certain finite-dimensional DG-algebras from being phantoms.
ER  -