An Abstract Index Theorem via Rees Algebras

Abstract

We develop a purely algebraic framework for index-type theorems based on the Rees construction for filtered differential graded algebras (FDGAs). Alongside the classical Rees module we introduce a smooth variant C^ω_ℛA, adapted to analytic arguments, and we study traces and their pointwise and coefficient-wise extensions to Rees algebras. The main result (Corollary 3.11) is an abstract index theorem: given a compatible datum of filtered differential graded associative algebras with traces A, B, G with a morphism ϕ: A→ B, an action ρ: G⊗ A→ A and a morphism i: G→ B, together with a graded-central element, one has tr_B(e^f₀+f₁) = tr_gr̂A(e^γh), where the left-hand side is the trace in B and the right-hand side the trace in the Laurent series associated graded gr̂A of A. Here f₀+f₁ is a curvature-type element, i.e., an element of the form d_Bβ+β² for some odd-degree element β, while γ and h are certain elements in gr̂A. The formalism is modelled on the Getzler rescaling technique and on the derivation of the localization formula for the loop space Chern character by Ludewig and Yi.

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BibTeX RIS

BibTeX

@misc{landi2026abstract,
  title = {{An Abstract Index Theorem via Rees Algebras}},
  author = {Eugenio Landi},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.30587},
  eprint = {2608.30587},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.30587},
}

RIS

TY  - UNPB
TI  - An Abstract Index Theorem via Rees Algebras
AU  - Eugenio Landi
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2608.30587
UR  - https://arxiv.org/abs/2608.30587
AB  - We develop a purely algebraic framework for index-type theorems based on the Rees construction for filtered differential graded algebras (FDGAs). Alongside the classical Rees module we introduce a smooth variant C^ω_ℛA, adapted to analytic arguments, and we study traces and their pointwise and coefficient-wise extensions to Rees algebras. The main result (Corollary 3.11) is an abstract index theorem: given a compatible datum of filtered differential graded associative algebras with traces A, B, G with a morphism ϕ: A→ B, an action ρ: G⊗ A→ A and a morphism i: G→ B, together with a graded-central element, one has tr_B(e^f_0+f_1) = tr_gr̂A(e^γh), where the left-hand side is the trace in B and the right-hand side the trace in the Laurent series associated graded gr̂A of A. Here f_0+f_1 is a curvature-type element, i.e., an element of the form d_Bβ+β^2 for some odd-degree element β, while γ and h are certain elements in gr̂A. The formalism is modelled on the Getzler rescaling technique and on the derivation of the localization formula for the loop space Chern character by Ludewig and Yi.
ER  -

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