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For a net of C*-algebras on a discrete metric space, we introduce a bimodule version of the DHR tensor category and show that it is an invariant of quasi-local algebras under isomorphisms with bounded spread. For abstract spin systems on a latticeL\subseteq \mathbb{R}^{n}satisfying a weak version of Haag duality, we construct a braiding on these categories. Applying the general theory to quasi-local algebrasAof operators on a lattice invariant under a (categorical) symmetry, we obtain a homomorphism from the group of symmetric QCA to\mathbf{Aut}_{\mathrm{br}}(\mathbf{DHR}(A)), containing symmetric finite-depth circuits in the kernel. For a spin chain with fusion categorical symmetry\mathcal{D}, we show that the DHR category of the quasi-local algebra of symmetric operators is equivalent to the Drinfeld center\mathcal{Z}(\mathcal{D}). We use this to show that, for the double spin-flip action\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}\curvearrowright \mathbb{C}^{2}\otimes \mathbb{C}^{2}, the group of symmetric QCA modulo symmetric finite-depth circuits in 1D contains a copy ofS_{3}; hence, it is non-abelian, in contrast to the case with no symmetry.
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This content will become publicly available on April 9, 2026
The Hilbert scheme of infinite affine space and algebraic K-theory
We study the Hilbert scheme\mathrm{Hilb}_{d}(\mathbf{A}^{\infty})from an\mathbf{A}^{1}-homotopical viewpoint and obtain applications to algebraic K-theory. We show that the Hilbert scheme\mathrm{Hilb}_{d}(\mathbf{A}^{\infty})is\mathbf{A}^{1}-equivalent to the Grassmannian of(d-1)-planes in\mathbf{A}^{\infty}. We then describe the\mathbf{A}^{1}-homotopy type of\mathrm{Hilb}_{d}(\mathbf{A}^{n})in a certain range, fornlarge compared tod. For example, we compute the integral cohomology of\mathrm{Hilb}_{d}(\mathbf{A}^{n})(\mathbf{C})in a range. We also deduce that the forgetful map\mathcal{FF}\mathrm{lat}\to\mathcal{V}\mathrm{ect}from the moduli stack of finite locally free schemes to that of finite locally free sheaves is an\mathbf{A}^{1}-equivalence after group completion. This implies that the moduli stack\mathcal{FF}\mathrm{lat}, viewed as a presheaf with framed transfers, is a model for the effective motivic spectrum\mathrm{kgl}representing algebraic K-theory. Combining our techniques with the recent work of Bachmann, we obtain Hilbert scheme models for the\mathrm{kgl}-homology of smooth proper schemes over a perfect field.
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- Award ID(s):
- 2054553
- PAR ID:
- 10612055
- Publisher / Repository:
- EMS Press
- Date Published:
- Journal Name:
- Journal of the European Mathematical Society
- Volume:
- 27
- Issue:
- 5
- ISSN:
- 1435-9855
- Page Range / eLocation ID:
- 2037 to 2071
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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