Affineness and reconstruction in higher Zariski geometry
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arXiv
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2025
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| author | Chedalavada, Anish |
| author_facet | Chedalavada, Anish |
| contents | We explain how the geometric framework introduced in arXiv:2508.11621 [math.AG] provides a universal property for the 2-rings of perfect complexes on qcqs spectral or Dirac spectral schemes. As an application, given a qcqs spectral or Dirac spectral scheme $X$ this produces a comparison morphism from $\operatorname{Spec} \mathrm{Perf}_{X}$ to $X$ itself, which is moreover natural in $X$. When $X$ is an ordinary qcqs scheme, this construction supplies a new proof of the Balmer-Thomason reconstruction of $X$ from its space of thick subcategories, assuming the result for noetherian rings due to Neeman. As another application, we find spectral and Dirac spectral enhancements of support varieties arising for 2-rings in representation theory which "geometrize" the 2-rings that produce them. For example, given a finite group $G$ over a field $k$, this produces a "spectral support variety" $\mathcal{V}_{G}$ such that $\mathrm{Perf}_{\mathcal{V}_{G}}$ maps into the stable module category of $kG$. We derive these results as a corollary of a general affineness criterion for 2-schemes which are covered by the Zariski spectra of rigid 2-rings: this states that such 2-schemes are affine if and only if they are quasicompact and quasiseparated. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_16964 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Affineness and reconstruction in higher Zariski geometry Chedalavada, Anish Algebraic Geometry Algebraic Topology Representation Theory We explain how the geometric framework introduced in arXiv:2508.11621 [math.AG] provides a universal property for the 2-rings of perfect complexes on qcqs spectral or Dirac spectral schemes. As an application, given a qcqs spectral or Dirac spectral scheme $X$ this produces a comparison morphism from $\operatorname{Spec} \mathrm{Perf}_{X}$ to $X$ itself, which is moreover natural in $X$. When $X$ is an ordinary qcqs scheme, this construction supplies a new proof of the Balmer-Thomason reconstruction of $X$ from its space of thick subcategories, assuming the result for noetherian rings due to Neeman. As another application, we find spectral and Dirac spectral enhancements of support varieties arising for 2-rings in representation theory which "geometrize" the 2-rings that produce them. For example, given a finite group $G$ over a field $k$, this produces a "spectral support variety" $\mathcal{V}_{G}$ such that $\mathrm{Perf}_{\mathcal{V}_{G}}$ maps into the stable module category of $kG$. We derive these results as a corollary of a general affineness criterion for 2-schemes which are covered by the Zariski spectra of rigid 2-rings: this states that such 2-schemes are affine if and only if they are quasicompact and quasiseparated. |
| title | Affineness and reconstruction in higher Zariski geometry |
| topic | Algebraic Geometry Algebraic Topology Representation Theory |
| url | https://arxiv.org/abs/2510.16964 |