Higher Koszul duality and $n$-affineness
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| Format: | Preprint |
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2025
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| author | Pascaleff, James Pavia, Emanuele Sibilla, Nicolò |
| author_facet | Pascaleff, James Pavia, Emanuele Sibilla, Nicolò |
| contents | We study $\mathbb{E}_n$-Koszul duality for pairs of algebras of the form $\mathrm{C}_{\bullet}(Ω^{n}_*X;\Bbbk) \leftrightarrow \mathrm{C}^{\bullet}(X;\Bbbk)$, and the closely related question of $n$-affineness for Betti stacks. It was expected, but not known, that $\mathbb{E}_n$-Koszul duality should induce a kind of Morita equivalence between categories of iterated modules. We establish this rigorously by proving that the $(\infty,n)$-category of iterated modules over $\mathrm{C}_{\bullet}(Ω_*^{n+1}X;\Bbbk)$ is equivalent to the $(\infty,n)$-category of quasi-coherent sheaves of $(\infty,n-1)$-categories on $\mathrm{cSpec}(\mathrm{C}^{\bullet}(X;\Bbbk))$, where $\mathrm{cSpec}(\mathrm{C}^{\bullet}(X;\Bbbk))$ is the cospectrum of $\mathrm{C}^{\bullet}(X;\Bbbk)$. By the monodromy equivalence, these categories are also equivalent to the category of higher local systems on $X$, $n\mathbf{LocSysCat}^{n-1}(X;\Bbbk)$. Our result is new already in the classical case $n=1$, although it can be seen to recover well known formulations of $\mathbb{E}_1$-Koszul duality as a Morita equivalence of module categories (up to appropriate completions of the $t$-structures). We also investigate (higher) affineness properties of Betti stacks. We give a complete characterization of $n$-affine Betti stacks, in terms of the $0$-affineness of their iterated loop space. As a consequence, we prove that $n$-truncated Betti stacks are $n$-affine; and that $π_{n+1}(X)$ is an obstruction to $n$-affineness. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_16935 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Higher Koszul duality and $n$-affineness Pascaleff, James Pavia, Emanuele Sibilla, Nicolò Algebraic Geometry Algebraic Topology Category Theory K-Theory and Homology We study $\mathbb{E}_n$-Koszul duality for pairs of algebras of the form $\mathrm{C}_{\bullet}(Ω^{n}_*X;\Bbbk) \leftrightarrow \mathrm{C}^{\bullet}(X;\Bbbk)$, and the closely related question of $n$-affineness for Betti stacks. It was expected, but not known, that $\mathbb{E}_n$-Koszul duality should induce a kind of Morita equivalence between categories of iterated modules. We establish this rigorously by proving that the $(\infty,n)$-category of iterated modules over $\mathrm{C}_{\bullet}(Ω_*^{n+1}X;\Bbbk)$ is equivalent to the $(\infty,n)$-category of quasi-coherent sheaves of $(\infty,n-1)$-categories on $\mathrm{cSpec}(\mathrm{C}^{\bullet}(X;\Bbbk))$, where $\mathrm{cSpec}(\mathrm{C}^{\bullet}(X;\Bbbk))$ is the cospectrum of $\mathrm{C}^{\bullet}(X;\Bbbk)$. By the monodromy equivalence, these categories are also equivalent to the category of higher local systems on $X$, $n\mathbf{LocSysCat}^{n-1}(X;\Bbbk)$. Our result is new already in the classical case $n=1$, although it can be seen to recover well known formulations of $\mathbb{E}_1$-Koszul duality as a Morita equivalence of module categories (up to appropriate completions of the $t$-structures). We also investigate (higher) affineness properties of Betti stacks. We give a complete characterization of $n$-affine Betti stacks, in terms of the $0$-affineness of their iterated loop space. As a consequence, we prove that $n$-truncated Betti stacks are $n$-affine; and that $π_{n+1}(X)$ is an obstruction to $n$-affineness. |
| title | Higher Koszul duality and $n$-affineness |
| topic | Algebraic Geometry Algebraic Topology Category Theory K-Theory and Homology |
| url | https://arxiv.org/abs/2504.16935 |