Chromatic Purity in Hermitian K-Theory at $p=2$
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| Format: | Preprint |
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2025
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| _version_ | 1866915105830076416 |
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| author | Levin, Jordan |
| author_facet | Levin, Jordan |
| contents | In this article we investigate the question of chromatic purity of L-theory. To do so, we utilize the theory of additive GW and L-theory in the language of Poincaré categories as laid out in the series of papers by Calmès et al. We apply this theory to chromatically localised L-theory at the prime $p=2$ and recover the L-theoretic analogues of chromatic purity for $E_1$-rings with involution. From this, we deduce that L-theory does not exhibit chromatic redshift. We deduce the higher chromatic vanishing of quadratic L-theory of arbitrary idempotent complete categories, thereby allowing the use of Hermitian trace methods to probe chromatic behaviour of GW and L-theory. Finally, we show that for $T(n+1)$-acyclic rings with involution, $T(n+1)$-local GW-theory depends only on $T(n+1)$-local K-theory and the associated duality, thereby proving a chromatic analogue of the homotopy limit problem for GW-theory. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_09633 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Chromatic Purity in Hermitian K-Theory at $p=2$ Levin, Jordan K-Theory and Homology Algebraic Topology In this article we investigate the question of chromatic purity of L-theory. To do so, we utilize the theory of additive GW and L-theory in the language of Poincaré categories as laid out in the series of papers by Calmès et al. We apply this theory to chromatically localised L-theory at the prime $p=2$ and recover the L-theoretic analogues of chromatic purity for $E_1$-rings with involution. From this, we deduce that L-theory does not exhibit chromatic redshift. We deduce the higher chromatic vanishing of quadratic L-theory of arbitrary idempotent complete categories, thereby allowing the use of Hermitian trace methods to probe chromatic behaviour of GW and L-theory. Finally, we show that for $T(n+1)$-acyclic rings with involution, $T(n+1)$-local GW-theory depends only on $T(n+1)$-local K-theory and the associated duality, thereby proving a chromatic analogue of the homotopy limit problem for GW-theory. |
| title | Chromatic Purity in Hermitian K-Theory at $p=2$ |
| topic | K-Theory and Homology Algebraic Topology |
| url | https://arxiv.org/abs/2501.09633 |