Trace methods for coHochschild homology
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917954139979776 |
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| author | Klanderman, Sarah Péroux, Maximilien |
| author_facet | Klanderman, Sarah Péroux, Maximilien |
| contents | Hochschild homology is a classical invariant of rings that plays an important role because of its connection to algebraic $K$-theory via the Dennis trace. At level zero, the Dennis trace is induced by the Hattori-Stallings trace. In this paper, we introduce new algebraic $K$-theories of coalgebras and obtain coalgebraic refinements of the Hattori-Stallings trace that connect these algebraic $K$-theories to coHochschild homology (the invariant analogous to Hochschild homology but for coalgebras). We employ bicategorical methods of Ponto to show that coHochschild homology is a shadow. Consequently, we obtain that coHochschild homology is Morita-Takeuchi invariant. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2301_11346 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Trace methods for coHochschild homology Klanderman, Sarah Péroux, Maximilien Algebraic Topology Category Theory K-Theory and Homology Primary: 16E40, 16T15, 18N10, 19A99, 55P43, 55U15. Secondary: 16D90, 18F30, 18M70, 18N60, 57T30 Hochschild homology is a classical invariant of rings that plays an important role because of its connection to algebraic $K$-theory via the Dennis trace. At level zero, the Dennis trace is induced by the Hattori-Stallings trace. In this paper, we introduce new algebraic $K$-theories of coalgebras and obtain coalgebraic refinements of the Hattori-Stallings trace that connect these algebraic $K$-theories to coHochschild homology (the invariant analogous to Hochschild homology but for coalgebras). We employ bicategorical methods of Ponto to show that coHochschild homology is a shadow. Consequently, we obtain that coHochschild homology is Morita-Takeuchi invariant. |
| title | Trace methods for coHochschild homology |
| topic | Algebraic Topology Category Theory K-Theory and Homology Primary: 16E40, 16T15, 18N10, 19A99, 55P43, 55U15. Secondary: 16D90, 18F30, 18M70, 18N60, 57T30 |
| url | https://arxiv.org/abs/2301.11346 |