Algebraic $K$-theory for squares categories
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866908826260733952 |
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| author | Campbell, Jonathan Kuijper, Josefien Merling, Mona Zakharevich, Inna |
| author_facet | Campbell, Jonathan Kuijper, Josefien Merling, Mona Zakharevich, Inna |
| contents | In this paper we introduce a new formalism for $K$-theory, called squares $K$-theory. This formalism allows us to simultaneously generalize the usual three-term relation $[B] = [A] + [C]$ for an exact sequence $A \hookrightarrow B \twoheadrightarrow C$ or for a subtractive sequence $A\hookrightarrow B \leftarrow C$, by defining $K_0$ of a squares category to satisfy a four-term relation $[A]+[D]= [C] + [B]$ for a ``good'' square diagram with these corners. Examples that rely on this formalism are $K$-theory of smooth manifolds of a fixed dimension and $K$-theory of (smooth and) complete varieties. Another application we give of this theory is the construction of a derived motivic measure taking value in the $K$-theory of homotopy sheaves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_02852 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Algebraic $K$-theory for squares categories Campbell, Jonathan Kuijper, Josefien Merling, Mona Zakharevich, Inna K-Theory and Homology Algebraic Geometry Algebraic Topology In this paper we introduce a new formalism for $K$-theory, called squares $K$-theory. This formalism allows us to simultaneously generalize the usual three-term relation $[B] = [A] + [C]$ for an exact sequence $A \hookrightarrow B \twoheadrightarrow C$ or for a subtractive sequence $A\hookrightarrow B \leftarrow C$, by defining $K_0$ of a squares category to satisfy a four-term relation $[A]+[D]= [C] + [B]$ for a ``good'' square diagram with these corners. Examples that rely on this formalism are $K$-theory of smooth manifolds of a fixed dimension and $K$-theory of (smooth and) complete varieties. Another application we give of this theory is the construction of a derived motivic measure taking value in the $K$-theory of homotopy sheaves. |
| title | Algebraic $K$-theory for squares categories |
| topic | K-Theory and Homology Algebraic Geometry Algebraic Topology |
| url | https://arxiv.org/abs/2310.02852 |