Hermitian K-theory for stable $\infty$-categories II: Cobordism categories and additivity - Archive ouverte HAL Access content directly
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## Hermitian K-theory for stable $\infty$-categories II: Cobordism categories and additivity

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Baptiste Calmès
Emanuele Dotto
• Function : Author
Yonatan Harpaz
Fabian Hebestreit
• Function : Author
Markus Land
• Function : Author
Kristian Moi
• Function : Author
Denis Nardin
• Function : Author
Thomas Nikolaus
• Function : Author
Wolfgang Steimle
• Function : Author

#### Abstract

We define Grothendieck-Witt spectra in the setting of Poincar\'e $\infty$-categories, show that they fit into an extension with a K- and an L-theoretic part and deduce localisation sequences for Verdier quotients. As special cases we obtain generalisations of Karoubi's fundamental and periodicity theorems for rings in which 2 need not be invertible. A novel feature of our approach is the systematic use of ideas from cobordism theory by interpreting the hermitian Q-construction as an algebraic cobordism category. We also use this to give a new description of the LA-spectra of Weiss and Williams.

#### Domains

Mathematics [math] Algebraic Geometry [math.AG]

### Dates and versions

hal-02941256 , version 1 (16-09-2020)

### Identifiers

• HAL Id : hal-02941256 , version 1
• ARXIV :

### Cite

Baptiste Calmès, Emanuele Dotto, Yonatan Harpaz, Fabian Hebestreit, Markus Land, et al.. Hermitian K-theory for stable $\infty$-categories II: Cobordism categories and additivity. 2020. ⟨hal-02941256⟩

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