arXiv · 0709.3169
We propose a new look on triangulated categories, which is based on the second Hochschild cohomology.
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Teimuraz Pirashvili. 2008-02-21. Singular extensions and triangulated categories. https://arxiv.org/abs/0709.3169
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In this paper, we obtain an exact sequence connecting $H_2(\rm{SL}_2(\mathcal{O}_{K,S}), \mathbb{Z})$ to $H_2(\rm{SL}_2(\mathcal{O}_{K,T}), \mathbb{Z})$, where $\mathcal{O}_{K,S}$ is a ring of $S$-integers and $T$ is a set of primes containing $S$. We apply this sequence to establish a relation between $H_2(\rm{SL}_2(\mathcal{O}_{K,S}), \mathbb{Z})$ with the second $K$-group $K_2(\mathcal{O}_{K,S})$, for $S$ large enough. This leads to a description of the rank and size of the torsion of $H_2(\rm{SL}_2(\mathcal{O}_{K,S}), \mathbb{Z})$. As an application, we propose a homological version of the Birch-Tate formula (conjecture) under these assumptions.
In their seminal work, Baum-Douglas-Taylor showed that under a certain compactness assumption, local boundary conditions for first-order elliptic operators over compact manifolds with boundary define cycles for relative K-homology. Using a systematic approach to regular boundary conditions, we generalize their result to elliptic operators of arbitrary order and to non-compact manifolds with potentially non-compact boundary. In addition, we show that under stronger regularity conditions, local boundary conditions even define cycles for absolute K-homology. In contrast to the relative classes, the resulting absolute K-homology classes are in general not independent of the choice of boundary condition.
We build on work of Muro-Raptis in [Ann. K-Theory 2 (2017), no. 2, 303-340] and Cisinski-Neeman in [Adv. Math. 217 (2008), no. 4, 1381-1475] to prove that the additivity of derivator K-theory holds for a large class of derivators that we call left pointed derivators, which includes all triangulated derivators. The proof methodology is an adaptation of the combinatorial methods of Grayson in [Doc. Math. 16 (2011), 457-464]. As a corollary, we prove that derivator K-theory is an infinite loop space. Finally, we speculate on the role of derivator K-theory as a trace from the algebraic K-theory of a stable $\infty$-category à la Blumberg-Gepner-Tabuada in [Geom. Topol. 17 (2013), no. 2, 733-838].