The low-dimensional homology of projective linear group of degree two

Behrooz Mirzaii; Elvis Torres Pérez

doi:10.4171/dm/1014

JournalsdmVol. 30, No. 5pp. 1157–1199

The low-dimensional homology of projective linear group of degree two

Behrooz Mirzaii
Universidade de São Paulo (USP), São Carlos, Brazil
ORCID
Elvis Torres Pérez
Pontifical Catholic University of Peru (PUCP), Lima, Peru
ORCID

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Abstract

In this article we study the low-dimensional homology of the projective linear group $PGL_{2} (A)$ over a commutative ring $A$ . In particular, we prove a Bloch–Wigner type exact sequence over local domains. As application we prove that

H_{2} (PGL_{2} (A), Z [\frac{1}{2}]) ≃ K_{2} (A) [\frac{1}{2}] and H_{3} (PGL_{2} (A), Z [\frac{1}{2}]) ≃ K_{3}^{ind} (A) [\frac{1}{2}],

provided $∣ A / m_{A} ∣ \neq = 2, 3, 4, 8$ .

Cite this article

Behrooz Mirzaii, Elvis Torres Pérez, The low-dimensional homology of projective linear group of degree two. Doc. Math. 30 (2025), no. 5, pp. 1157–1199

DOI 10.4171/DM/1014