Suslin homology

In mathematics, the Suslin homology is a homology theory attached to algebraic varieties. It was proposed by Suslin in 1987, and developed by Suslin and Voevodsky (1996). It is sometimes called singular homology as it is analogous to the singular homology of topological spaces.

By definition, given an abelian group A and a scheme X of finite type over a field k, the theory is given by

${\displaystyle H_{i}(X,A)=\operatorname {Tor} _{i}^{\mathbb {Z} }(C,A)}$

where C is a free graded abelian group whose degree n part is generated by integral subschemes of ${\displaystyle \triangle ^{n}\times X}$, where ${\displaystyle \triangle ^{n}}$ is an n-simplex, that are finite and surjective over ${\displaystyle \triangle ^{n}}$.

References

• Geisser, Thomas (2009), On Suslin's singular homology and cohomology, arXiv:0912.1168, Bibcode:2009arXiv0912.1168G
• Levine, Marc (1997), "Homology of algebraic varieties: an introduction to the works of Suslin and Voevodsky", Bull. Amer. Math. Soc. (N.S.), 34 (3): 293–312, doi:10.1090/s0273-0979-97-00723-4, MR 1432056
• Suslin, Andrei; Voevodsky, Vladimir (1996), "Singular homology of abstract algebraic varieties", Invent. Math., 123 (1): 61–94, Bibcode:1996InMat.123...61S, CiteSeerX 10.1.1.46.9175, doi:10.1007/bf01232367, MR 1376246