Singular homology of abstract algebraic varieties. (English) Zbl 0896.55002

The authors construct a reasonable singular homology theory on the category of schemes of finite type over an arbitrary field \(k\). Let \(X\) be a CW-complex. The theorem of A. Dold and R. Thom [Ann. Math., II. Ser. 67, 239-281 (1958; Zbl 0091.37102)] shows that \(H_i(X,{\mathbb{Z}})\) coincide with \(\pi_i\) of the simplicial Abelian group \(\operatorname{Hom}_{\text{top}} (\Delta_{\text{top}}^{\circ}, \coprod_{d=0}^\infty S^d(X))^{+},\) where \(S^d(X)\) is the \(d\)-th symmetric power of \(X\), \(\Delta_{\text{top}}^i\) is the usual \(i\)-dimensional topological simplex and for any Abelian monoid \(M\) denote by \(M^+\) the associated Abelian group.
Conjecture. If \(X\) is a variety over \({\mathbb{C}}\) then the evident homomorphism \[ \operatorname{Hom}(\Delta^\circ, \coprod_{d=0}^\infty S^d(X))^+\rightarrow \operatorname{Hom}_{\text{top}} (\Delta_{\text{top}}^{\circ}, \coprod_{d=0}^\infty S^d(X))^+ \] induces isomorphisms \(H_i^{\text{sing}}(X,{\mathbb{Z}}| n)\cong H_i(X({\mathbb{C}}),{\mathbb{Z}}| n)\).
The authors prove that the conjecture is true. Also, they prove a rather general version of the rigidity theorem of A. Suslin [Invent. Math. 73, 241-245 (1983; Zbl 0514.18008); Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 222-244 (1987; Zbl 0675.12005)], O. Gabber (unpublished), H. A. Gillet and R. W. Thomason [J. Pure Appl. Algebra 34, 241-254 (1984; Zbl 0577.13009)]. One of the main results of the paper is that if \(F\) is any \(qfh\)-sheaf on the category of schemes of finite type over an algebraically closed field \(k\) of characteristic zero, then \(H_{\text{sing}}^*(F,{\mathbb{Z}|}n)= \text{Ext}_{qfh}^*(F,{\mathbb{Z}|}n)\).


55N10 Singular homology and cohomology theory
14A10 Varieties and morphisms
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