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

Suslin, Andrei; Voevodsky, Vladimir

Invent. Math. 123, No. 1, 61-94 (1996).

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 Math., Berkeley/Calif. 1986, Vol. 1, 222-244 (1987; Zbl 0675.12005)], O. Gabber (unpublished), H. A. Gillet and R. W. Thomason [J. Pure Appl.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 \(H_{\text{sing}}^*(F,{\mathbb{Z}|}n)= \text{Ext}_{qfh}^*(F,{\mathbb{Z}|}n)\).

Reviewer: V.V.Chueshev (Kemerovo)

MSC:

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

Keywords:

singular homology; abstract algebraic varieties; category of schemes of finite type

Citations:

Zbl 0091.37102; Zbl 0514.18008; Zbl 0675.12005; Zbl 0577.13009

References:

[1]	S. Bloch: Algebraic cycles and higher K-theory. Adv. in Math.,61, 267-304 (1986) · Zbl 0608.14004 · doi:10.1016/0001-8708(86)90081-2
[2]	N. Bourbaki: Algébre commutative. Herman, Paris, 1964 · Zbl 0205.34302
[3]	A. Dold, R. Thom: Quasifaserungen und unendliche symmtrische Produkte. Ann. Math.67, 230-281 (1956) · Zbl 0091.37102
[4]	E. Friedlander: Some computations of algebraic cycle homology. (Preprint 1993).
[5]	E. Friedlander and H. Lawson. A theory of algebraic cocycles. Ann. Math.,136, 361-428 (1992) · Zbl 0788.14014 · doi:10.2307/2946609
[6]	E.M. Friedlander and B. Mazur. Filtrations on the homology of algebraic varieties. AMS, Providence, RI, 1994 · Zbl 0841.14019
[7]	H. Gillet and R. Thomason. The K-theory of strict hensel local rings and a theorem of Suslin. J. Pure Appl. Algebra,34, 241-254 (1984) · Zbl 0577.13009 · doi:10.1016/0022-4049(84)90037-9
[8]	A. Grothendieck. Revetements etale et groupe fondamental(SGA 1). Lecture Notes in Math. 224. Springer, Heidelberg, 1971
[9]	A. Grothendieck, M. Artin, and J.-L. Verdier. Theorie des topos et cohomologie etale des schemas (SGA 4). Lecture Notes in Math. 269,270,305. Springer, Heidelberg, 1972-73
[10]	A. Grothendieck and J. Dieudonne. Etude Cohomologique des Faisceaux Coherents (EGA 3). Publ. Math. IHES,11,17, 1961,1963
[11]	A. Grothendieck and J. Dieudonne. Etude Locale des Schemas et des Morphismes de Schemas (EGA 4). Publ. Math. IHES,20,24,28,32, 1964-67
[12]	J.F. Jardine. Simplicial objects in a Grothendieck topos. Contemp. Math.,55(1), 193-239 (1986) · Zbl 0606.18006
[13]	S. Landsburg. Relative cycles and algebraic K-theory. Am. J. Math.,111, 599-632 (1989) · Zbl 0722.14001 · doi:10.2307/2374815
[14]	H.B. Lawson. Algebraic cycles and homotopy theory. Ann. Math.,129, 599-632 (1989) · Zbl 0688.14006 · doi:10.2307/1971448
[15]	S. Lichtenbaum. Suslin homology and Deligne 1-motives. Preprint, 1992 · Zbl 0902.14004
[16]	S. MacLane. Homology. Springer-Verlag, 1963
[17]	J.S. Milne. Etale Cohomology. Princeton Univ. Press, Princeton, NJ, 1980 · Zbl 0433.14012
[18]	M. Raynaud and L. Gruson. Criteres de platitude et de projectivite. Inn. Math.,13, 1-89 (1971) · Zbl 0227.14010 · doi:10.1007/BF01390094
[19]	J.-P. Serre. Algebre locale. Multiplicities. Lecture Notes in Math., 11, 1965
[20]	I.R. Shafarevich. Basic algebraic geometry. Springer-Verlag, New York, 1974 · Zbl 0284.14001
[21]	A. Suslin. On the K-theory of algebraically closed fields. Invent. Math,73, 241-245 (1983) · Zbl 0514.18008 · doi:10.1007/BF01394024
[22]	A. Suslin. Algebraic K-theory of fields. Proceedings of ICM Berkeley, pages 222-244, 1986
[23]	A. Suslin. Higher Chow groups of affine varieties and etale cohomology. Preprint, 1993
[24]	V. Voevodsky. Homology of schemes and covariant motives. Harvard Univ. Ph.D. thesis, 1992