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\(T\)-spectra and Poincaré duality. (English) Zbl 1149.14015

Let \(E\) be a commutative ring spectrum with a distinguished element \(c\in E^{2}(\mathbb P^{\infty})\). The element \(c\) is called the complex orientation of \(E\) and the pair\((E,c)\) a complex oriented ring spectrum. Complex orientation \(c\) gives rise to the fundamental class \([X]\in E_{2d}(X)\) of a smooth complex projective variety, which has the property that the cap product: \[ \cap \, [X] : E^{*}(X)\rightarrow E_{2d-*}(X) \] is the isomorphism (Poincaré Duality isomorphism). Here \(d\) denotes the complex dimension of \(X\), \(E_{*}\) (resp. \( E^{*}\) ) is the homology (resp. cohomology) theory defined by the spectrum \(E\) [cf. J. F. Adams, Stable homotopy and generalised homology. Chicago Lectures in Mathematics. (Chicago - London): The University of Chicago Press. (1974; Zbl 0309.55016)]. The authors establish the analogue of this result in the context of algebraic geometry. (For a fixed field \(k\) they consider symmetric commutative ring \(T\)-spectrum \(\mathcal A\) in the sense of V. Voevodsky [Doc. Math., J. DMV, Extra Vol. ICM Berlin 1998, vol. I, 579–604 (1998; Zbl 0907.19002)] and also J. F. Jardine [Doc. Math., J. DMV 5, 445–553 (2000; Zbl 0969.19004)] for the definition of the \(T\) spectrum). The \(T\)-spectrum yields bi-graded homology and cohomology theories \(A^{*,*}\) and \(A_{*,*}\) on the category of algebraic varieties. There is also a notion of an orientation of \(\mathcal A\) and the fundamental class of a smooth projective equi-dimensional variety \(X/k\) of dimension \(d.\) The main result of the paper asserts that the cap product map: \[ \cap \, [X] : A^{*,*}(X) \rightarrow A_{2d-*,d-*} \] is an isomorphism. The authors also give examples of oriented symmetric commutative ring spectra. The first one is a symmetric model \({\mathbb M}{\mathbb G}{\mathbb L}\) of the algebraic cobordism \(T\)-spectrum \(MGL\) of Voevodsky and the second is the Eilenberg-Mac Lane \(T\)-spectrum \(H\) (cf. Zbl 0907.19002, p.601 and p.598 for the constructions).

MSC:

14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
14F42 Motivic cohomology; motivic homotopy theory
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
19E20 Relations of \(K\)-theory with cohomology theories
57R75 \(\mathrm{O}\)- and \(\mathrm{SO}\)-cobordism
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References:

[1] DOI: 10.1007/s00209-006-0049-4 · Zbl 1190.14010 · doi:10.1007/s00209-006-0049-4
[2] Jardine J. F., Doc. Math. 5 pp 445– (2000)
[3] Morel F., IHES Publ. Math. 90 pp 45– (1999)
[4] DOI: 10.1023/B:KTHE.0000019788.33790.cb · Zbl 1047.19001 · doi:10.1023/B:KTHE.0000019788.33790.cb
[5] DOI: 10.1016/S0022-4049(01)00134-7 · Zbl 1056.14027 · doi:10.1016/S0022-4049(01)00134-7
[6] Yagunov S., Rigidity II: Non-Orientable Case, Doc. Math. 9 pp 29– (2004) · Zbl 1056.14029
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