Panin, Ivan; Yagunov, Serge \(T\)-spectra and Poincaré duality. (English) Zbl 1149.14015 J. Reine Angew. Math. 617, 193-213 (2008). 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). Reviewer: Piotr Krasoń (Szczecin) Cited in 1 ReviewCited in 1 Document 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 Keywords:algebraic variety; symmetric commutative ring \(T\)-spectrum; orientation, fundamental class; Poincaré Duality Citations:Zbl 0309.55016; Zbl 0907.19002; Zbl 0969.19004 PDFBibTeX XMLCite \textit{I. Panin} and \textit{S. Yagunov}, J. Reine Angew. Math. 617, 193--213 (2008; Zbl 1149.14015) Full Text: DOI arXiv 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.