Koszul duality for tori (Q2763515)

The author investigates the relationship between nonequivariant and equivariant cohomology of spaces with torus actions. \textit{M. Goresky}, \textit{R. Kottwitz}, and \textit{R. MacPherson} [Invent. Math. 131, No. 1, 25-83 (1998; Zbl 0897.22009)] studied Koszul duality for subanalytic spaces and real coefficients; subject of the present work is extension of this result to more general coefficients. The author proves as a main result:NEWLINENEWLINENEWLINETheorem. Let \(R\) be a commutative ring with 1. Let \(X\) be a \(T\)-space, \(Y\) a space over \(BT\), \(S^*=H^*(BT)\), \(\Lambda=H(T)\). Let \(t\) and \(h\) denote the Koszul functors. ThenNEWLINENEWLINENEWLINE(1) The singular cochain complex \(C^*(X;R)\) is a differential \(\Lambda\)-module and \(H(tC^* (X;R)) \cong H^*(tX;R)\) as \(S^*\)-modules.NEWLINENEWLINENEWLINE(2) The singular cochain complex \(C^*(X;R)\) is a weak \(S^*\)-module and \(H(hC^* (Y;R))\cong H^*(hY;R)\) as \(\Lambda\)-modules.NEWLINENEWLINENEWLINEAs an application, the calculation of the cohomology of a toric variety is described.