Equivariant and Nonequivariant Module Spectra
نویسنده
چکیده
Let G be a compact Lie group, let RG be a commutative algebra over the sphere G-spectrum SG, and let R be its underlying nonequivariant algebra over the sphere spectrum S. When RG is split as an algebra, as holds for example for RG = MUG, we show how to “extend scalars” to construct a split RG-module MG = RG ∧R M from an R-module M . We also show how to compute the coefficients MG ∗ in terms of the coefficients RG ∗ , R∗, and M∗. This allows the wholesale construction of highly structured equivariant module spectra from highly structured nonequivariant module spectra. In particular, it applies to construct MUG-modules from MU-modules and therefore gives conceptual constructions of equivariant Brown-Peterson and Morava K-theory
منابع مشابه
Change of universe functors in equivariant stable homotopy theory
One striking difference between nonequivariant and equivariant stable homotopy is that, in the equivariant context, one must specify those representations with respect to which spectra are to be stable. One may specify stability with respect only to trivial representations (thereby obtaining what is often called the naive equivariant stable category), with respect to all representations (thereb...
متن کاملE∞-ring structures for Tate spectra
Let G be a compact Lie group and kG a G spectrum (as defined in [3, Section I.2]). Greenlees and May ([2]) have defined an associated G-spectrum t(kG) called the Tate spectrum. They observe that if kG is a ring G-spectrum then there is an induced ring G-spectrum structure on t(kG), and that if kG is homotopy-commutative then t(kG) will also be homotopycommutative (see [2, Proposition 3.5]). It ...
متن کاملThe equivariant Kazhdan-Lusztig polynomial of a matroid
We define the equivariant Kazhdan-Lusztig polynomial of a matroid equipped with a group of symmetries, generalizing the nonequivariant case. We compute this invariant for arbitrary uniform matroids and for braid matroids of small rank.
متن کاملRiemann-roch for Equivariant Chow Groups
Here ĜG(X) is the completion of the equivariant Grothendieck group of coherent sheaves along the augmentation ideal of the representation ring R(G), and the groups CH iG(X) are the equivariant Chow groups defined in [EG2]. The map τ G has the same functorial properties as the nonequivariant Riemann-Roch map of [BFM] and [F, Theorem 18.3]. IfG acts freely, then τ can be identified with the noneq...
متن کاملWeights in the cohomology of toric varieties
We describe the weight filtration in the cohomology of toric varieties. We present the role of the Frobenius automorphism in an elementary way. We prove that equivariant intersection homology of an arbitrary toric variety is pure. We also obtain a results concerning Koszul duality: nonequivariant intersection cohomology is equal to the cohomology of the Koszul complex IH∗ T (X)⊗ H ∗(T ).
متن کامل