## Algebraic & Geometric Topology

### On some adjunctions in equivariant stable homotopy theory

#### Abstract

We investigate certain adjunctions in derived categories of equivariant spectra, including a right adjoint to fixed points, a right adjoint to pullback by an isometry of universes, and a chain of two right adjoints to geometric fixed points. This leads to a variety of interesting other adjunctions, including a chain of six (sometimes seven) adjoints involving the restriction functor to a subgroup of a finite group on equivariant spectra indexed over the trivial universe.

#### Article information

Source
Algebr. Geom. Topol., Volume 18, Number 4 (2018), 2419-2442.

Dates
Revised: 18 January 2018
Accepted: 24 February 2018
First available in Project Euclid: 3 May 2018

https://projecteuclid.org/euclid.agt/1525312833

Digital Object Identifier
doi:10.2140/agt.2018.18.2419

Mathematical Reviews number (MathSciNet)
MR3797071

Zentralblatt MATH identifier
06867662

#### Citation

Hu, Po; Kriz, Igor; Somberg, Petr. On some adjunctions in equivariant stable homotopy theory. Algebr. Geom. Topol. 18 (2018), no. 4, 2419--2442. doi:10.2140/agt.2018.18.2419. https://projecteuclid.org/euclid.agt/1525312833

#### References

• P Balmer, I Dell'Ambrogio, B Sanders, Grothendieck–Neeman duality and the Wirthmüller isomorphism, Compos. Math. 152 (2016) 1740–1776
• P Balmer, B Sanders, The spectrum of the equivariant stable homotopy category of a finite group, Invent. Math. 208 (2017) 283–326
• A D Elmendorf, I Kriz, M A Mandell, J P May, Rings, modules, and algebras in stable homotopy theory, Math. Surveys Monogr. 47, Amer. Math. Soc., Providence, RI (1997)
• H Fausk, P Hu, J P May, Isomorphisms between left and right adjoints, Theory Appl. Categ. 11 (2003) 107–131
• J P C Greenlees, J P May, Generalized Tate cohomology, Mem. Amer. Math. Soc. 543, Amer. Math. Soc., Providence, RI (1995)
• M A Hill, M J Hopkins, D C Ravenel, On the nonexistence of elements of Kervaire invariant one, Ann. of Math. 184 (2016) 1–262
• P Hu, I Kriz, Real-oriented homotopy theory and an analogue of the Adams–Novikov spectral sequence, Topology 40 (2001) 317–399
• P Hu, I Kriz, P Somberg, Derived representation theory and stable homotopy categorification of $\mathrm{sl}_k$, preprint (2016) Available at \setbox0\makeatletter\@url http://www.math.lsa.umich.edu/~ikriz/drt16084.pdf {\unhbox0
• L G Lewis, Jr, J P May, M Steinberger, J E McClure, Equivariant stable homotopy theory, Lecture Notes in Math. 1213, Springer (1986)
• R J Milgram, The homology of symmetric products, Trans. Amer. Math. Soc. 138 (1969) 251–265
• M Nakaoka, Cohomology mod $p$ of the $p$–fold symmetric products of spheres, J. Math. Soc. Japan 9 (1957) 417–427
• B Sanders, The compactness locus of a geometric functor and the formal construction of the Adams isomorphism, preprint (2016)