A NOTE ON p-COMPLETION OF SPECTRA
GEOFFROY HOREL
ABSTRACT. We compare two notions ofp-completion for a spectrum.
There are two endofunctors of the category of spectra that can legitimately be called completion at the primep. The first one is the localization at the Moore spectrumSZ/pand is what is usually calledp-completion and the second one is the unit mapX7→Mat(bX)in an adjunction between spectra and pro-objects in the category of spectra whose homotopy groups are finite p-groups and almost all zero. The goal of this note is to compare these two functors. This will be achieved in Theorem2.3.
This result should not come as a surprise to experts in the field. However, it seems to be missing from the literature. The main reason is probably that, in order to formulate it precisely, one needs an∞-categorical notion of pro-categories.
Acknowledgement. I wish to thank Tobias Barthel and Drew Heard for helpful conversa- tions.
1. NOTATIONS
This note is written using the language of ∞-categories. All the categorical notion should be understood in the ∞-categorical sense. All the∞-categories that we consider are stable and we denote by Map their mapping spectrum.
ForEa spectrum, we denote bycpEthe localization ofEwith respect to the homology theory SZ/p. Note that by [Bou79, Theorem 3.1.], this coincides with theHZ/plocal- ization ifE is bounded below. ForE a bounded below spectrum with finitely generated homotopy groups, the mapE→cpEinduces the mapE∗→E∗⊗Zpon homotopy groups.
We denote bySpp−f inthe smallest full stable subcategory ofSpcontaining the spectrum HZ/p. This is also the full subcategory ofSpspanned by the spectra whose homotopy groups are finitep-groups and are almost all 0.
We denote bySpcpthe category Pro(Spp−f in). The inclusionSpp−f in→Spinduces a limit preserving functor Mat :Spcp→Sp. This has a left adjoint denotedX7→X. We oftenb use the notationX={Xi}i∈Ifor objects ofSpcp. This means thatX=limiXiinSpcpwith Icofiltered andXiinSpp−f in. Any object ofcSppadmits a presentation of this form. The functor Mat(X)is then given by the formula Mat(X) =limIXiwhere the limit is computed inSp.
We denote byτnthen-th Postnikov section endofunctor onSp. By the universal prop- erty of the pro-category, there is a unique endofunctor ofSpcpthat coincides withτn on Spp−f inand commutes with cofiltered limits. ForAa pro-pabelian group, we denote by HAb the object ofSpcpgiven by applying the Eilenberg-MacLane functor to an inverse sys- tem of finite abelian group whose limit isA. Note for instance thatHbZplives inSpcpwhile HZplives inSp. We obviously have a weak equivalenceHZp→Mat(HbZp).
We denote bySppf tthe full subcategory ofSpspanned by bounded below spectra whose homotopy groups are finitely generated Zp-modules. Note that ifX is a bounded below spectrum that has finitely generated homotopy groups, thencpX is inSpf tp. Similarly, we denote bySpcpf tthe full subcategory ofcSppspanned by pro-spectra that are bounded below
1
2 GEOFFROY HOREL
and whose homotopy groups are finitely generatedZp-modules (given a pro-spectrumX= {Xi}i∈I, itsn-th homotopy group is the pro-abelian group{πn(Xi)}i∈I).
2. PROOFS
Lemma 2.1. The map HZp→Mat(HbZp)is adjoint to a weak equivalenceHdZp→HbZp. Proof. It suffices to show that for any spectrumFinSpp−f in, the map
Map
cSpp(HbZp,F)→Map(HZp,F)
is a weak equivalence. Since both sides of the equation are exact inF, it suffices to do it forF=HZ/p. Hence we are reduced to proving that the map
colimnHk(HZ/pn,Z/p)→Hk(HZp,Z/p)
is an isomorphism for each k. SinceZ/p is a field, cohomology is dual to homology and it suffices to prove that Hk(HZp,Zp) is isomorphic to {Hk(HZ/pn,Z/p)}n in the category of pro-abelian groups. In [Lur11, Proposition 3.3.10.], Lurie shows that there is an isomorphism of pro-abelian groups:
Hk(Σ−mΣ∞K(Zp,m),Z/p)∼={Hk(Σ−mΣ∞K(Z/pn,m),Z/p)}n
By Freudenthal suspension theorem, for any abelian groupA, the mapΣ−mΣ∞K(A,m)→ HA is about m-connected. Thus, taking m large enough, Lurie’s result gives what we
need.
Proposition 2.2. Let Y be an object ofSppf t. Then the unit map Y →Mat(bY)is a weak equivalence.
Proof. Let us call a spectrumY good if this is the case. The good spectra form a triangu- lated subcategory ofSp. This subcategory containsHZ/p. According to Lemma2.1, it also containsHZp. Hence, it containsτnY for anynand anyY inSppf t.
Thus, forY inSppf t, there is an equivalenceτnY →Mat(dτnY)for eachn. In order to prove thatY is good, it will be enough to prove that the map
Mat(bY)→limnMat(dτnY)
is a weak equivalence. Since Mat is a right adjoint, it is enough to prove that the obvious map
Yb→limndτnY
is a weak equivalence. As in the previous lemma, it is enough to prove that for eachkthe map
colimnHk(τnY,Z/p)→Hk(Y,Z/p)
is an isomorphism which is straightforward.
We can now prove our main result.
Theorem 2.3. There is a natural transformation from cptoMat(−)b that is a weak equiva- lence when restricted to spectra X such that cpX is inSppf t. In particular, it is a weak equiv- alence on spectra that are bounded below and have finitely generated homotopy groups.
Proof. We first make the observation that for any spectrumX, the obvious mapXb→cdpX is a weak equivalence. Indeed, it suffices to prove that for any F inSpp−f in, the map X→cpX induces a weak equivalence
Map(cpX,F)→Map(X,F) but this follows from the fact thatF is local with respect toSZ/p.
Thus, there is a natural transformation of endofunctors ofSp:
α(X):cpX→Mat(dcpX)'Mat(Xb)
A NOTE ONp-COMPLETION OF SPECTRA 3
Proposition2.2tells us thatα(X)is a weak equivalence whenevercp(X)is inSppf t as
desired.
3. THEADAMS SPECTRAL SEQUENCE
As an application of Theorem 2.3, we give an alternative construction of the Adams spectral sequence. We denote byH the Eilenberg-MacLane spectrumHZ/p. We denote byA the ring spectrum Map(H,H). Note thatA∗=π−∗A is the Steenrod algebra. There is a functorSpop→ModA sendingX to Map(X,H).
Proposition 3.1. Let X be any spectrum and Y be an object ofSppf t. Then, the obvious map
Map(X,Y)→MapA(Map(Y,H),Map(X,H)) is a weak equivalence.
Proof. SinceHis inSpp−f in, we have an equivalence Map(X,H)'Map
Spcp(Xb,H)for any spectrumX. By2.2, the map
Map(X,Y)→Map
Spcp(X,b Yb)
is an equivalence. Hence, we are reduced to proving that the obvious map Map
Spc(X,bYb)→MapA(Map(bY,H),Map(bX,H))
is an equivalence. We claim more generally that for any objectZinSpcp, the map Map
Spc(bX,Z)→MapA(Map(Z,H),Map(Xb,H))
is an equivalence. Indeed, since both sides are limit preserving in the variableZ, it suffices
to prove it forZ=HZ/pwhich is tautological.
According to [EKMM97, Theorem IV.4.1.], we get a conditionally convergent spectral sequence
Exts,tA∗(H∗(Y),H∗(X)) =⇒ πs+tMap(X,Y) forY a spectrum inSpf tp.
REFERENCES
[Bou79] A. Bousfield. The localization of spectra with respect to homology.Topology, 18(4):257–281, 1979.
[EKMM97] A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May.Rings, modules, and algebras in stable homotopy theory, volume 47 ofMathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1997. With an appendix by M. Cole.
[Lur11] J. Lurie. Derived algebraic geometry XIII.available at http://www.math.harvard.edu/ lurie/, 2011.
MAXPLANCKINSTITUTE FORMATHEMATICS, VIVATSGASSE7, 53111 BONN, DEUTSCHLAND E-mail address:geoffroy.horel@gmail.com