A note on p-completion of spectra

(1)

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 byc_pEthe 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 bySp_{p−f in}the 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 bySpc_pthe category Pro(Sp_{p−f in}). The inclusionSp_{p−f in}→Spinduces a limit preserving functor Mat :Spc_p→Sp. This has a left adjoint denotedX7→X. We oftenb use the notationX={X_i}_i∈Ifor objects ofSpc_p. This means thatX=lim_iXiinSpc_pwith Icofiltered andX_iinSp_{p−f in}. Any object ofcSp_padmits 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 ofSpc_pthat coincides withτn on Sp_p−_{f in}and commutes with cofiltered limits. ForAa pro-pabelian group, we denote by HAb the object ofSpc_pgiven by applying the Eilenberg-MacLane functor to an inverse sys- tem of finite abelian group whose limit isA. Note for instance thatHbZplives inSpc_pwhile HZplives inSp. We obviously have a weak equivalenceHZp→Mat(HbZp).

We denote bySp_p^{f t}the 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, thenc_pX is inSp^{f t}_p. Similarly, we denote bySpc_p^{f t}the full subcategory ofcSp_pspanned by pro-spectra that are bounded below

1

(2)

2 GEOFFROY HOREL

and whose homotopy groups are finitely generatedZp-modules (given a pro-spectrumX= {X_i}i∈I, itsn-th homotopy group is the pro-abelian group{π_n(X_i)}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 spectrumFinSp_p−_{f in}, the map

Map

cSp_p(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

colim_nH^k(HZ/pⁿ,Z/p)→H^k(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 H_k(HZp,Zp) is isomorphic to {H_k(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:

H_k(Σ^−mΣ^∞K(Zp,m),Z/p)∼={H_k(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 ofSp_p^{f 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 inSp_p^{f t}.

Thus, forY inSp_p^{f 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)→lim_nMat(dτnY)

is a weak equivalence. Since Mat is a right adjoint, it is enough to prove that the obvious map

Yb→lim_ndτ_nY

is a weak equivalence. As in the previous lemma, it is enough to prove that for eachkthe map

colim_nH^k(τ_nY,Z/p)→H^k(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 c_ptoMat(−)b that is a weak equivalence when restricted to spectra X such that c_pX is inSp_p^{f t}. In particular, it is a weak equivalence 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→cd_pX is a weak equivalence. Indeed, it suffices to prove that for any F inSp_{p−f in}, the map X→c_pX induces a weak equivalence

Map(c_pX,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):c_pX→Mat(dc_pX)'Mat(Xb)

(3)

A NOTE ONp-COMPLETION OF SPECTRA 3

Proposition2.2tells us thatα(X)is a weak equivalence whenevercp(X)is inSp_p^{f 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 functorSp^op→Mod_A sendingX to Map(X,H).

Proposition 3.1. Let X be any spectrum and Y be an object ofSp_p^{f t}. Then, the obvious map

Map(X,Y)→Map_A(Map(Y,H),Map(X,H)) is a weak equivalence.

Proof. SinceHis inSp_{p−f in}, we have an equivalence Map(X,H)'Map

Spc_p(Xb,H)for any spectrumX. By2.2, the map

Map(X,Y)→Map

Spc_p(X,b Yb)

is an equivalence. Hence, we are reduced to proving that the obvious map Map

Spc(X,bYb)→Map_A(Map(bY,H),Map(bX,H))

is an equivalence. We claim more generally that for any objectZinSpc_p, the map Map

Spc(bX,Z)→Map_A(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

Ext^s,t_A∗(H^∗(Y),H^∗(X)) =⇒ πs+tMap(X,Y) forY a spectrum inSp^{f t}_p.

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:[email protected]