CONSECUENCIAS DE LAS RELACIONES COMERCIALES CON IRAN

copies ofL∧Xn. So iff :X −→Y is a monomorphism respectively a level equivalence of symmetric spectra of simplicial sets, then so is Id∧f :GmL∧X −→GmL∧Y.

Some other properties of flat spectra are fairly straightforward from the definition:

Proposition 5.7. (i)A wedge of flat symmetric spectra is flat.

(ii) A filtered colimit of flat symmetric spectra is flat.

(iii) The smash product of two flat symmetric spectra is flat.

Proof. Properties (i) and (ii) follow from the two facts that the smash product commutes with colimits

and that a filtered colimit or a wedge of monomorphisms is a monomorphism.

(iii) LetAand beB flat symmetric spectra of simplicial sets. Iff :X −→Y is a monomorphism, then Id∧f :B∧X −→B∧Y is a monomorphism sinceB is flat; then Id∧Id∧f :A∧B∧X−→A∧B∧Y is a monomorphism sinceAis flat (where we have implicitly used the associativity isomorphisms). ThusA∧B

is flat.

An example of a non-flat symmetric spectrum is ¯S, the subspectrum of the sphere spectrum given by

(5.8) _S¯n =

(

∗ forn= 0

Sn forn≥1.

So the difference between ¯S and S is only one missing vertex in level 0, but that missing vertex makes a

huge difference for the flatness property. Indeed, since ¯Sis trivial in level 0 we have

(¯S∧S¯)2 = Σ+2 ∧S¯1∧¯S1 = Σ+2 ∧S 2

while (¯S∧S)2∼= ¯S2=S2. So ¯S∧ − does not take the inclusion ¯S−→Sto a monomorphism, hence ¯Sis not

flat.

Now we develop a convenient criterion for recognizing flat symmetric spectra which involves latching spaces.

Definition 5.9. The nth latching space LnA of a symmetric spectrum A is the Σn-simplicial set (A∧¯S)n where ¯Sis the subspectrum of the sphere spectrum defined in (5.8). Thenth level of the morphism

Id∧i: A∧S¯ −→ A∧S ∼=A, for i : ¯S −→S the inclusion, provides a natural map of pointed Σn-spaces νn:LnA−→An.

Since the latching spaces play important roles in what follows, we make their definition more explicit. Specializing the construction of the smash product (compare Section I.3) to A∧¯_S displays LnA as the coequalizer, in the category of pointed Σn-simplicial sets, of two maps

n−2 _ p=0 Σ+_n ∧Σp×Σ1×Σn−p−1Ap∧S 1 ∧Sn−p−1 −→ n−1 _ p=0 Σ+_n ∧Σp×Σn−pAp∧S n−p _.

(in the target we have discarded the wedge summand which would contributeAn∧S¯0, since that is just a point, and similarly in the source). One of the maps takes the wedge summand indexed bypto the wedge summand indexed by p+ 1 using the map

σp∧Id : Ap∧S1∧Sn−p−1 −→ Ap+1∧Sn−p−1

and inducing up. The other map takes the wedge summand indexed by pto the wedge summand indexed bypusing the canonical isomorphism

Ap∧S1∧Sn−p−1

∼

=

−→ Ap∧Sn−p and inducing up.

For example, L0Ais a one-point simplicial set,L1A=A0∧S1andL2Ais the pushout of the diagram

A0∧S2

act onS2

←−−−−−−− Σ+₂ ∧A0∧S2

Id∧σ0∧Id

ThusL2Ais the quotient of Σ+2 ∧A1∧S1 by the equivalence relation generated by

γ∧σ0(a∧x)∧y ∼ (γ(1,2))∧σ0(a∧y)∧x for a∈A0 and x, y∈S1. In general,LnAis a quotient of Σ+n ∧Σn−1An−1∧S

1 _{by a suitable equivalence} relation.

Proposition 5.10. A symmetric spectrum of simplicial setsAis flat if and only if for everyn≥1the map of Σn-simplicial setsνn:LnA−→An is injective.

The proof of Proposition 5.10 uses a certain natural ‘filtration’ for symmetric spectra which shows how a general symmetric spectrum is built from semifree ones. We have put the term ‘filtration’ in quotes since in general this only is a natural sequence of symmetric spectra and morphisms with colimit the given spectrum. In the special case of flat symmetric spectra, the morphisms are injective.

For any integer k we denote by_S[k] _{the sphere spectrum truncated below level k, i.e., the symmetric} subspectrum of_S with level

(S[k])n = (

∗ forn < k Sn _forn≥k.

For example we haveS[1]= ¯SandS[k] =Sfor allk≤0. Note that two consecutive truncated sphere spectra

are related by the equation S1_∧

S[k] = sh(S[1+k]). Given a symmetric spectrumA we define a sequence Fm_{Aof symmetric spectra by}

(FmA)n = (_S[n−m]∧A)n

as a Σn-space. The structure map (Fm_A)n∧S1_−→(Fm_{A)n+1 is given by the composite} (_S[n−m]∧A)n∧S1 twist −−−→ S1∧(_S[n−m]∧A)n = (S1∧S[n−m]∧A)n = (sh(S[1+n−m])∧A)n ξ S[1+n−m],A −−−−−−−−→ (sh(S[1+n−m]∧A))n = (S[1+n−m]∧A)1+n χ1,n −−−→(S[n+1−m]∧A)n+1 .

For example we have

F0A= Σ∞A , (Fn−1A)n=LnA and (FmA)n =An form≥n.

In general the spaces of (FmA)n for m < n are a kind of ‘generalized latching objects’. The inclusions

S[k+1] −→ S[k] induce morphisms jm : Fm−1A −→ FmA and in a fixed level n, the system stabilizes to (A∧_S)n which is isomorphic to An. So the colimit of the sequence of symmetric spectra FmA over the morphismsjm is isomorphic toA.

Proposition 5.11. For every symmetric spectrumA and everym≥0 the commutative square

(5.12) GmLmA Gmνm // GmAm Fm−1A _j m / /_Fm_A

is a pushout square, where the vertical morphisms are adjoint to the identityLmA= (Fm−1A)mrespectively

the isomorphism(Fm_A)m∼_=A m.

Proof. The commutative square of symmetric spectra

Sk_. kS¯ // Sk_. kS=GkSk S[k+1] //S[k]

5. DERIVED SMASH PRODUCT 117