Quotients of MGL, their slices and their geometric parts

0.38 MB·English

by  Marc Levine

QUOTIENTS OF MGL, THEIR SLICES AND THEIR
GEOMETRIC PARTS
5
1
MARCLEVINEANDGIRJASHANKERTRIPATHI
0
2

n Abstract. Letx1,x2,...beasystemofhomogeneouspolynomialgenerators
a for the Lazard ring L∗ = MU2∗ and let MGLS denote Voevodsky’s alge-
J braic cobordism spectrum in the motivic stable homotopy category over a
1 base-scheme S [Vo98]. Take S essentially smooth over a field k. Relying on
1 the Hopkins-Morel-Hoyois isomorphism [Hoy] of the 0th slice s0MGLS for
Voevodsky’s slice tower with MGLS/(x1,x2,...) (after inverting the char-
] acteristic of k), Spitzweck [S10] computes the remaining slices of MGLS as
G snMGLS = ΣnTHZ⊗L−n (again, after inverting the characteristic of k).
We apply Spitzweck’s method to compute the slices of a quotient spectrum
A
MGLS/({xi : i ∈ I}) for I an arbitrary subset of N, as well as the mod p
h. version MGLS/({p,xi : i ∈ I}) and localizations with respect to a system
t of homogeneous elements in Z[{xj :j 6∈I}]. In case S =Speck, k a field of
a characteristiczero,weapplythistoshowthatforEalocalizationofaquotient
m ofMGLasabove,thereisanaturalisomorphismforthetheorywithsupport
[ Ω∗(X)⊗L−∗E−2∗,−∗(k)→EX2m−2∗,m−∗(M)
for X a closed subscheme of a smooth quasi-projective k-scheme M, m =
1
v dimkM.
6
3
4
2
Contents
0
1. Introduction 1
0 1. Quotients and homotopy colimits in a model category 3
5 2. Slices of effective motivic module spectra 10
1
3. The slice spectral sequence 16
:
v 4. Slices of quotients of MGL 18
i 5. Modules for oriented theories 21
X
6. Applications to quotients of MGL 27
r
a References 29
Introduction
This paper has a two-fold purpose. We consider Voevodsky’s slice tower on the
motivic stable homotopy category SH(S) over a base-scheme S [Vo00]. For E in
SH(S), we have the nth layer s E in the slice tower for E. Let MGL denote
n
Voevodsky’s algebraic cobordism spectrum in SH(S) [Vo98] and let x ,x ,... be
1 2
a system of homogeneous polynomial generators for the Lazard ring L . Via the
∗
1991 Mathematics Subject Classification. Primary 14C25, 19E15; Secondary 19E08 14F42,
55P42.
BothauthorswishtothanktheHumboldtFoundationforfinancialsupport.
1
2 MARCLEVINEANDGIRJASHANKERTRIPATHI
classifying map for the formal group law for MGL, we may consider x as an
i
element of MGL2i,i(S), and thereby as a map x :Σ2i,iMGL→MGL, giving the
i
quotient MGL/(x ,x ,...). Spitzweck [S10] shows how to build on the Hopkins-
1 2
Morel-Hoyoisisomorphism [Hoy]
MGL/(x ,x ,...)∼=s MGL
1 2 0
to compute all the slices s MGL of MGL. Our first goal here is to extend
n
Spitzweck’s method to handle quotients of MGL by a subset of {x ,x ,...}, as
1 2
well as localizations with respect to a system of homogeneous elements in the ring
generated by the remaining variables; we also consider quotients of such spectra
by an integer. Some of these spectra are Landweber exact, and the slices are thus
computable by the results of Spitzweck on the slices of Landweber exact spectra
[S12], but many of these, such as the truncatedBrown-Petersonspectra orMorava
K-theory, are not.
The second goal is to extend results of [DL14, L09, L15], which consider the
“geometricpart”X 7→E2∗,∗(X)ofthebi-gradedcohomologydefinedbyanoriented
weakcommutativeringT-spectrumE andraisethequestion: istheclassifyingmap
E∗(k)⊗L∗ Ω∗ →E∗
an isomorphism of oriented cohomology theories, that is, is the theory E∗ a theory
of rational type in the sense of Vishik [Vi12]? Starting with the case E = MGL,
discussed in [L09], which immediately yields the Landweber exact case, we have
answered this affirmatively for “slice effective” algebraic K-theory in [DL14], and
extended to the case of slice-effective covers of a Landweber exact theory in [L15].
In this paper, we use our computation of the slices of a quotient of MGL to show
that the classifying map is an isomorphism for the quotients and localizations of
MGL described above.
The paper is organized as follows: in §1 and §2, we axiomatize Spitzweck’s
methodfrom[S10]toamoregeneralsetting. In§1wegiveadescriptionofquotients
in a suitable symmetric monoidal model category in terms of a certain homotopy
colimit. In §2 we begin by recalling some basic facts and the slice tower and its
construction. We then apply the results of §1 to the category of R-modules in a
symmetricmonoidalmodelcategory(withsomeadditionaltechnicalassumptions),
developingamethodforcomputingtheslicesofanR-moduleM,assumingthatR
andM are effective and that the 0th slice s M is of the form M/({x :i∈I}) for
0 i
some collection {[xi] ∈ R−2di,−di(S),di < 0} of elements in R-cohomology of the
base-scheme S; see theorem 2.3. We also discuss localizations of such R-modules
andthemodpcase(corollary2.4andcorollary2.5). Wediscusstheassociatedslice
spectralsequenceforsuchMandits convergencepropertiesin§3,andapplythese
results to our examples of interest: truncated Brown-Petersonspectra, Morava K-
theoryandconnectiveMoravaK-theory,aswellasthe Landweberexactexamples,
the Brown-Petersonspectra BP and the Johnson-Wilson spectra E(n), in §4.
The remainder of the paper discusses the classifying map from algebraic cobor-
dism Ω and proves our results on the rationality of certain theories. This is es-
∗
sentially taken from [L15], but we need to deal with a technical problem, namely,
that it is not at presentclear if the theories [MGL/({x :i∈I})]2∗,∗ have a multi-
i
plicative structure. For this reason,we extend the setting used in [L15] to theories
that are modules over ring-valued theories. This extension is taken up in §5 and
we apply this theory to quotients and localizations of MGL in §6.
QUOTIENTS OF MGL, THEIR SLICES AND THEIR GEOMETRIC PARTS 3
1. Quotients and homotopy colimits in a model category
In this section we consider certain quotients in a model category and give a
description of these quotients as a homotopy colimit (see proposition 1.9). This is
anabstractionofthe methods developedin[S10]forcomputing the slicesofMGL.
Let (C,⊗,1) be a closed symmetric monoidal simplicial pointed model category
with cofibrant unit 1. We assume that 1 admits a fibrant replacement α : 1 → 1
such that 1 is a 1-algebra in C, that is, there is an associative multiplication map
µ1 :1⊗1→1suchthatµ1◦(α⊗id)andµ1◦(id⊗α)aretherespectivemultiplication
isomorphisms 1⊗1→1, 1⊗1→1.
For a cofibrant object T in C, the map T ∼= T ⊗1 −i−d⊗−→α T ⊗1 is a cofibration
andweak equivalence. Indeed, the functor T ⊗(−) preservescofibrations,and also
maps that are both a cofibration and a weak equivalence, whence the assertion.
Remark 1.1. We will be applying the results of this section to the following situa-
tion: M is a cofibrantly generated symmetric monoidal simplicial model category
satisfying the monoid axiom [ScSh, definition 3.3], R is a commutative monoid in
M, cofibrant in M and C is the categoryof R-modules in C, with model structure
as in [ScSh, §4], that is, a map is a fibration or a weak equivalence in C if and only
if it is so as a mapin M, andcofibrations aredetermined by the LLP with respect
to acyclic fibrations. By [ScSh, theorem 4.1(3)], the category R-Alg of monoids in
C has the structure of a cofibrantly generated model category, with fibrations and
weak equivalence those maps which become a fibration or weak equivalence in M,
and each cofibration in R-Alg is a cofibration in C. The unit 1 is C is just R and
we may take α:1→1 to be a fibrant replacement in R-Alg.
Let {x : T → 1 | i∈ I} be a set of maps with cofibrant sources T . We assign
i i i
each T an integer degree d >0.
i i
Let1/(x )bethehomotopycofiber(i.e.,mappingcone)ofthemapx :1⊗T →
i i i
1 and let p :1→1/(x ) be the canonical map.
i i
Let A={i ,...,i } be a finite subset of I and define 1/({x :i∈A}) as
1 k i
1/({x :i∈A}):=1/(x )⊗...⊗1/(x ).
i i1 ik
Of course, the object 1/({x : i ∈ A}) depends on a choice of ordering of the
i
elements in A, but only up to a canonical symmetry isomorphism. We could for
examplefixtheparticularchoicebyfixingatotalorderonAandtakingtheproduct
in the proper order.The canonical maps p , i ∈ I composed with the map 1 → 1
i
give rise to the canonical map
p :1→1/({x :i∈A})
I i
defined as the composition
1−µ−−→1 1⊗k →1⊗k −p−i1−⊗−.−..⊗−−pi→k 1/({x :i∈A}).
i
For finite subsets A⊂B ⊂I, define the map
ρ :1/({x :i∈A})→1/({x :i∈B})
A⊂B i i
as the composition
µ−1
1/({x :i∈A})−−→1/({x :i∈A})⊗1
i i
−i−d−⊗−pB−\−→A 1/({x :i∈A})⊗1/({x :i∈B\A})∼=1/({x :i∈B}).
i i i
4 MARCLEVINEANDGIRJASHANKERTRIPATHI
where the last isomorphism is again the symmetry isomorphism.
Because C is a symmetric monoidalcategorywith unit 1, we havea well-defined
functor from the category P (I) of finite subsets of I to C:
fin
1/(−):P (I)→C
fin
sending A⊂I to 1/({x :i∈A}) and sending each inclusion A⊂B to ρ .
i A⊂B
Definition 1.2. The object 1/({x :i∈I}) of C is defined by
i
1/({x })= hocolim1/({x :i∈A}).
i i
A∈Pfin(I)
More generally, for M ∈C, we define M/({x :i∈I}) as
i
M/({x :i∈I}):=1/({x :i∈I})⊗QM,
i i
where QM → M is a cofibrant replacement for M. In case the index set I is
understood, we often write these simply as 1/({x }) or M/({x }).
i i
Remark 1.3. 1. The object 1/(x ) is cofibrant and hence the objects 1/({x : i ∈
i i
A})arecofibrantforallfinitesetsA. Asapointwisecofibrantdiagramhascofibrant
homotopycolimit[Hir03,corollary14.8.1,example18.3.6,corollary18.4.3],1/({x :
i
i ∈ I}) is cofibrant. Thus M/({x : i ∈ I}) := 1/({x : i ∈ I})⊗QM is also
i i
cofibrant.
2. We often select a single cofibrant object T and take Ti := T⊗di for certain
integers di > 0. As T is cofibrant, so is T⊗di. In this case we set degT = 1,
degT⊗di =di.
We let [n] denote the set {0,...,n} with the standardorderand ∆the category
with objects [n], n = 0,1,..., and morphisms the order-preserving maps of sets.
For a small category A and a functor F : A → C, we let hocolim F denote the
A ∗
standard simplicial object of C whose geometric realization is hocolim F, that is
A
hocolimF = F(σ(0)).
n
A
σ:[an]→A
Lemma 1.4. Let {x :T →1:i∈I }, {x :T →1:i∈I } be two sets of maps
i i 1 i i 2
in C, with cofibrant sources T . Then there is a canonical isomorphism
i
1/({x :i∈I ∐I })∼=1/({x :i∈I })⊗1/({x :i∈I }).
i 1 2 i 1 i 2
Proof. ThecategoryP (I ∐I )isclearlyequaltoP (I )×P (I ). Forfunctors
fin 1 2 fin 1 fin 2
F : A → C, i = 1,2, [hocolim F ⊗F ] is the diagonal simplicial space
i i A1×A2 1 2 ∗
associated to the bisimplicial space (n,m) 7→ [hocolim F ] ⊗[hocolim F ] .
A1 1 n A2 2 m
Thus
hocolimF ⊗F ∼=hocolim[hocolimF ]⊗F .
1 2 1 2
A1×A2 A2 A1
This gives us the isomorphism
1/({x :i∈I ∐I })
i 1 2
= hocolim 1/({x :i∈A })⊗1/({x :i∈A })
i 1 i 2
(A1,A2)∈Pfin(I1)×Pfin(I2)
∼= hocolim 1/({x :i∈A })⊗ hocolim 1/({x :i∈A })
i 1 i 2
A1∈Pfin(I1) A2∈Pfin(I2)
=1/({x :i∈I })⊗1/({x :i∈I })
i 1 i 2
(cid:3)
QUOTIENTS OF MGL, THEIR SLICES AND THEIR GEOMETRIC PARTS 5
Remark 1.5. Via this lemma, we have the isomorphism for all M ∈C,
M/({x :i∈I ∐I })∼=(M/({x :i∈I })/({x :i∈I }).
i 1 2 i 1 i 2
Let I be the category of formal monomials in {x }, that is, the category of
i
maps N : I → N, i 7→ N , such that N = 0 for all but finitely many i ∈ I, and
i i
with a unique map N → M if N ≥ M for all i ∈ I. As usual, the monomial
i i
in the x corresponding to a given N is xNi, written xN. The index N = 0,
i i∈I i
corresponding to x0 =1, is the final object of I.
Q
Take an i∈I. For m>k ≥0 integers, define the map
×xm−k :1⊗T⊗m →1⊗T⊗k
i i i
as the composition
1⊗T⊗m =1⊗T⊗m−k⊗T⊗k −i−d−1⊗−x−⊗i−m−−−k−⊗−i−dT−i⊗−→k 1⊗m−k+1⊗T⊗k −µ−⊗−i→d 1⊗T⊗k.
i i i i i
Incasek =0,weuse1insteadof1⊗1forthetarget;wedefine×x0tobetheidentity
map. Theassociativityofthemapsµ1 showsthat×xm−k◦×xn−m =×xn−k,hence
i i i
the maps ×xn all commute with each other.
i
Now suppose we have a monomial in the x ; to simplify the notation, we write
i
the indices occurring in the monomial as {1,...,r} rather than {i ,...,i }. This
1 r
gives us the monomial xN :=xN1 ·...·xNr. Define
1 r
TN :=1⊗T⊗N1⊗...⊗1⊗T⊗Nr ⊗1;
∗ 1 r
incaseN =0,wereplace...⊗1⊗1⊗1⊗T⊗Mi+1⊗...with...⊗1⊗T⊗Mi+1⊗...,
i i+1 i+1
and we set T0 :=1.
∗
Let N → M be a map in I, that is N ≥ M ≥ 0 for all i. We again write the
i i
relevant index set as {1,...,r}. Define the map
×xN−M :TN →TM
∗ ∗
as the composition
TN −N−−rj=−1−×−x−jN−j−−−M→j 1⊗T⊗M1 ⊗...⊗1⊗T⊗Mr⊗1 −µ−M→T⊗M;
∗ 1 r ∗
the map µM is a composition of ⊗-product of multiplication maps µ1 :1⊗1→1,
with these occurring in those spots with M =0. In case N =M =0, we simply
j i i
delete the term ×x0 from the expression.
i
The fact that the maps µ1 satisfy associativity yields the relation
×xM−K ◦×xN−M =×xN−K
and thus the maps ×xN−M all commute with each other.
Defining D (N):=TN and D (N →M)=×xN−M gives us the I-diagram
x ∗ x
D :I →C.
x
We consider the following full subcategories of I. For a monomial M let I
≥M
denote the subcategory of monomials which are divisible by M, and for a positive
integer n, recalling that we have assigned each T a positive integral degree d ,
i i
let I denote the subcategory of monomials of degree at least n, where the
deg≥n
degree of N := (N ,...,N ) is N d +···+N d . One defines similarly the full
1 k 1 1 k k
subcategories I and I .
>M deg>n
LetI◦ bethefullsubcategoryofI ofmonomialsN 6=0andI◦ ⊂I◦ bethefull
≤1
subcategoryofmonomialsN forwhichN ≤1foralli. Wehavethecorresponding
i
6 MARCLEVINEANDGIRJASHANKERTRIPATHI
subdiagrams D : I◦ → C and D : I◦ → C of D . For J ⊂ I a subset, we
x x ≤1 x
have the corresponding full subcategories J ⊂ I, J◦ ⊂ I◦ and J◦ ⊂ I◦ and
≤1 ≤1
correspondingsubdiagramsD . Ifthecollectionofmapsx isunderstood,wewrite
x i
simply D for D .
x
Let F : A → C be a functor, a an object in C, c : A → C the constant functor
a
withvalue a andϕ:F →c a naturaltransformation. Thenϕ induces a canonical
a
mapϕ˜:hocolim F →ainC. Asintheproofof[S10,Proposition4.3],letC(A)be
A
the category A with a final object ∗ adjoined and C(F,ϕ) : C(A)→C the functor
with value a on ∗, with restriction to A being F and which sends the unique map
y → ∗ in C(A), y ∈ A, to ϕ(y). Let [0,1] be the category with objects 0,1 and
a unique non-identity morphism, 0 → 1 and let C(A)Γ be the full subcategory of
C(A)×[0,1]formed by removingthe object∗×1. We extend C(F,ϕ) to a functor
C(F,ϕ)Γ :C(A)Γ →C by C(F,ϕ)Γ(y×1)=pt, where pt is the initial/final object
in C.
Lemma 1.6. There is a natural isomorphism in C
hocolimC(F,ϕ)Γ ∼=hocofib(ϕ˜:hocolimF →a).
C(A)Γ A
Proof. ForacategoryAwe letN(A) denote the simplicialnerveofA. We havean
isomorphismofsimplicial sets N(C(A))∼=Cone(N(A),∗), where Cone(N(A),∗) is
theconeoverN(A)withvertex∗. Similarly,thefullsubcategoryA×[0,1]ofC(A)Γ
hasnerveisomorphictoN(A)×∆[1]. ThisgivesanisomorphismofN(C(A)Γ)with
the push-out in the diagram
(cid:31)(cid:127) //
N(A) Cone(N(A),∗)
(cid:127)_
id×δ0
(cid:15)(cid:15)
N(A)×∆[1]
This in turn gives an isomorphism of the simplicial object hocolim C(F,ϕ)Γ
C(A)Γ ∗
with the pushout in the diagram
(cid:31)(cid:127) //
hocolim(cid:127)_ AF C(hocolimAF,a)
(cid:15)(cid:15)
C(hocolim F,pt).
A
This gives the desired isomorphism. (cid:3)
Lemma 1.7. Let J ⊂K ⊂I be finite subsets of I. Then the map
hocolimD →hocolimD
x x
J◦ K◦
≤1 ≤1
induced by the inclusion J ⊂K is a cofibration in C.
Proof. We give the category of simplicial objects in C, C∆op, the Reedy model
structure, using the standard structure of a Reedy category on ∆op. By [Hir03,
theorem 19.7.2(1), definition 19.8.1(1)], it suffices to show that
hocolimD →hocolimD
∗ ∗
J◦ K◦
≤1 ≤1
QUOTIENTS OF MGL, THEIR SLICES AND THEIR GEOMETRIC PARTS 7
is a cofibration in C∆op, that is, for each n, the map
ϕ :hocolimD ∐ LnhocolimD →hocolimD
n J≤◦1 n LnhocolimJ≤◦1D∗ K◦≤1 ∗ K◦≤1 n
is a cofibration in C, where Ln is the nth latching space.
We note that
hocolimD = D(σ(0))
n
J◦
≤1 σ∈N_(J≤◦1)n
where we view σ ∈ N(J◦ ) as a functor σ : [n] → J◦ ; we have a similar
≤1 n ≤1
description of hocolimK◦ Dn. The latching space is
≤1
LnhocolimD = D(σ(0)),
∗
J◦
≤1 σ∈N(_J≤◦1)dneg
where N(J◦ )deg is the subset of N(J◦ ) consisting of those σ which contain an
≤1 n ≤1 n
identity morphism; LnhocolimK◦ D∗ has a similar description. The maps
≤1
LnhocolimD →hocolimD ,LnhocolimD →LnhocolimD∗,
∗ n ∗
J◦ J◦ J◦ K◦
≤1 ≤1 ≤1 ≤1
LnhocolimD →hocolimD ,hocolimD →hocolimD
∗ n n n
K◦ K◦ J◦ K◦
≤1 ≤1 ≤1 ≤1
are the unions of identity maps on D(σ(0)) over the respective inclusions of the
index sets. As N(K◦ )deg∩N(J◦ ) =N(J◦ )deg, we have
≤1 n ≤1 n ≤1 n
hoJc≤o◦l1imDn∐LnhocolimJ≤◦1D∗ LnhoKco◦≤l1imD∗ ∼=hoJc≤o◦l1imDn C,
_
where
C = D(σ(0)),
σ∈N(K◦≤1)dn_eg\N(J≤◦1)dneg
and the map to hocolimK◦ Dn is the evident inclusion. As D(N) is cofibrant for
all N, this map is clearly ≤a1cofibration, completing the proof. (cid:3)
We have the n-cube (cid:3)n, the category associated to the partially ordered set of
subsets of {1,...,n}, ordered under inclusion, and the punctured n-cube (cid:3)n of
0
proper subsets. We have the two inclusion functors i+,i− : (cid:3)n−1 → (cid:3)n, i+(I) :=
n n n
I ∪{n}, i−(I) = I and the natural transformation ψ : i− → i+ given as the
n n n n
collection of inclusions I ⊂ I ∪ {n}. The functor i− induces the functor i− :
n n0
(cid:3)n−1 →(cid:3)n.
0
For a functor F : (cid:3)n → C, we have the iterated homotopy cofiber, hocofib F,
n
defined inductively as the homotopy cofiber of hocofib (F(ψ )) : hocofib(F ◦
n−1 n
i−) → hocofib(F ◦ i+). Using this inductive construction, it is easy to define
n n
a natural isomorphism hocofibnF ∼= hocolim(cid:3)n+1Fˆ, where Fˆ ◦ i−n+10 = F and
0
Fˆ(I)=pt if n∈I.
The following result is proved in [S10, Lemma 4.2 and Proposition 4.3].
Lemma 1.8. Assume that I is countable. Then there is a canonical isomorphism
in HoC
1/({x | i∈I})∼=hocofib[hocolimD →hocolimD ]
i x x
I◦ I
8 MARCLEVINEANDGIRJASHANKERTRIPATHI
Proof. As 1 is the final object in I, the collection of maps ×xN : TN → 1 defines
∗
a weak equivalence π : hocolim D → 1. In addition, for each N ∈ I◦, the
I x
comma categoryN/I◦ has initial object the map N →N¯, where N¯ =1 if N >0,
1 i i
and N¯ = 0 otherwise. Thus I◦ is homotopy right cofinal in I◦ (see e.g. [Hir03,
i 1
definition 19.6.1]). Since D is a diagram of cofibrant objects in C, it follows from
x
[Hir03, theorem 19.6.7] that the map hocolimI◦Dx → hocolimI◦Dx is a weak
1
equivalence. This reduces us to identifying 1/({x }) with the homotopy cofiber of
i
π≤◦1 : hocolimI1◦Dx →1, where π≤◦1 is the composition of π with the natural map
:hocolimI◦Dx →hocolimIDx.
Next, w1e reduce to the case of a finite set I. Take I = N. Let P (I) be the
fin
category of finite subsets of I, ordered by inclusion, consider the full subcategory
PO (I) of P (I) consisting of the subsets I := {1,...,n} and let I◦ ⊂ I◦
fin fin n n,≤1 ≤1
be the full subcategory with all indices in I . As PO (I) is cofinal in P (I), we
n fin fin
have
colimhocolimD ∼=hocolimD .
x x
n I◦ I◦
n,≤1 ≤1
Take n≤m. By lemma 1.7 the the map hocolimI◦ Dx →hocolimI◦ Dx is
n,≤1 m,≤1
acofibrationinC. Thus,usingthe ReedymodelstructureonCN withNconsidered
asadirectcategory,theN-diagraminC,n7→hocolimI◦ Dx,isacofibrantobject
n,≤1
in CN. As N is a direct category, the fibrations in CN are the pointwise ones, hence
N has pointwise constants [Hir03, definition 15.10.1]and therefore [Hir03, theorem
19.9.1] the canonical map
hocolimhocolimD →colimhocolimD
x x
n∈N I◦ n∈N I◦
n,≤1 n,≤1
is a weak equivalence in C. This gives us the weak equivalence in C
hocolimhocolimD →hocolimD .
x x
n I◦ I◦
n,≤1 ≤1
Since N iscontractible,the canonicalmaphocolimN1→1 isa weakequivalencein
C, giving us the weak equivalences
hocofib[hocolimD →1]
x
I◦
≤1
∼hocofib[hocolimhocolimD →hocolim1]
x
n∈N I◦ n∈N
n,≤1
∼hocolim[hocofib[hocolimD →1]].
x
n∈N I◦
n,≤1
Thus, we need only exhibit isomorphisms in HoC
ρ :hocofib[hocolimD →1]→1/(x ,...,x ):=1/(x )⊗...⊗1/(x ),
n x 1 n 1 n
I◦
n,≤1
which are natural in n∈N.
By lemma 1.6 we have a natural isomorphism in C,
hocofib[hocolimD →1]∼= hocolim C(D ,π)Γ.
x x
I◦ C(I◦ )Γ
n,≤1 n,≤1
However, I◦ is isomorphic to (cid:3)n by sending N = (N ,...,N ) to I(N) :=
n,≤1 0 1 n
{i|N =0}. Similarly,C(I◦ )isisomorphicto(cid:3)n,andC(I◦ )Γisthusisomor-
i n,≤1 n,≤1
phic to (cid:3)n+1. From our discussion above, we see that hocolim C(D ,π)Γ
0 C(I◦ )Γ x
n,≤1
QUOTIENTS OF MGL, THEIR SLICES AND THEIR GEOMETRIC PARTS 9
is isomorphic to hocofib C(D ,π), so we need only exhibit isomorphisms in HoC
n x
ρ :hocofib C(D ,π)→1/(x )⊗...⊗1/(x )
n n x 1 n
which are natural in n∈N.
We do this inductively as follows. To include the index n in the notation, we
write C(D ,π) for the functor C(D ,π) : (cid:3)n →C. For n = 1, hocofib C(D ,π)
x n x 1 x 1
isthemappingconeofµ ◦(×x ⊗id):1⊗T ⊗1→1,whichisisomorphicinHoC
1 1 1
to the homotopy cofiber of ×x : 1⊗T → 1. As this latter is equal to 1/(x ),
1 1 1
so we take ρ :hocofib C(D ,π) →1/(x ) to be this isomorphism. We note that
1 1 x 1 1
C(D ,π) ◦i+ =C(D ,π) and C(D ,π) ◦i− =C(D ,π) ⊗T ⊗1.
x n n x n−1 x n n x n−1 n
DefineC(D ,π)′ byC(D ,π)′ ◦i− =C(D ,π) ⊗1⊗T ⊗1,C(D ,π)′ ◦i+ =
x n x n n x n−1 n x n n
C(D ,π) ⊗1, with the natural transformationC(D ,π)′ ◦ψ given as
x n−1 x n n
C(D ,π) ⊗1⊗T ⊗1−(−id−⊗−µ−)◦−(−id−⊗−×−x−n−⊗−id−1→) C(D ,π) ⊗1.
x n−1 n x n−1
The evident multiplication maps give a weak equivalence C(D ,π)′ →C(D ,π) ,
x n x n
giving us the isomorphism in HoC
ρ :hocofib C(D ,π) →1/(x )⊗...⊗1/(x )
n n x n 1 n
defined as the composition
hocofib C(D ,π) ∼=hocofib C(D ,π)′
n x n n x n
∼=hocofib(hocofib (C(D ,π) ⊗1⊗T )
n−1 x n−1 n
−h−o−co−fi−b−n−−1−(−id−⊗−×−x−n→) hocofib (C(D ,π) ⊗1))
n−1 x n−1
∼=hocofib(hocofib (C(D ,π) )⊗1⊗T
n−1 x n−1 n
−i−d⊗−−×−x→n hocofib (C(D ,π) )⊗1)
n−1 x n−1
∼=hocofib (C(D ,π) )⊗hocofib(×x :1⊗T →1)
n−1 x n−1 n n
=hocofib (C(D ,π) )⊗1/(x )
n−1 x n−1 n
−ρ−n−−1−⊗−i→d 1/(x )⊗...⊗1/(x )⊗1/(x ).
1 n−1 n
Via the definition of hocofib ,
n
hocofib C(D ,π) =hocofib[hocofib (C(D ,π) ◦i−)
n x n n−1 x n n
−h−o−co−fi−b−n−−−1(−C−(−D−x,−π−)n−−−1−(ψ−n−)→) hocofib (C(D ,π) ◦i+]
n x n n
and the identification C(D ,π) ◦i+ = C(D ,π) , we have the canonical map
x n n x n−1
hocofib (C(D ,π) )→hocofib (C(D ,π) . One easily sees that the diagram
n−1 x n−1 n x n
hocofib[hocolimI◦ Dx →1] //hocofib[hocolimI◦ Dx →1]
n−1,≤1 n,≤1
∼ ∼
(cid:15)(cid:15) (cid:15)(cid:15)
//
hocofib (C(D ,π) ) hocofib (C(D ,π) )
n−1 x n−1 n x n
ρn−1 ρn
(cid:15)(cid:15) (cid:15)(cid:15)
1/(x )⊗...⊗1/(x ) //1/(x )⊗...⊗1/(x )
1 n−1 ρ{1,...,n−1}⊂{1,...,n} 1 n
commutes in HoC, giving the desired naturality in n. (cid:3)
10 MARCLEVINEANDGIRJASHANKERTRIPATHI
NowletM beanobjectinC,letQM →M beacofibrantreplacementandform
the I-diagram D ⊗QM :I →C, (D ⊗QM)(N)=D (N)⊗QM.
x x x
Proposition 1.9. Assume that I is countable. Let M be an object in C. Then
there is a canonical isomorphism in HoC
M/({x | i∈I})∼=hocofib[hocolimD ⊗QM →hocolimD ⊗QM]
i x x
I◦ I
Proof. This follows directly from lemma 1.8, noting the definition of M/({x | i ∈
i
I}) as [1/({x | i∈I})]⊗QM and the canonical isomorphism
i
hocofib[hocolimD ⊗QM →hocolimD ⊗QM]
x x
I◦ I
∼=hocofib[hocolimD →hocolimD ]⊗QM
x x
I◦ I
(cid:3)
Proposition 1.10. Let F : I → C be a diagram in a cofibrantly generated
deg≥n
model category C. Suppose for every monomial M of degree n the natural map
hocolimF| →F(M) is a weak equivalence. Then the natural map
I>M
hocolimF| →hocolimF
deg≥n+1
is a weak equivalence.
Proof. Thisisjust[S10,lemma4.4],withthefollowingcorrections: thestatementof
the lemma in loc. cit. has “hocolimF| →F(M) is a weak equivalence” rather
I≥M
than the correct assumption “hocolimF| →F(M) is a weak equivalence” and
I>M
in the proof, one should replace the object Q(M) with colimQ| rather than
I>M
with colimQ| . (cid:3)
I≥M
2. Slices of effective motivic module spectra
In this section we will describe the slices for modules for a commutative and
effective ring T-spectrum R that satisfies certain additional conditions. We adapt
the constructions used in describing slices of MGL in [S10], which go through
without significant change in this more general setting.
Let us first recall the definition of the slice tower in SH(S). We will use
the standard model category Mot := Mot(S) of symmetric T-spectra over S,
T := A1/A1 \{0}, with the motivic model structure as in [J00], for defining the
triangulated tensor category SH(S):=HoMot(S).
For an integer q, let ΣqSHeff(S) denote the localizing subcategory of SH(S)
T
generated by S := {ΣqΣ∞X | p ≥ q,X ∈ Sm/S}, that is ΣqSHeff(S) is the
q T T + T
smallest triangulated subcategory of SH(S) which contains S and is closed under
q
direct sums and isomorphisms in SH(S). This gives a filtration on SH(S) by full
localizing subcategories
···⊂Σq+1SHeff(S)⊂ΣqSHeff(S)⊂Σq−1SHeff(S)⊂···⊂SH(S).
T T T
The set S is a set of compact generators of ΣqSH(S) and the set ∪ S is
q T q q
similarlyasetofcompactgeneratorsforSH(S). ByNeeman’striangulatedversion
of Brown representability theorem [N97], the inclusion i :ΣqSHeff(S)→SH(S)
q T
has a right adjoint r :SH(S)→ΣqSHeff(S). We let f :=i ◦r . The inclusion
q T q q q