The slices of a Landweber exact theory
Scribe notes from a talk by Lorenzo Mantovani
20 Mar 2014
Notation. —
• SHeff (S) is the localizing triangulated subcategory of SH(S) generated
by Σ∞
T X+ for X ∈ Sm/S .
• SH(S)T is the Tate subcategory, the full localizing triangulated subcategory of SH(S) generated by S p,q = Sp−q ∧ Gqm .
• SH(S)T ≥0 is like the above, but generated by S p,q for q ≥ 0.
1
s0 M GL
The main theorem is
Theorem. The cofiber cofib(1 → M GL) belongs to ΣT SH(S)T ≥0 .
Corollary. s0 1 → s0 M GL is an isomorphism in SH(S). (In particular, if
S = Spec k, then s0 M GL = HZ.)
2
Description of s0 M GL
Idea. We have seen a map Z[x1 , . . . , xn ] = M U∗ → M GL2∗,∗ . We want this
to define a map “multiplication by xi ” on M GL. We will need to bifibrantly
replace M GL and cofibrantly replace S 2n,n . We can form the object
M GL/(x1 , x2 , . . .)M GL.
Definition. A multi-index is a function f : N → N almost everywhere zero;
these will encode exponents for monomials. In particular, multi-indices are not
of fixed length. We let I be the set of such multi-indices.
We define a map D : I → SptΣ
P1 (Sm/S ) by
Da = Σ2,1 M GL∧M GL a1
∧M GL
...
∧M GL
Σ2m,m M GL∧M GL am .
Taking two multi-indices a and b which differ only in that ai ≥ bi for some i,
we have a map Db → Da given by id
. . ∧ id} ∧ xi ∧ . . . ∧ xi .
| ∧ .{z
|
{z
}
bi
1
ai −bi
Proposition. If F : Ideg≥n
→ C is a diagram in a cofibrantly generated model
category, and hocolim F deg I≥a = F (a) for all a of degree m, then hocolim F =
hocolim F .
deg I≥n+1
Proposition. fi : SH(S) → ΣiT SH(S)eff ⊆ SH(S) preserves homotopy colimits.
Theorem. Suppose that M GL → Q := M GL/(x1 , . . .)M GL is the map to
the zero slice. Then si M GL ' ΣiT s0 M GL ⊗ M U2i , and this isomorphism is
compatible to M U∗ → M GL∗,∗ .
Sketch proof. —
• The map Ddeg I≥1 ,→ D induces a map on hocolims, and this is the
map M GL → Q. So hocolim Ddeg I≥1 → f1 M GL is an isomorphism in
SH(S). Here f1 is the slice truncation functor introduced in a previous
talk.
• Next show that f2 hocolim Ddeg I≥1 ' hocolim F2 Ddeg i≥2 , where Fn :
Σ
SptΣ
P1 (−) → SptP1 (−) lifts fn .
• By induction one shows that fn M GL ' hocolim Ddeg I≥n .
•
sn M GL ' cofib(fn+1 M GL → fn M GL)
' hocolim cofib(Fn+1 D
→ Fn D deg I≥n
deg I≥n
).
• Now cofib(a) = 0 if deg(a) ≥ n + 1, and
cofib(a) = ΣT s0 M GL (deg(a) = n),
M
hocolim(−) =
ΣnT s0 M GL = Σnt s0 M GL ⊗ M U2n .
quotients of N2n
We use in this last step that sn ΣT E = ΣT sn−1 E.
3
Computation of the slices of a Landweber exact spectrum
Theorem. If M∗ is a Landweber exact M U∗ -module, and E ∈ SH(S) is the
element representing the (co)homology theory defined by M∗ , namely
(M GL ∧ −)∗,∗ ⊗M U∗ M∗
: SH(S) → GrAb,
then
si E ' ΣiT s0 M GL ⊗ M2i .
2
Main idea. We take a projective resolution
0 → A∗ → B ∗ → M ∗ → 0
of M∗ by M U∗ -modules. (Landweber exactness guarantees homological dimension 1.)
3