MODULARITY LIFTING THEOREMS - NOTES FOR POSTECH ...

20 TOBY GEE
(ii) Suppose that A is a catenary local O-algebra, that each irreducible
component of Spec A has the same dimension, and that each generic
point of Spec A has characteristic 0. Suppose that every prime of A
minimal over λA contains a unique minimal prime of A. Let M be
a finitely generated O-torsion free A-module. If M/λM is a nearly
faithful A/λA-module, then M is a nearly faithful A-module.
(iii) Suppose that A is a Noetherian local ring, that Spec A is irreducible,
and that M is a finitely generated A-module. Suppose that the mAdepth
of M is greater than or equal to the Krull dimension of A.
Then M is a nearly faithful A-module.
Proof. (i) We need to show that (M/IM)P �= 0 for every prime I ⊂ P .
If not, then we have MP /IP MP = 0, and by Nakayama’s lemma we
conclude that MP = 0, a contradiction.
(ii) This requires slightly more work, but is still elementary. See Lemma
2.2(2) of [Tay08].
(iii) This uses some homological algebra - see Lemma 2.3 of [Tay08].
�
We now apply these results as follows. Consider firstly S � ζ,∞
as a mod-
ule for R loc
ζ [[X1, . . . , Xu]]. We employ the third criterion above with A =
Rloc ζ [[X1, . . . , Xu]]. Since Spec Rloc ζ,l
is irreducible, we see that Spec Rloc
ζ [[X1, . . . , Xu]]
is irreducible, so that it is enough to check that the mA-depth of S� ζ,∞ is at
least dim A. But A has the same dimension as O[[T1, . . . , Tu, A1, . . . , A7]],
considered as an
so it is enough to prove the corresponding assertion for S� ζ,∞
O[[T1, . . . , Tu, A1, . . . , A7]]-module. But S� ζ,∞ is a finite free O[[T1, . . . , Tu, A1, . . . , A7]]module,
so this is immediate, and we conclude that S� ζ,∞ is a nearly faithful
Rloc ζ [[X1, . . . , Xu]]-module.
From the third criterion, we see that S� ζ,∞ /λ is a nearly faithful (Rloc
ζ λ)[[X1, . . . , Xu]]module,
so that S� 1,∞ /λ is a nearly faithful (Rloc 1 λ)[[X1, . . . , Xu]]-module.
We now use the second criterion to conclude that S� 1,∞ is a nearly faithful
Rloc 1 [[X1, . . . , Xu]]-module (note that any complete local noetherian ring is
catenary (even excellent)). Finally, using the first criterion again we see that
S� 1,∞ /(T1, . . . , A1, . . . , A7) ∼ = S1,∅ is a nearly faithful Runiv 1 -module.
As above, this means that ker(Runiv 1 → T1,∅) is nilpotent, so any O-point
of Runiv 1 is a O-point of T1,∅ and is thus modular. In particular ρ is modular,
as required.
References
[Tay08] Richard Taylor, Automorphy for some l-adic lifts of automorphic mod l Galois
representations. II, Pub. Math. IHES 108 (2008), 183–239.
Department of Mathematics, Northwestern University, 2033 Sheridan Road,
Evanston, IL 60208-2730, USA
E-mail address: gee@math.northwestern.edu