We fix a profinite groupG. Let G acts on Gnby left multiplication. The cohomology forG arises from the diagram
· · · ########
G×G×G ######G×G ####G,
the arrows being the projections di : Gn+1 → Gn for each i = 0, . . . , n given by di(σ0, . . . , σi−1,σ!i, σi+1, . . . , σn), where by σ!i we indicate that we have omitted σi from the (n+1)-tuple(σ0, . . . , σn).
We assume that allG-modules to be discrete. For every G-module M, we form the abelian groupXn = Xn(G, M) = Map(Gn+1, M)of all continuous mapsx : Gn+1 → M. Xn is a G-module by(σx)(σ0, . . . , σn) = σx(σ−1σ0, . . . σ−1σn). The mapsdi : Gn+1 → Gn induce G-module homomorphisms di∗ : Xn−1→ Xn, and we form the alternating sum
∂n =
n
∑
i=0
(−1)id∗i : Xn−1→ Xn.
Fact B.2.1. The sequence
0 −→ M−→∂0 X0 −→∂1 X1 −→∂2 X2 −→ · · ·∂3
is exact.
§B.2 Cohomology for profinite groups ·46·
We apply the functor “fixed module”, and set forn≥0
Cn(G, M) = Xn(G, M)G.
Cn(G, M)consists of the continuous functionsx : Gn+1 → M such that x(σσ0, . . . , σσn) = σx(σ0, . . . , σn)for allσ ∈ G. These functions are called the n-cochains of G with coefficients in M.
From the Fact B.2.1, we obtained a sequence
C0(G, M)−→∂1 C1(G, M) −→∂2 C2(G, M) −→ · · ·∂3 ,
which is no longer exact in general; but it is still a complex, i.e.,∂n+1◦∂n =0. We now set
Zn(G, M) = ker(Cn(G, M) −→∂n+1 Cn+1(G, M)), Bn(G, M) = im(Cn−1(G, M) −→∂n Cn(G, M)),
andB0(G, A) = 0. Since ∂n+1◦∂n = 0, we have Bn(G, M) ⊂ Zn(G, M). The elements of Zn(G, M)andBn(G, M)are called then-cocycles and n-coboundaries respectively.
Definition B.2.2. For eachn≥0, the quotient group
Hn(G, M) = Zn(G, M)/Bn(G, M)
is called then-dimensional cohomology group of G with coefficients in M.
Fact B.2.3. If G is a finite group and M is a finite G-module, then Hn(G, M)is a finite module for each n ≥0.
Fact B.2.4. For n=0, 1, and 2, the groups Hn(G, M)admit the following interpretations:
(i) For n=0, we have H0(G, M) = MG; (ii) For n=1, we have
H1(G, M) = {x : G →M | x(στ) = σx(τ) +x(σ) ∀σ, τ∈ G} {x : σ#→ (σ−1)m |m ∈ M} ;
§B.2 Cohomology for profinite groups ·47·
(iii) For n=2, we have
H2(G, M)
= {x : G2 → M| x(στ, ρ) +x(σ, τ) = x(σ, τρ) +σx(τ, ρ) ∀σ, τ, ρ∈ G} {x : G2 →M | x(σ, τ) = y(σ) −y(στ) +σy(τ)} , with arbitrary y : G → M∈ C1(G, M).
Let{I,≤}be a directed set. Let{Ti}i∈I be a family of objects indexed byI and fij : Ti →Tj be a homomorphism for alli ≤ j with the following properties:
(a) fiiis the identity ofTi, and (b) fik = fjk◦ fijfor alli≤ j≤k.
The triple I = {I, Ti, fij} is called adirected system. The underlying set of the direct limit, T, of the direct systemI = {I, Ti, fij}is defined as the disjoint union of theTi’s modulo a certain equivalence relation∼:
lim−→
I
Ti = 4
i∈I
Ti 5
∼.
Here, ifti ∈ Tiandtj ∈ Tj,ti ∼tjif there is somek ∈ I such that fik(ti) = fjk(tj).
LetU, V runs through the open normal subgroups of G. If V ⊆ U, then the projections Gn+1 → (G/V)n+1 → (G/U)n+1induce homomorphisms
Cn(G/U, MU) → Cn(G/V, MV) → Cn(G, M),
which commute with the operators∂n+1. We thus obtain homomorphisms
Hn(G/U, MU) → Hn(G/V, MV) →Hn(G, M).
The groups Hn(G/U, MU) form a direct system and we have a canonical homomorphism lim−→UHn(G/U, MU) →Hn(G, M).
§B.2 Cohomology for profinite groups ·48·
Fact B.2.5. The above homomorphism is an isomorphism:
lim−→
U
Hn(G/U, MU) −→/ Hn(G, M).
Fact B.2.6 (The inflation-restriction exact sequence). Let U be a closed normal subgroup of G, and suppose that Hm(G, M) = 0 for all m =1, 2, . . . , n−1. Then the following sequence is exact:
0−→Hn(G/U, MU) −→Hn(G, M) −→H0(G/U, Hn(U, M))
−→Hn+1(G/U, MU).
§B.2 Cohomology for profinite groups ·49·
Bibliography
[1] M. F. Atiyah and I. G. Macdonald,Introduction to Commutative Algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969.
[2] C. Breuil, B. Conrad, F. Diamond, and R. Taylor,On the modularity of elliptic curves over Q:
wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), no. 4, 843–939.
[3] J. W. S. Cassels and A. Fr¨ohlich (eds.), Algebraic Number Theory (2nd edition), London, United Kingdom, London Mathematical Society, 2010, Reprint of the 1967 original. MR 911121 (88h:11073)
[4] L. Clozel, M. Harris, and R. Taylor, Automorphy for some l-adic lifts of automorphic mod l Galois representations, Publ. Math. Inst. Hautes ´Etudes Sci. (2008), no. 108, 1–181, With Appendix A, summarizing unpublished work of Russ Mann, and Appendix B by Marie-France Vign´eras.
[5] B. Conrad, F. Diamond, and R. Taylor,Modularity of certain potentially Barsotti-Tate Galois representations, J. Amer. Math. Soc. 12 (1999), no. 3, 521–567.
[6] B. de Smit and H. W. Lenstra, Jr. , Explicit construction of universal deformation rings, Modular forms and Fermat’s last theorem (Boston, MA, 1995), Springer, 1997, pp. 313–326.
[7] F. Diamond, On deformation rings and Hecke rings, Ann. of Math. (2) 144 (1996), no. 1, 137–166.
[8] J.-M. Fontaine and B. Mazur, Geometric Galois representations, Elliptic curves, modular forms, & Fermat’s last theorem (Hong Kong, 1993), Ser. Number Theory, I, Int. Press, Cambridge, MA, 1995, pp. 41–78.
©Hui-Wen Chou 2012 ·50·
[9] A. Grothendieck,Technique de descente et th´eor`emes d’existence en g´eom´etrie alg´ebrique. II.
Le th´eor`eme d’existence en th´eorie formelle des modules, S´eminaire Bourbaki, vol. 5, Soci´et´e Math´ematique de France, 1995, pp. 369–390.
[10] A. Grothendieck and J. Dieudonn´e,El´ements de G´eom´etrie Alg´ebrique, Publ. Math. IHES, 4 (1960), 8 (1961), 11 (1961), 17 (1963), 20 (1964), 24 (1965), 28 (1966), 32 (1967).
[11] K. Haberlan, Galois Cohomology of Algebraic Number Fields, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978, With two appendices by Helmut Koch and Thomas Zink.
[12] M. Harris, N. Shepherd-Barron, and R. Taylor,A family of Calabi-Yau varieties and potential automorphy, Ann. of Math. (2) 171 (2010), no. 2, 779–813.
[13] H. Hida,Galois representations into GL2(Zp[[X]])attached to ordinary cusp forms, Invent.
Math. 85 (1986), no. 3, 545–613.
[14] C. Khare and J.-P. Wintenberger,Serre’s modularity conjecture. I, Invent. Math. 178 (2009), no. 3, 485–504.
[15] ,Serre’s modularity conjecture. II, Invent. Math. 178 (2009), no. 3, 505–586.
[16] M. Kisin, The Fontaine-Mazur conjecture for GL2, J. Amer. Math. Soc. 22 (2009), no. 3, 641–690.
[17] , Lecture Notes on Deformations of Galois Representations, Clay Mathematics Institute 2009 Summer School on Galois Representations (University of Hawaii at Manoa, Honolulu, Hawaii), June 15 - July 10 2009.
[18] ,Moduli of finite flat group schemes and modularity, Ann. of Math. (2) 170 (2009), no. 3, 1085–1180.
[19] S. Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, 1998.
[20] B. Mazur, Deforming of Galois Representations, Galois Groups over Q (Y. Ihara, K. Ribet, and J.-P. Serre, eds.), Mathematical Sciences Research Institute Publications, no. 16, Springer-Verlag, 1987, pp. 385–437.
BIBLIOGRAPHY ·51·
[21] ,An introduction to the deformation theory of Galois representations, Modular forms and Fermat’s last theorem (Boston, MA, 1995), Springer-Verlag, New York, 1997, pp. 243–311.
[22] J. Neukirch,Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften, vol. 322, Springer-Verlag, 1999.
[23] R. Ramakrishna, On a variation of Mazur’s deformation functor, Compositio Math. 87 (1993), 269–286.
[24] M. Schlessinger,Functors of Artin rings, Trans. A.M.S. 130 (1968), 208–222.
[25] J.-P. Serre,Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, 1979.
[26] , Linear representations of finite groups, Graduate Texts in Mathematics, vol. 42, Springer-Verlag, New York, 1996.
[27] ,Galois cohomology, Springer Monographs in Mathematics, Springer-Verlag, 1997.
[28] R. Taylor,Automorphy for some l-adic lifts of automorphic mod l Galois representations. II, Publ. Math. Inst. Hautes ´Etudes Sci. (208), no. 108, 183–239.
[29] R. Taylor and A. Wiles,Ring-theoretic properties of certain Hecke algebras, Ann. of Math. 141 (1995), no. 3, 553–572.
[30] J. Tilouine, Deformations of Galois representations and Hecke algebras, Published for The Mehta Research Institute of Mathematics and Mathematical Physics, Allahabad, 1996.
[31] A. Wiles,Modular elliptic curves and Fermat’s Last Theorem, Ann. of Math. 141 (1995), no. 3, 443–551.
BIBLIOGRAPHY ·52·
Index
irreducible, 36 reducible, 36 ring of Witt vectors, 16 small morphism, 24 strictly equivalent, 19 subcategory, 41
Teichm¨uller representative, 15 Theorem
Burnside Basis, 12 Grothendieck, 24 Hermite-Minkowski, 13
Inflation-restriction exact sequence, 49 Mazur, Ramakrishna, 36
Schlessinger, 24 Schur, 37 Yoneda, 44
topologicalG-module, 45 topological group, 7 transitive, 6
universal, 7
universal deformation, 23 universal deformation ring, 23 universal element, 44
universal mapping property, 44 universal pair, 23
Witt vectors, 16
Zariski cotangent space, 27 Zariski tangent space, 27
INDEX ·54·