Almost CpC_p-representation

  • Jean-Marc Fontaine


Let Qp\overline{\Bbb Q_p} be an algebraic closure of Q\Bbb Q and CC its pp-adic completion. Let KK be a finite extension of Q\Bbb Q contained in Qp\overline{\Bbb Q_p} and set GK=Gal(Qp/K)G_K=\mathrm{Gal}(\overline{\Bbb Q_p}/K). A Qp\Bbb Q_p-representation (resp. a CC-representation) of GKG_K is a finite dimensional Q\Bbb Q-vector space (resp. CC-vector space) equipped with a linear (resp. semi-linear) continuous action of GKG_K. A Banach representation of GKG_K is a topological Q\Bbb Q-vector space, whose topology may be defined by a norm with respect to which it is complete, equipped with a linear and continuous action of GKG_K. An almost CC-representation of GKG_K is a Banach representation XX which is almost isomorphic to a CC-representation, i.e. such that there exists a CC-representation WW, finite dimensional sub-Q\Bbb Q-vector spaces VV of XX and VV' of WW stable under GKG_K and an isomorphism X/VW/VX/V\to W/V'. The almost CC-representations of GKG_K form an abelian category C(GK)\cal C(G_K). There is a unique additive function dh:ObC(GK)N×Zdh: {Ob}\cal C(G_K)\to \Bbb N\times\Bbb Z such that dh(W)=(dimCW,0)dh(W)=(\dim_{C}W,0) if WW is a CC-representation and dh(V)=(0,dimQpV)dh(V)=(0,\dim_{\Bbb Q_p}V) if VV is a Qp\Bbb Q_p-representation. If XX and YY are objects of C(GK)\cal C(G_K), the Qp\Bbb Q_p-vector spaces ExtC(GK)i(X,Y)\mathrm{Ext}^{i}_{\cal C(G_K)}(X,Y) are finite dimensional and are zero for i∉{0,1,2}i\not\in\{0,1,2\}. One gets i=02(1)idimQExtC(GK)i(X,Y)=[K:Q]h(X)h(Y)\sum_{i=0}^{2}(-1)^{i}\dim_{\Bbb Q}\mathrm{Ext}^{i}_{\cal C(G_K)}(X,Y)=-[K:\Bbb Q]h(X)h(Y). Moreover, there is a natural duality between ExtC(GK)i(X,Y)\mathrm{Ext}^i_{\cal C(G_K)}(X,Y) and ExtC(GK)2i(Y,X(1))\mathrm{Ext}^{2-i}_{\cal C(G_K)}(Y,X(1)).