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- W4361793153 abstract "Let $overline{Bbb Q_p}$ be an algebraic closure of $Bbb Q$ and $C$ its $p$-adic completion. Let $K$ be a finite extension of $Bbb Q$ contained in $overline{Bbb Q_p}$ and set $G_K=mathrm{Gal}(overline{Bbb Q_p}/K)$. A $Bbb Q_p$-representation (resp. a $C$-representation) of $G_K$ is a finite dimensional $Bbb Q$-vector space (resp. $C$-vector space) equipped with a linear (resp. semi-linear) continuous action of $G_K$. A Banach representation of $G_K$ is a topological $Bbb Q$-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 $G_K$. An almost $C$-representation of $G_K$ is a Banach representation $X$ which is almost isomorphic to a $C$-representation, i.e. such that there exists a $C$-representation $W$, finite dimensional sub-$Bbb Q$-vector spaces $V$ of $X$ and $V'$ of $W$ stable under $G_K$ and an isomorphism $X/Vto W/V'$. The almost $C$-representations of $G_K$ form an abelian category $cal C(G_K)$. There is a unique additive function $dh: {Ob}cal C(G_K)to Bbb NtimesBbb Z$ such that $dh(W)=(dim_{C}W,0)$ if $W$ is a $C$-representation and $dh(V)=(0,dim_{Bbb Q_p}V)$ if $V$ is a $Bbb Q_p$-representation. If $X$ and $Y$ are objects of $cal C(G_K)$, the $Bbb Q_p$-vector spaces $mathrm{Ext}^{i}{cal C(G_K)}(X,Y)$ are finite dimensional and are zero for $inotin{0,1,2}$. One gets $sum{i=0}^{2}(-1)^{i}dim_{Bbb Q}mathrm{Ext}^{i}{cal C(G_K)}(X,Y)=-[K:Bbb Q]h(X)h(Y)$. Moreover, there is a natural duality between $mathrm{Ext}^i{cal C(G_K)}(X,Y)$ and $mathrm{Ext}^{2-i}_{cal C(G_K)}(Y,X(1))$." @default.
- W4361793153 created "2023-04-05" @default.
- W4361793153 creator A5057767888 @default.
- W4361793153 date "2003-01-01" @default.
- W4361793153 modified "2023-09-27" @default.
- W4361793153 title "Almost $C_p$-representation" @default.
- W4361793153 doi "https://doi.org/10.4171/dms/3/10" @default.
- W4361793153 hasPublicationYear "2003" @default.
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