Ogus Derivations and the Explicit Syzygy Class Maps
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概要
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Let X be a given closed subscheme with the arithmetic D_2 condition in a complex projective space P=P^N(C)=Proj(S) (S:=C[Z_O,...,Z_N]). To study locally along X the sheaf of q-th syzygy module Z^<(q)>_<X/P>(q≥1) of the homogeneous coordinate ring R_X of X as an S-module, here we construct and calculate explicitly the q-syzygy class map for degree m part ρ^(q,m) : Γ(P,Z^<(q)>_<X/P>(m))→H^1(X,Ω^q_P&otime;N^V_<X/P>(m)) in a local expression form with using local frames and local coordinates. This map induces an isomorphism onto the space of infinitesimal obstructions <ρ(q,m)>^^^- : Γ(P,Z^<(q)>_<X/P>(m))/Σ^N_<i=0>Z_i・Γ(P,Z^<(q)>_<X/P>(m-1))≌Tor^S_q(Rx,S/S_+)_<(m)>→Im<[δ_<LFT>]>^^^-⊊︀H^1(X,Ω^q_P&otime;N^V__<X/P>(m)). In other words, we give good Cech representatives for the image of the C-linear homomorphism ρ^<(q,m)>.