ScalingStacks

Proposition 12 [03SZ]

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Proposition 12

Let Ei=F⁡(Xi),0≤i≤k,k≥1E_{i}=F(X_{i}),0\leq i\leq k,k\geq 1 be locally free rank one 𝒪Y{\cal O}_{Y}-modules (vector bundles) corresponding to objects Xi=(Li,ρi)∈F​O​(X∨),0≤i≤kX_{i}=(L_{i},\rho_{i})\in FO(X^{\vee}),0\leq i\leq k. Then the formulas for

mkF​O​(X∨):⊗0≤i≤kHom(Ei,Ei+1)→Hom(E0,Ek)[2−k]m_{k}^{FO(X^{\vee})}:\otimes_{0\leq i\leq k}Hom(E_{i},E_{i+1})\to Hom(E_{0},E_{k})[2-k]

coincide (after the extension of scalars from 𝐂ε{{\bf C}}_{\varepsilon} to 𝐂ε​⊗^​Ω0∗{{\bf C}}_{\varepsilon}\widehat{\otimes}\Omega^{\ast}_{0}) with the formulas for

mk𝒞u​n​r​a​m,0t​r,Π​(Y):⊗0≤i≤kHom(Xi,Xi+1)→Hom(X0,Xk)[2−k]m_{k}^{{\cal C}_{unram,0}^{tr,\Pi}(Y)}:\otimes_{0\leq i\leq k}Hom(X_{i},X_{i+1})\to Hom(X_{0},X_{k})[2-k]

when the spaces of morphisms are identified via the maps ν⁡(Xi,Xj)\nu(X_{i},X_{j}).

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