In SGA2,Grothendieck introduced two important examples（EXP X 2.1） which satisfy Grothendieck-Lefschetz condition.My question is what's the intuition for the "local" version of Lefschetz's theory ?
I know that for projective algebraic scheme $X$, $H^d(X,\oplus M(n))=H^{d+1}_m(A,M_m)$ where $A$ is the affine cone for $X$( c.f Kunz,Residue and duality for projective algebraic varieities proposition 6.3).Is it correct to use local Lefschetz theory for this isomorphism to get the global one?
Also local Lefschetz theory is used in SGA4 to prove the theorem of smooth base change which is essentially a homotopy theorem.What is the intuition for this?
