Endomorphisms of Koszul complexes: formality and application to deformation theory

We study the differential graded Lie algebra of endomorphisms of the Koszul resolution of a regular sequence on a unitary commutative K-algebra R and we prove that it is homotopy abelian over K but not over R (except trivial cases). We apply this result to prove an annihilation theorem for obstructions of (derived) deformations of locally complete intersection ideal sheaves on projective schemes.