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356 P. Bressler, A. Gorokhovsky, R. Nest, and B. Tsygan3.2.1 Hochschild cochains. Suppose that A is a k-vector space. The k-vector spaceC n .A/ of Hochschild cochains of degree n 0 is defined byC n .A/ WD Hom k .A˝n ; A/:The graded vector space g.A/ WD C .A/Œ1 has a canonical structure of a graded Liealgebra under the Gerstenhaber bracket denoted by Œ;below. Namely, C .A/Œ1 iscanonically isomorphic to the (graded) Lie algebra of derivations of the free associativeco-algebra generated by AŒ1.Suppose in addition that A is equipped with a bilinear operation W A ˝ A ! A,i.e. 2 C 2 .A/ D g 1 .A/. The condition Œ; D 0 is equivalent to the associativityof .Suppose that A is an associative k-algebra with the product . For a 2 g.A/ letı.a/ D Œ; a. Thus, ı is a derivation of the graded Lie algebra g.A/. The associativityof implies that ı 2 D 0, i.e. ı defines a differential on g.A/ called the Hochschilddifferential.For a unital algebra the subspace of normalized cochains xC n .A/ C n .A/ is definedbyxC n .A/ WD Hom k ..A=k 1/˝n ; A/:The subspace xC .A/Œ1 is closed under the Gerstenhaber bracket and the action of theHochschild differential and the inclusion xC .A/Œ1 ,! C .A/Œ1 is a quasi-isomorphismof DGLA.Suppose in addition that R is a commutative Artin k-algebra with the nilpotentmaximal ideal m R The DGLA g.A/ ˝k m R is nilpotent and satisfies g i .A/ ˝k m R D0 for i < 1. Therefore, the Deligne 2-groupoid MC 2 .g.A/ ˝k m R / is defined.Moreover, it is clear that the assignment R 7! MC 2 .g.A/ ˝k m R / extends to a functoron the category of commutative Artin algebras.3.2.2 Star products. Suppose that A is an associative unital k-algebra. Let m denotethe product on A.Let R be a commutative Artin k-algebra with maximal ideal m R . There is a canonicalisomorphism R=m R Š k.An (R-)star product on A is an associative R-bilinear product on A ˝k R such thatthe canonical isomorphism of k-vector spaces .A ˝k R/ ˝R k Š A is an isomorphismof algebras. Thus, a star product is an R-deformation of A.The 2-category of R-star products on A, denoted Def.A/.R/, is defined as thesubcategory of the 2-category Alg 2 Rof R-algebras (see 4.1.1) with• Objects: R-star products on A,• 1-morphisms W m 1 ! m 2 between the star products i those R-algebra homomorphisms W .A ˝k R;m 1 / ! .A ˝k R;m 2 / which reduce to the identity mapmodulo m R , i.e. ˝R k D Id A ,