Hochschild cohomology, the characteristic morphism and derived deformations
نویسنده
چکیده
A notion of Hochschild cohomology HH∗(A) of an abelian category A was defined by Lowen and Van den Bergh (2005) and they showed the existence of a characteristic morphism χ from the Hochschild cohomology of A into the graded centre Z∗(Db(A)) of the bounded derived category of A. An element c ∈ HH2(A) corresponds to a first order deformation Ac of A (Lowen and Van den Bergh, 2006). The problem of deforming an object M ∈ Db(A) to D(Ac) was treated by Lowen (2005). In this paper we show that the element χ(c)M ∈ Ext 2 A(M,M) is precisely the obstruction to deforming M to D (Ac). Hence this paper provides a missing link between the above works. Finally we discuss some implications of these facts in the direction of a “derived deformation theory”.
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