Abelian and derived deformations in the presence of Z-generating geometric helices

For a Grothendieck category C which, via a Z-generating sequence (O(n))_{n in Z}, is equivalent to the category of quasi-coherent modules over an associated Z-algebra A, we show that under suitable cohomological conditions taking quasi-coherent modules defines an equivalence between linear deformations of A and abelian deformations of C. If (O(n))_{n in Z} is at the same time a geometric helix in the derived category, we show that restricting a (deformed) Z-algebra to a thread of objects defines a further equivalence with linear deformations of the associated matrix algebra.