Realizability of modules over Tate cohomology

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A-infinity algebras, modules and functor categories (2005, background): “Let Cof be the class of morphisms i such that in each commutative square of solid arrows in Alg // i ˜ // ?? there is a dotted arrow which makes the two triangles commutative.”
Cohomological quotients and smashing localizations (2003, unknown): “Under some additional assumptions, this extension is induced by a Hochschild cocycle of degree (3,−1); it plays a crucial role in [4].”
Representations of Algebras and Related Topics (2011, background): “hat Hˆ∗(G,M) = Hom∗ kG(k,M) is isomorphic to the graded ﬁeld of fractions of H∗(G,k)/p. Such an object does not always exist, as can be seen using the obstruction theory of Benson, Krause and Schwede [14, 15]. There is one case where there are ﬁeld objects, and that is the case of an elementary abel”
On triangulated orbit categories (2005, background): “2 T b splits,then V ishereditary and each objectX 2 T is(non canonically)isom orphic to the sum ofthe S nH n V (X ),n 2 Z. Theproofisan exercise on t-structureswhich we leave to thereader.”
Weight structures and simple dg modules for positive dg algebras (2010, background): “Now realizing modules over the homology H∗(A) as homologies of dg modules over A is in general a hard problem, cf. for example [5].”