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Suppose we have an Abelian category $\mathfrak A$ and a ring $R$. From this data we can form a new Abelian category $\mathfrak A[R]$ whose objects are objects $A\in\mathfrak A$ together with a ring homomorphism $R\to\mathfrak A(A,A)$ and whose morphisms are morphisms in $\mathfrak A$ commuting with the $R$-actions.

Let's assume that we understand the category $\mathfrak A$ and the ring $R$ well in the sense that we know the projective objects in $\mathfrak A$ and in the category of $R$-modules. Can we use this to characterise projective objects in $\mathfrak A[R]$ in a nice way?

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    available at hopf.math.purdue.edu/Neeman/triangulatedcats.pdf2011-05-05

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