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I'm not sure this question is research level question. Sorry in advance. Hypothesis $k$ is a commutative ring. $A$ is an augmented $k$-algebra. $A^e$ is defined as the $k$-algebra $A\otimes_{k}A^{op}$...

Let $C$ be an abelian category with enough projectives and $\underline{C}$ the stable category of $C$ that is obtained by factoring out projective modules. When is the functor $\Omega^1 : \underline{...