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Lets take a ring $R$ and an ideal $\mathfrak p \subset R$, and call them an L-pair (just for brevity) if $\mathfrak p$-adic completion of any projective module is again projective (as R-module); and L-...

Let $\mathcal{C}$ be an abelian category and $\mathcal{S}$ be a full subcategory of $\mathcal{C}$. We call $\mathcal{S}$ a Serre subcategory of $\mathcal{C}$ if $\mathcal{S}$ is closed under ...

Let $R$ be a commutative Noetherian ring. An acyclic complex $P$ of projective $R$-modules is called totally acyclic if for every projective $R$-module $Q$, the complex Hom$_R(P, Q)$ is also acyclic. ...