The system of fixed elements of a projectivity contains with any two points the line connecting them and with any two lines their intersection. It is, therefore, in its structure very much like a subplane of the plane under consideration ; and thus one may expect the structure of the projectivity to be dominated by the structure of the system of its fixed elements, provided this system is not "...

Let X be a (quasi-projective) variety over k. We follow [Sha13, Chap. 6, §1.2].

We will generalize the projective model structure in the category of unbounded complexes of modules over a commutative ring to the category of unbounded complexes of quasi-coherent sheaves over the projective line. Concretely we will define a locally projective model structure in the category of complexes of quasicoherent sheaves on the projective line. In this model structure the cofibrant obj...

Let $R$ be a right artinian ring or a perfect commutative‎‎ring‎. ‎Let $M$ be a noncosingular self-generator $sum$-lifting‎‎module‎. ‎Then $M$ has a direct decomposition $M=oplus_{iin I} M_i$‎,‎where each $M_i$ is noetherian quasi-projective and each‎‎endomorphism ring $End(M_i)$ is local‎.

We compare several quasi-Frobenius-type properties for corings that appeared recently in literature and provide several new characterizations for each of these properties. By applying the theory of Galois comodules with a firm coinvariant ring, we can characterize a locally quasi-Frobenius (quasi-co-Frobenius) coring as a locally projective generator in its category of comodules.