The Transformation of Adulthood in the Digital Age

The article is devoted to the phenomenon of modern youth’s infantilization, attracting more and more attention of researchers. The view of infantilism as a kind of youth disease generated by a change in socio-cultural reality and digitalization is opposed the interpretation of infantilism as an artificially derived category (media product). The data of a pilot “image of an adult “ study (in young people perception) are presented. The thesis that in modern realities the “adult image” is at the stage of critical rethinking is proposed.

Conflict between decision-making subjects that arises during the investigation of careless crimes committed by medical staff

The diversity of the interpretation of the cause-and-effect relationship by various subjects of law enforcement activity leads to the birth of an absolutely newthing in itself a new process. Then the causal relationship, being a derived category, acquires a new meaning and a new essence, which leads to its modification, contradictions in judicial practice and conflicts between decision-making subjects.

Ring Constructions and Generation of the Unbounded Derived Module Category

We consider the smallest triangulated subcategory of the unbounded derived module category of a ring that contains the injective modules and is closed under set indexed coproducts. If this subcategory is the entire derived category, then we say that injectives generate for the ring. In particular, we ask whether, if injectives generate for a collection of rings, do injectives generate for related ring constructions, and vice versa. We provide sufficient conditions for this statement to hold for various constructions including recollements, ring extensions and module category equivalences.

Rank functions on triangulated categories

We introduce the notion of a rank function on a triangulated category 𝒞 {\mathcal{C}} which generalizes the Sylvester rank function in the case when 𝒞 = 𝖯𝖾𝗋𝖿 ⁢ ( A ) {\mathcal{C}=\mathsf{Perf}(A)} is the perfect derived category of a ring A. We show that rank functions are closely related to functors into simple triangulated categories and classify Verdier quotients into simple triangulated categories in terms of particular rank functions called localizing. If 𝒞 = 𝖯𝖾𝗋𝖿 ⁢ ( A ) {\mathcal{C}=\mathsf{Perf}(A)} as above, localizing rank functions also classify finite homological epimorphisms from A into differential graded skew-fields or, more generally, differential graded Artinian rings. To establish these results, we develop the theory of derived localization of differential graded algebras at thick subcategories of their perfect derived categories. This is a far-reaching generalization of Cohn’s matrix localization of rings and has independent interest.

Quasiexcellence implies strong generation

We prove that the bounded derived category of coherent sheaves on a quasicompact separated quasiexcellent scheme of finite dimension has a strong generator in the sense of Bondal–Van den Bergh. This simultaneously extends two results of Iyengar–Takahashi and Neeman and is new even in the affine case. The main ingredient includes Gabber’s weak local uniformization theorem and the notions of boundedness and descendability of a morphism of schemes.

Parametrizing torsion pairs in derived categories

We investigate parametrizations of compactly generated t-structures, or more generally, t-structures with a definable coaisle, in the unbounded derived category D ( M o d - A ) \mathrm {D}({\mathrm {Mod}}\text {-}A) of a ring A A . To this end, we provide a construction of t-structures from chains in the lattice of ring epimorphisms starting in A A , which is a natural extension of the construction of compactly generated t-structures from chains of subsets of the Zariski spectrum known for the commutative noetherian case. We also provide constructions of silting and cosilting objects in D ( M o d - A ) \mathrm {D}({\mathrm {Mod}}\text {-}A) . This leads us to classification results over some classes of commutative rings and over finite dimensional hereditary algebras.

PROJECTIVE GENERATION FOR EQUIVARIANT -MODULES

We investigate compact projective generators in the category of equivariant "Image missing"-modules on a smooth affine variety. For a reductive group G acting on a smooth affine variety X, there is a natural countable set of compact projective generators indexed by finite dimensional representations of G. We show that only finitely many of these objects are required to generate; thus the category has a single compact projective generator. The proof goes via an analogous statement about compact generators in the equivariant derived category, which holds in much greater generality and may be of independent interest.