Twists of graded algebras in monoidal categories (Q6632092)

Twists of graded algebas over a field were introduced by \textit{J. J. Zhang} [Proc. Lond. Math. Soc. (3) 72, No. 2, 281--311 (1996; Zbl 0852.16005)] to study certain categories of graded modules over graded algebras in Noncommutative Projective Algebraic Geometry (NCPAG), and since then the framework of twisted algebas has proved useful in its own right. The procedure known as Zhang twist turned out to preserve many important properties of graded algebras such as several growth measures (e.g., Gelfand-Kirillov dimension), ring-theoretic notions (e.g., Noetherian domains), and homological conditions and measures (e.g., \ global dimensions, Gorenstein conditions). On the other hand, properties of algebras are preserved via equivalent categories of modules (e.g., via Morita equivalence). The main result of \textit{J. J. Zhang} [Proc. Lond. Math. Soc. (3) 72, No. 2, 281--311 (1996; Zbl 0852.16005)] is that twisting algebras are related to having a type of Morita equivalence. This paper aims to generalize Zhang's twisting method for graded algebras, and to study the corresponding categories of graded modules in the monoidal setting.\N\NThe synopsis of the paper goes as follows:\N\N\begin{itemize}\N\item[\S 2] provides background and preliminary results on graded algebraic structures in monoidal categories.\N\N\item[\S 3] addresses closed monoidal categories and enrichment in the graded setting, examining endomorphism algebras in categories of modules in the monoidal setting and discussing gradings on enriched categories.\N\N\item[\S 4] introduces twisted systems and twisted algebras for the monoidal setting, defining twist equivalence.\N\N\item[\S 5] extends the main results of the paper.\N\N\item[Appendix] is concerned with lengthy diagrammatic arguments deferred,\N\end{itemize}