Gluing compactly generated t-structures over stalks of affine schemes (Q6635456)

This paper offers a detailed investigation into the properties of compactly generated \(t\)-structures in the derived category, particularly focusing on the bijection between such \(t\)-structures over a commutative ring \(R\) and certain families of \(t\)-structures over the local rings \(R_m\), where \(m\) runs through the maximal ideals in the Zariski spectrum of \(R\). More precisely, the paper reveals the gluing conditions that these families must satisfy, and applies this framework to check the compact generation of homotopically smashing \(t\)-structures locally via localizations at maximal ideals. The authors also utilize a result by \textit{P. Balmer} and \textit{G. Favi} [Proc. Lond. Math. Soc. (3) 102, No. 6, 1161--1185 (2011; Zbl 1220.18009)] to conclude that the \(\otimes\)-Telescope Conjecture for quasi-coherent and quasi-separated schemes is a stalk-local property, which is a significant contribution to algebraic geometry and homotopy theory. Moreover, the paper extends the work of \textit{J. Trlifaj} and \textit{S. Şahinkaya} [J. Algebra 408, 28--41 (2014; Zbl 1304.13019)] by establishing an explicit bijection between cosilting objects of cofinite type over \(R\) and compatible families of cosilting objects of cofinite type over all localizations \(R_m\) at maximal primes. This not only deepens our understanding of cosilting objects but also provides a new perspective and tools for module theory in homological algebra.\N\NOverall, this paper is both theoretically profound and highly applicable, offering an important framework that enriches our understanding of the deep connections between algebraic geometry, category theory, and homotopy theory.