Vychylující teorie v algebraické geometrii

• Název práce v češtině: Vychylující teorie v algebraické geometrii
• Název v anglickém jazyce: Tilting theory in algebraic geometry
• Akademický rok vypsání: 2023/2024
• Typ práce: disertační práce
• Ústav: Katedra algebry (32-KA)
• Vedoucí / školitel: RNDr. Michal Hrbek, Ph.D.

Zásady pro vypracování

While the classical derived Morita theory represented by compact tilting complexes trivializes in case of commutative rings, it has been known for some years that large tilting and cotilting complexes are ubiquitous in commutative algebra. Tilting theory in commutative algebra has seen several breakthrough in the recent years, including a complete classification of the corresponding t-structures ([1],[2]), and more recently, the topological algebra explanation of the generalized derived Morita theory induced by large tilting objects ([3],[4]). In special but important cases, the tilting and cotilting complexes have even been constructed explicitly ([5]), which allowed to obtained much deeper insight.

The aim of this thesis is to extend this recent progress from the affine case of a commutative ring to the general algebro-geometric setting of a separated Noetherian scheme. The relevant class of t-structures has been recently described in [8], while the good local behavior of tilting and cotilting objects has been established in [6],[7]. Apart from focusing on derived categories of quasi-coherent sheaves, a more abstract approach for generalizations of some of the classification results can be considered for tensor triangulated categories endowed with a suitable t-structure.

Seznam odborné literatury

[1] - Angeleri Hügel, L., Pospíšil, D., Šťovíček, J., & Trlifaj, J. (2014). Tilting, cotilting, and spectra of commutative Noetherian rings. Transactions of the American Mathematical Society, 366(7), 3487-3517.
[2] - Angeleri Hügel, L., & Hrbek, M. (2021). Parametrizing torsion pairs in derived categories. Representation Theory of the American Mathematical Society, 25(23), 679-731.
[3] - Positselski, L., & Šťovíček, J. (2021). The tilting–cotilting correspondence. International Mathematics Research Notices, 2021(1), 189-274.
[4] - Hrbek, M. (2022). Topological endomorphism rings of tilting complexes. arXiv preprint arXiv:2205.11105.
[5] - M. Hrbek, T. Nakamura, J. Šťovíček. Tilting complexes and codimension functions over commutative noetherian rings. Nagoya Mathematical Journal. Published online 2024:1-76. doi:10.1017/nmj.2024.1
[6] - Hrbek, M., Šťovíček, J., & Trlifaj, J. (2020). Zariski locality of quasi-coherent sheaves associated with tilting. Indiana University Mathematics Journal, 69(5), 1733-1762.
[7] - Breaz, S., Hrbek, M., & Modoi, G. C. (2022). Silting, cosilting, and extensions of commutative ring. arXiv preprint arXiv:2204.01374.
[8] - Dubey, U. V., & Sahoo, G. (2023). Compactly generated tensor t-structures on the derived categories of Noetherian schemes. Mathematische Zeitschrift, 303(4), 100.