Tilting theory has a long history originating in the representation theory of finite-dimensional algebra [Hap] and algebraic geometry [Bei]. Later, infinitely generated tilting modules were introduced as a formal generalization of tilting objects representing derived equivalences. It turned out recently that in the context of commutative rings, there is a very geometric classification of tilting modules up to additive equivalence in terms of support and using very well-known homological theories [APST], [HS]. If time allows, I will touch recent results which elaborate the relevance of the geometric point of view further - the Zariski locality tilting modules [HST].
[APST] L. Angeleri Hugel, D. Pospisil, J. Stovicek, J. Trlifaj: Tilting, cotilting, and spectra of commutative noetherian rings, Trans. Amer. Math. Soc. 366 (2014), 3487-3517.
[Bei] A. A. Beilinson: Coherent sheaves on P^n and problems in linear algebra, Funktsional. Anal. i Prilozhen. 12, 68-69, 1978.
[Hap] D. Happel: On the derived category of a finite-dimensional algebra, Comment. Math. Helv. 62, 339-389, 1987.
[HS] M. Hrbek, J. Stovicek: Tilting classes over commutative rings, preprint, arXiv:1701.05534.
[HST] M. Hrbek, J. Stovicek, J. Trlifaj: Zariski locality of quasi-coherent sheaves associated with tilting, preprint, arXiv:1712.08899.