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Hyeongkwan Kim : Homological methods in commutative algebra

Beyond Krull dimension, rings and modules have various "dimensions": depth, height, projective/injective dimensions, flat dimension, global dimension, weak dimension, among others. These notions are defined homologically in terms of lengths of resolutions, and Ext and Tor functors provide a way to measure them. I'll talk about how they are related with each other. I'll start from geometic interpretation of Krull dimension and height, followed by regular sequences and depth. Then I'll introduce Cohen-Macaulay modules and Gorenstein modules as modules having particularly nice homological properties. Just as in the case of completion where analysis is introduced to algebra to prove some highly nontrivial results, homological techniques have proved to be very powerful in modern commutative algebra, producing such surprising results as homological characterization of regular rings (Serre, Auslander, Buchsbaum). I'll briefly introduce the notion of canonical modules and the question of finiteness of injective resolution. Finally, I'll talk about how these notions can be globalized to scheme and sheaves, which makes geometry "kind of equivalent" to algebra.

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