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In this talk I shall report two recent results, joint with Haoyang Guo and Tong Liu respectively, concerning derived de Rham cohomology and p-adic Hodge theory. Firstly, I hope to convey the belief that many constructions in p-adic Hodge theory (e.g. Fontaine's period rings and Scholze's period sheaves) can be uniformly constructed using derived de Rham theory. Secondly I'll make sense of tensor product, as a filtered object, of filtered modules over a filtered ring. Lastly we use this notion to (re)formulate the so-called p-adic Poincaré sequence as well as the relation between Nygaard filtration on prismatic cohomology and Hodge filtration on derived de Rham cohomology.
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