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CMI PhD Research Seminar Date: Tuesday, 22 October 2024 Time: 10:30 - 11:30 AM Venue: Seminar Hall The Formal Functions Theorem and some applications Pranoy Sarkar Chennai Mathematical Institute. 22-10-24 Abstract The Formal Functions Theorem gives a basic comparison result between the operations of taking cohomology and taking completions. It can be stated as follows: Let fâ â be a projective morphism of noetherian schemes from X to Y, and F a coherent sheaf on X. Let y be a point of Y. Define X_n to be the n-th infinitesimal thickening of the fibre of f over y, and let F_n be the restriction of F to X_n. Let R^i be the completion of the i-th higher direct image of F at the point y, and let H^i be the inverse limit of the system {H^i(X_n, F_n)}. Then, there is a natural isomorphism from R^i to H^i for all i. We shall outline the proof and also some important applications like Zariski Main Theorem, Stein Factorization etc.
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