Takeshi Saito - 4/4 Singular Supports in Equal and Mixed Characteristics

Beilinson defined the singular support of a constructible sheaf on a smooth scheme over a field as a closed conical subset on the cotangent bundle. He further proved its existence and fundamental properties, using Radon transform as a crucial tool. In first lectures, we formulate the definition in a slightly different but equivalent way, using an interpretation by Braverman—Gaitsgory of the local acyclicity. We also recall Beilinson's proof of existence. In mixed characteristics, the theory is still far from complete. As a replacement of the cotangent bundle, we introduce the Frobenius—Witt cotangent bundle, that has the correct rank but defined only on the characteristic p fiber. Using it, we define the singular support and its relative variant. Finally, we show that Beilinson's argument using the Radon transform gives a proof of the existence of the saturation of the relative variant.

Takeshi Saito (The university of Tokyo)

Lecture notes :
https://www.ms.u-tokyo.ac.jp/~t-saito...
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