Singular Supports in Equal and Mixed Characteristics - 4/4

Explore advanced concepts in algebraic geometry through this lecture that examines singular supports of constructible sheaves in both equal and mixed characteristics. Learn about Beilinson's definition of singular support as a closed conical subset on the cotangent bundle of smooth schemes over fields, and understand the fundamental properties proven using Radon transform techniques. Discover the Braverman-Gaitsgory interpretation of local acyclicity and follow Beilinson's proof of existence in the equal characteristic case. Investigate the challenges and developments in mixed characteristic theory, including the introduction of the Frobenius-Witt cotangent bundle as a replacement for the traditional cotangent bundle in characteristic p fibers. Examine the definition of singular support and its relative variant in this context, and understand how Beilinson's Radon transform argument provides proof for the existence of saturation of the relative variant. This fourth and final lecture in the series provides comprehensive coverage of these sophisticated mathematical concepts with detailed theoretical foundations and proofs.