Microlocal composition and pure sheaves

Two lessons explain how kernel germs compose at prescribed covectors and how closed half-space tests define pure and simple sheaves with their exact shifts. Fourteen exercises have complete solutions.

• Microlocal composition at prescribed covectors · editable source
• Pure and simple sheaves from directional tests · editable source

The composition lesson gives the cutoff, compactness and denominator arguments for a bounded microlocal composition, its compatibilities, and the graph condition needed to compare germ actions with regional operators. Its prerequisite table links the exact directional cutoff, boundary control, isolated direct-image representation and ordinary kernel calculus proofs, together with the tensor and localization results used in the argument.

The purity lesson derives closed half-space tests, compares their normalized types under contact kernels, and treats zero covectors and arbitrary bounded coefficients. Its prerequisite table links the conormal coefficient model, microlocal stalk formula, ordered-index calculation and contact-kernel selection proofs, together with the remaining normalization and kernel-composition arguments. Simplicity and nonvanishing examples assume a nonzero ring. The external prerequisite proofs remain accessible through their linked programme passages.

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