Activities for part 3

Choose a chapter activity below. Each page turns one mathematical idea into a finite example that you can change.

• 0 · Compare the two exponent testsAn exponent controls both the shape of the unit region and the effect of merging boxes. Convexity and stable merging are different conditions, although both change at exponent one.
• 1 · Check the parts of a measure ruleA proposed rule needs a condensed ring and a module of allowed measures for every extremally disconnected probe. Point masses must be included, and separate pieces must receive independent data.
• 2 · Compare two analytic measure rulesThe p-adic rule and the solid rule over a discrete ring both pass the analytic test. They support different infinite operations because their coefficient rings and measure modules differ.
• 3 · Measure what happens when boxes mergeA fine-stage measure must stay within its size limit after its boxes are merged at a coarser stage. This chart calculates the change for equal weights.
• 4 · Separate global and boundary termsNear infinity, functions are expanded in the reciprocal coordinate. Polynomial terms extend across the whole line, while negative powers describe a tail that lives near the boundary.
• 5 · Compute a boundary quotientCompactly supported pushforward compares functions near the boundary with functions defined everywhere. A line and a coordinate cross give small models of this quotient.
• 6 · Match bounded functions with valuationsA chosen subring of bounded functions selects the valuations that give every one of those functions size at most one. Under the theorem's assumptions, the selected region also recovers the subring.
• 7 · Follow the six operationsThe six operations are tensor product, internal Hom, pullback, ordinary pushforward, compactly supported pushforward, and its right adjoint. They form three adjoint pairs.
• 8 · Pair complementary degreesCoherent duality pairs a class with one in the complementary degree. Their product reaches the top degree, and a trace map then produces a scalar on the base.

Every activity is self-contained and uses no network connection or external library. You can therefore save and run the pages offline.