Item 6 (01KH) — The Clowder Project

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Style	Class	Font	Theorem Environments
Style 1	`book`	Alegreya Sans	`tcbthm`
Style 2	`book`	Alegreya Sans	`amsthm`
Style 3	`book`	Arno*	`amsthm`
Style 4	`book`	Computer Modern	`amsthm`

*To be replaced with Linus Romer's Elemaints when it is released.

Interaction With the Empty Set III. Let $f\colon X\to Y$ be a function.
1. Interaction With Direct Images. The diagram
  
  commutes, i.e. we have
  
  \[ f_{*}\webleft (D_{X}\webleft (U\webright )\webright )=D_{Y}\webleft (f_{!}\webleft (U\webright )\webright ) \]
  
  for each $U\in \mathcal{P}\webleft (X\webright )$.
2. Interaction With Inverse Images. The diagram
  
  commutes, i.e. we have
  
  \[ f^{-1}\webleft (D_{Y}\webleft (U\webright )\webright )=D_{X}\webleft (f^{-1}\webleft (U\webright )\webright ) \]
  
  for each $U\in \mathcal{P}\webleft (X\webright )$.
3. Interaction With Direct Images With Compact Support. The diagram
  
  commutes, i.e. we have
  
  \[ f_{!}\webleft (D_{X}\webleft (U\webright )\webright )=D_{Y}\webleft (f_{*}\webleft (U\webright )\webright ) \]
  
  for each $U\in \mathcal{P}\webleft (X\webright )$.

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