Efficient searches for electromagnetic counterparts to gravitational wave, high-energy neutrino, and gamma-ray burst events demand rapid processing of image arithmetic and geometry set operations in a database to cross-match galaxy catalogs, observation footprints, and all-sky images. Here we introduce HEALPix Alchemy, an open-source, pure Python implementation of a set of methods that enables rapid all-sky geometry calculations. HEALPix Alchemy is built upon HEALPix, a spatial indexing strategy that is widely used in astronomical databases as well as the native format of LIGO-Virgo-KAGRA gravitational-wave sky localization maps. Our approach leverages new multirange types built into the PostgreSQL 14 database engine. This enables fast all-sky queries against probabilistic multimessenger event localizations and telescope survey footprints. Questions such as What are the galaxies contained within the 90% credible region of an event?'' and What is the rank-ordered list of the fields within an observing footprint with the highest probability of containing the event?’’ can be performed in less than a few seconds on commodity hardware using off-the-shelf cloud-managed database implementations without server-side database extensions. Common queries scale roughly linearly with the number of telescope pointings. As the number of fields grows into the hundreds or thousands, HEALPix Alchemy is orders of magnitude faster than other implementations. HEALPix Alchemy is now used as the spatial geometry engine within SkyPortal, which forms the basis of the Zwicky Transient Facility transient marshal, called Fritz.