local anesthesia vs anaesthesia

local anesthesia and anaesthesia both are nouns.

Word	Noun	Adjective	Verb	Adverb
local anesthesia	Yes	No	No	No
anaesthesia	Yes	No	No	No

As nouns, anaesthesia is a hypernym of local anesthesia; that is, anaesthesia is a word with a broader meaning than local anesthesia:

local anesthesia: loss of sensation in a small area of the body (as when a local anesthetic is injected for a tooth extraction)
anaesthesia: loss of bodily sensation with or without loss of consciousness

Other hypernyms of local anesthesia include anesthesia.

local anesthesia (noun)	anaesthesia (noun)
loss of sensation in a small area of the body (as when a local anesthetic is injected for a tooth extraction)	loss of bodily sensation with or without loss of consciousness