local government vs government

local government and government both are nouns.

Word	Noun	Adjective	Verb	Adverb
local government	Yes	No	No	No
government	Yes	No	No	No

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

local government: the government of a local area
government: the organization that is the governing authority of a political unit

Other hypernyms of local government include authorities, regime.

local government (noun)	government (noun)
the government of a local area	the act of governing; exercising authority
	(government) the system or form by which a community or other political unit is governed
	the study of government of states and other political units
	the organization that is the governing authority of a political unit