luxury vs indulgence

luxury and indulgence both are nouns.

Word	Noun	Adjective	Verb	Adverb
luxury	Yes	No	No	No
indulgence	Yes	No	No	No

As nouns, indulgence is a hypernym of luxury; that is, indulgence is a word with a broader meaning than luxury:

Other hypernyms of luxury include self-indulgence.

luxury (noun)	indulgence (noun)
something that is an indulgence rather than a necessity	the act of indulging or gratifying a desire
the quality possessed by something that is excessively expensive	foolish or senseless behavior
wealth as evidenced by sumptuous living	the remission by the pope of the temporal punishment in purgatory that is still due for sins even after absolution
	an inability to resist the gratification of whims and desires
	a disposition to yield to the wishes of someone