Let me more specific here
For the last two decades, people's interest in their ancestor's family, cultural and living background have been increasing.
There are two questions.
The first question is about the correct usage of a phrase: For the last two decades is being used as a phrase. Is it correct to use like this?. I think there are no specific rules for phrase-making.
The second question is on the usage of apostrophe rules: I'm trying to use apostrophe rules in the next phrase. But the Possession now belongs to three items (family, cultural, living). Is it correct to use in this manner?:
people's interest in their ancestor's family, cultural and living background have been increasing.
Or in the presence of possessive adjective (their), I simply should write as:
people's interest in their ancestor family, living, cultural background have been increasing.
Best Answer
You are talking about several things that are possessed.
People's interest (a plural word ending in a letter other than s, accepts 's to show possession)
Their [people] ancestors (a possessive plural pronoun needing no modification)
Ancestors' background[s] (many ancestors as opposed to one ancestor, pluralized by adding s' -- and many backgrounds, since you refer to many ancestors)
Additionally you need to list the adjectives describing the types of backgrounds so the reader can easily identify the list as describing the backgrounds as opposed to being a separate phrase.