Learn Turkish: Şirket müdürünün gözaltı kararı üzerine kurul onu açığa aldı. - Upon the company manager's custody decision, the board suspended him.

Here, üzerine means 'upon' or 'as a result of'. It is used right after a noun or a possessive phrase, which in this case is gözaltı kararı (custody decision). It shows that the board's action was an immediate and direct response to the decision.

Could we just say kararından sonra instead of kararı üzerine?

You could, but it changes the meaning slightly. Using -den sonra simply means 'after' in a chronological sense. Using üzerine highlights a strong cause-and-effect relationship, emphasizing that the board suspended him because of the decision, not just after it.

How do we break down the long noun chain Şirket müdürünün gözaltı kararı?

This is a nested noun compound. First, we have şirket müdürü (company manager). Then, the genitive suffix -nün is added to show possession, meaning 'the company manager's'. This links to the second compound, gözaltı kararı (custody decision).

Why is there an -ı at the end of gözaltı kararı?

It is the third-person possessive suffix used to form a noun compound. When combining the nouns gözaltı (custody) and karar (decision) to make the specific concept 'custody decision', the second noun must take this possessive suffix.