It is important that the intonation of the short question rises
if you want to show interest in what the other person is saying and sound polite.
'I'm going to the pub after work.'
'Are you? Can I come too?'
I bought a new computer at the weekend.
'Did you? That's nice.'
In the second example, a falling or flat intonation with the follow up That's nice
suggests the second speaker is uninterested or busy and wouldn't encourage the first speaker to say more.