Short exchange is a type of jazz (4)
I believe the answer is:
trad
'a type of jazz' is the definition.
(I've seen this before)
'short exchange' is the wordplay.
'short' means to remove the last letter.
'exchange' becomes 'trade' (trade is a kind of exchange).
'trade' with its final letter removed is 'TRAD'.
'is' acts as a link.
(Other definitions for trad that I've seen before include "Style of jazz" , "Type of jazz or folk" , "music" , "Form of folk or jazz" , "(Of jazz) traditional" .)