I witnessed a cigar-shaped object being "chased" by a military jet aircraft at dusk.

At dusk, I saw a whitish cigar-shaped object west of Auburn, at about 20 degrees above the horizon, moving very rapidly south to north. About 30 seconds into the event, I then saw a military jet fighter approach from north to south (and it appeared much closer than the object),  25 degrees above the horizon, make a sharp turnaround  and head towards the object. An afterburner "puff" of smoke was seen that remained in the clear sky for sometime afterwards. The unknown object was fairly sharply defined and appeared to be reflecting the (setted) sunlight, and I saw no condensation trail.  I saw the object by naked eye and through 7x35 binoculars. I admit I was looking for UFOs at the time. Many had been reported in the area. I was inside looking through a window pane.