Creighton University law perofessor Patrick Borchers says the Senate clearly has authority to try the case, especially given that the President was impeached by the House while in office.
"The ability of the Senate to bar an impeached officer the right to hold future office would make little sense if the officer could resign and avoid that consequence. The best historical evidence is Grant’s Secretary of War William Belknap who resigned hoping to avoid impeachment on bribery grounds. Even though he had resigned, the House impeached him and he was tried in the Senate, though acquitted narrowly. Most state constitutions at the time had provisions for impeachment of persons no longer holding the office."
Patrick Borchers served from 1999-2007 as the eighth Dean of Creighton School of Law.