Can I leave my computer on downloading over Shabbos?
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It is generally permitted to begin a task on erev Shabbos which will not be completed until after Shabbos has begun when it will be completed by itself. (Shulchan Aruch Orach Chaim 252:1
One would be required to have the sound off and I believe contemporary poskim would require the monitor to be off as well.
One should note, however that there are restrictions about what one can, and what one should, be doing on erev Shabbos that I haven't addressed.
Your question relates to the concept of Shvitat Kelim, resting equipment. Must our equipment rest on Shabbat or Yom Tov? Chazal decided the answer is NO.