Kate Spade was depressed over her business problems and separation from her husband, which was causing her to drink a lot ... this according to cops.

Law enforcement sources tell us, Kate was alone in her apartment when she tied a scarf around her neck, attached it to her closet door knob and hanged herself in her bedroom.

We're told her husband, Andy Spade, was not in the apartment. As we reported, the couple had separated and he was living in a nearby building. Sources say Andy told cops he was separated from Kate for a couple of months. Andy added he was at his apartment when he learned Kate had committed suicide.

Andy told cops Kate "was drinking a lot," and was depressed by business problems and their separation. As we reported, he wanted a divorce but she did not.

It puts her suicide note in a somewhat different light. In the note, she tells her 13-year-old daughter it was not her fault, adding, "Ask Daddy!" One interpretation is that she blames Andy.