Kate Spade was extremely depressed in her last days of life, because her husband wanted a divorce ... law enforcement sources tell TMZ.

We've learned Andy Spade was not living at the family home ... he and Kate had separated and he was living in a nearby apartment.

Our law enforcement sources say after their interview Tuesday they were clear ... Andy wanted a divorce but Kate did not.

It makes her suicide note particularly interesting and possibly revealing. She wrote to her 13-year-old daughter, Frances Beatrix, "Bea - I have always loved you. This is not your fault. Ask Daddy!" It's unclear what Kate meant by the words "Ask Daddy!" One interpretation is that he knows why she ended her life.

The couple married in 1994.

As we reported, Kate hanged herself in her bedroom.