Your Ex may turn into a valued contact in the future, or might be your future boss, you never know. If they lied then it will really depend on whether they can do the job or not, and a resume only gets you in the door, interviews are what gets you jobs anyway.
That said, I once received a resume from someone who listed as one of their past employers a company I'd also worked for. He listed his work there as being on a project that I'd worked on, and partially led. One with less than 6 people on it. One that happened without him being involved at all!
I was amazed at the nerve. Would have been easy to call the company if we'd been reference checking and found out, but worse, I'd worked there so I knew right away it was a lie.
We uninvited him from his interview.