My Mom's doctor has said that you cannot put a person in a nursing home against her will. Is this true?

It is true, unless you are the court-appointed guardian. Getting guardianship usually involves proof that the person is not mentally competent to handle her own affairs. What does your mother's doctor say about that?

While you cannot control where she lives, you do have control over your own actions. You can refuse to live with her. You can refuse to provide help. You can involve agencies that are charged with the welfare and safety of vulnerable adults.

In my state, what the doctor said is true. With some qualifications: if the person is determined to be incompetent and a guardian is appointed, the guardian can consent to the person's admission to a nursing home, even if the ward (the person subject to the guardianship) does not consent.