If my ex-wife has been living with another man, can I now stop paying alimony? 

"While there is relatively little authority on the relevance of
premarital cohabitation to spousal support, there is a great deal of
statutory and case law relating to the significance of post marital
cohabitation with a third party. Many states now have legislation which
provides for the modification or termination of alimony under certain
circumstances where an alimony recipient is cohabiting with a third
party. The common law of many other states also permits such
modification or termination of alimony where cohabitation reduces the
recipient spouse's need for alimony. In other cases, alimony awards can
be modified or terminated upon cohabitation based upon the language of
a settlement agreement or divorce decree.

As of November 1994, 12 states had enacted statutes which allow courts
to reduce, suspend, or terminate alimony in some situations where the
alimony recipient is living with a third party. Those states are
Alabama, California, Connecticut, Delaware, Georgia, Illinois,
Louisiana, New York, Oklahoma, Pennsylvania, Tennessee, and Utah. The
relevant language from these statutes is set forth in the box on the
following page.

These statutes generally provide that postmarital cohabitation is a
sufficient basis for termination of a spousal support award. The exact
type of cohabitation which is necessary varies from state to state. The
Alabama, California, Oklahoma, Pennsylvania, and Utah statutes still
speak of cohabiting with a member of the opposite sex. The New York law
is even narrower, only addressing the situation of a recipient wife's
"habitually living with another man and holding herself out as his
wife." N.Y. Dom. Rel. Law 248. A similar gender-based provision in the
Official Code of Georgia was held to violate the equal protection
clause of the fourteenth amendment to the United States Constitution.
Sims v. Sims, 243 Ga. 275, 253 S.E.2d 762 (1979).

In addition, some of the statutes permit the parties to a divorce
decree to agree that cohabitation will not have the effect stated in
the statute. Such provisions can be found in the California, Delaware,
and Illinois statutes.

There has been substantial litigation under the more broadly phrased
statutes, typically concerning whether there has been sufficient proof
of the triggering event, i.e., "cohabitation." The Alabama judiciary,
in determining whether a former spouse is "living openly or cohabiting
with a member of the opposite sex," Ala. Code 30-2-55, has required
both a sexual element and some degree of permanence in the
relationship. See, e.g., Ayers v. Ayers, 643 So. 2d 1375, 1377 (Ala.
Civ. App.), cert. denied, 643 So. 2d 1377 (Ala. 1994) (stating that
"`[c]ohabitation has been defined as some permanency of relationship
coupled with more than occasional sexual activity between the
cohabitants'"; trial and appellate courts found only evidence of a
"three-year romantic friendship" between the former wife and her
alleged cohabitant and denied the former husband's request to terminate
alimony (quoting Tucker v. Tucker, 416 So. 2d 1053, 1055 (Ala. Civ.
App. 1982)); Perkins v. Perkins, 643 So. 2d 992, 993 (Ala. Civ. App.
1994) (evidence supported finding of cohabitation where "wife's
daughters testified that their mother was living with a man, that his
clothes and toiletries were located at her house, that they were
`always together,' that they spent practically every night together,
and that he ate his meals at her house and drove her car" and where
wife admitted to sexual relationship with the man and to discussions of
marriage). Louisiana's Court of Appeal has also required both a sexual
element and a permanency element. See Polk v. Polk, 626 So. 2d 1233
(La. Ct. App. 1993), cert. denied, 634 So. 2d 381 (La. 1994) (mere fact
that wife had a child by her paramour held insufficient to terminate
support where there was no common residence). Factors which indicate a
permanent relationship include occupying the same dwelling, ceasing to
date other members of the opposite sex, and sharing household expenses.
Perkins v. Perkins, supra. The Alabama courts have not defined exactly
how strong the sexual element of the relationship must be, but mere
occasional sexual intercourse is not sufficient. See Roe v. Roe, 611
So. 2d 380 (Ala. Civ. App. 1992). However, some sexual activity in the
relationship must be established. See id. (wife lived with 75-year-old
man and provided transportation and housekeeping services for him; not
sufficient to constitute cohabitation). "

You should seek legal advice as to your situation and how to go about
stopping the alimony in the most legal way - by court order.  You do
not want to get yourself in to trouble later on should she stop living
with the other man and then you are sued for arrears.