How do I find out if somethings in public domain I would like to use some of lovectaft's work like Arkam and his university but how do know if it's safe to use
Anything published prior to 1923 is most likely in the public domain in the U.S. (unauthorized publications may be an exception). Anything 1923 or later you can't be certain without doing some research.
The Copyright Law of the United States tries to encourage the creation of art and culture by rewarding authors and artists with a set of exclusive rights. Copyright law grants authors and artists the exclusive right to make and sell copies of their works, the right to create derivative works, and the right to perform or display their works publicly. These exclusive rights are subject to a time limit, and generally expire 70 years after the author's death. United States copyright law is governed by the federal Copyright Act of 1976. The United States Constitution explicitly grants Congress the power to create copyright law. Specifically, Congress has the power: H. P. Lovecraft Lovecraft in 1934 Born Howard Phillips Lovecraft (1890-08-20)August 20, 1890 Providence, Rhode Island, USA Died March 15, 1937(1937-03-15) (aged 46) Taking US copyright law as running for 70 years from author's death, and Lovecraft's death in 1937, his works are comfortably within the public domain.
@Shadowfax that analysis is wrong. For post-1923 publications you have to research a couple of issues to determine copyright status.
It's difficult. There is argument about whether any of Lovecraft's works are currently still under copyright. The apparent right-holders will argue that they are, but from what I've seen there is no proof the copyrights were ever renewed. Unfortunately, the only way to research renewals is to hire an agency to do it or to go to the Copyright Office or one of a few libraries that have complete records and do it yourself. It'll be time consuming. The other potential issue is that even if you decide they're all in the public domain, if the rights-holders disagree and sue you, you still have to defend their claim of copyright infringement. Lastly, there are various trademarks around Lovecraft's IP, for various goods. On a positive note, simply referencing Arkham or Miskatonic University may not be much of an issue. You may want to look for trademark filings for those names, depending on exactly how you're going to be using them, but the mere reference to them isn't likely to be a copyright problem.
I tried searching Project Gutenburg for public domain e-books by Lovecraft and found these. https://www.gutenberg.org/ebooks/search/?query=lovecraft