Can you go to the federal court system to appeal a state supreme court ruling?

Generally, state court decisions can only be appealed to the US Supreme Court (not lower federal courts) if all avenues of appeal have been exhausted within the state court system and if the case involves a preserved federal question. A federal question is a matter that involves federal law, the US Constitution, or a US treaty. In order for a question to be preserved, it must be raised at the trial and at every appellate level.