To remove the authentication cookie, simply use the ASP.NET Core SignOutAsync extension method on the HttpContext. You will need to pass the scheme used (which is provided by IdentityServerConstants.DefaultCookieAuthenticationScheme unless you have changed it):
Or you can use the overload that will simply sign-out of the default authentication scheme:
Typically you should prompt the user to logout which requires a POST to remove the cookie. Otherwise an attacker could hotlink to your logout page causing the user to be automatically logged out. This means you will need a page to prompt the user to logout.
If a logoutId is passed to the login page and the returned LogoutRequest’s ShowSignoutPrompt is false then it is safe to skip the prompt. This would occur when the logout page is requested due to a validated client initiated logout via the end session endpoint. Your logout page process can continue as if they user submitted the post back to logout, in essence calling SignOutAsync.
If your user has signed in with an external login, then it’s likely that they should perform an external logout of the external provider as well.