.NET Core Identity cookie cannot be verified by another .NET API using database-stored DataProtectionKeys

I have a .NET Core Identity API that is not the direct internet facing API.

The Yarp API is the one accessible from the Internet and I want to use the Identity cookie to know if user is logged in and if not redirect him to login page.

I stored in DB the Identity API DataProtection keys to be accessible by all Identity instances:

services.AddDataProtection().PersistKeysToDbContext<ApplicationDbContext>();

All works fine I can see the key in the DataProtectionKey table in the Identity DB.

Now I want in the Yarp API to have a middleware that check the Identity cookie .AspNetCore.Identity.Application if it is present and if it is ok to know if otherwise I should challenge the request to redirect towards login in Identity (login path is allowed by the middleware). In the Yarp api I configured the same line towards same db and the authentication with cookie:

services.AddDataProtection().PersistKeysToDbContext<ApplicationDbContext>();
services.AddAuthentication(options =>
{
    options.DefaultScheme = CookieAuthenticationDefaults.AuthenticationScheme;
    options.DefaultChallengeScheme = CookieAuthenticationDefaults.AuthenticationScheme;
}).AddCookie();

In the middleware I attempted both the:

var authResult = await context.AuthenticateAsync(CookieAuthenticationDefaults.AuthenticationScheme);

and the:

TicketDataFormat ticketDataFormat = new TicketDataFormat(dataProtector);
AuthenticationTicket ticket = ticketDataFormat.Unprotect(cookieContent); 
//I can see in the cookieContent the encrypted string

but none of them are successful.

Any idea what else I should do ?

Indeed RuikaiFeng suggestion was correct:

services.AddAuthentication(options =>    
{
 options.DefaultAuthenticateScheme = IdentityConstants.ApplicationScheme;        
 options.DefaultChallengeScheme = IdentityConstants.ApplicationScheme;    
}).AddCookie(IdentityConstants.ApplicationScheme);

but there was also something else: the SetApplicationName that need to be the same value in both apis:

services.AddDataProtection().PersistKeysToDbContext<ApplicationDbContext>()
.SetApplicationName("SameApiName");

And now the httpContext will show the needed information directly:

context?.User?.Identity?.IsAuthenticated