That's right. Scope name does not have any common thing with VLANs. DHCP helper on switches redirects DHCP queris to the proper IP address.
The most important thing is to set proper pool address, network mask and default gateway in each scope. If you have common DNS servers for your domain you can set options 006 and 015 in server options and 003 in each scope's scope options.
Then you don't have to configure it in each scope, it will be inherited.