What is this part of the sentence?

"You can’t be serious about this schedule of battles."

First note that this is a simple sentence with the independent clause You can’t be serious. Tacked on to this clause is the prepositional phrase about this schedule of battles, which acts adverbially to modify the adjective "serious."

By definition, a sentence fragment is a group of words that begins with a capital letter and ends with a period, question mark, or exclamation point but is grammatically incomplete. Thus, as you can see, the highlighted part of the sentence is clearly not a sentence fragment. Nor is it a dependent clause, because it can stand on its own meaningfully— which a dependent clause is incapable of, by definition. It therefore is a clause with the subject as "You", the verb "can't be", and the adjective "serious" acting as the subject complement. I suppose that's all there's to it.