Usually “sans” is placed before a noun without the definite article (sans doute or sans alcool). But occasionally I see sans with the definite article (sans la foi). Is there a rule here?
There are MANY examples of prepositional constructions lacking articles in French: à cheval, sous clef, contre nature (which are usually used as some sort of adverbials). “sans doute” falls into that category, note that qualifying them (as Grevisse notes) usually causes the article to reappear: sans le moindre doute, sans un seul doute (aucun being a determiner itself and thus preventing articles, sans aucun doute is an obvious exception).
Grevisse, oddly enough, seems not to take notice that the partitive form of the indefinite article (de, du, des) seems to be disallowed after sans (this covers the “sans alcool” case). I think that means de is not allowed to follow other prepositions (at §1039 1o c, he discusses sequences of prepositions: when de is present, it always comes first).