No. f(x,y) = (x,xy).

abx and Jason Starr, thank you. If I understand correctly, the entire argument can be repeated for schemes over a field of char not dividing n by taking the exact sequence $ 0 \rightarrow Z/nZ \rightarrow \mathbb{G}_m \rightarrow \mathbb{G}_m \rightarrow 0 $ on a fiber in the etale topology, where the second map is multiplying by n. Is that correct? What happens if char k divides n, out of curiosity?

This is also proposition 9.7, Hartshorne chapter 2, where he proves (rather tersely) that a punctured one parameter flat family fills up uniquely as the (scheme theoretic) closure.

Ok, if someone posts an answer with a short summary of keywords from which it follows, I'll accept it.

@Sasha could you elaborate please? I thought Borel-Weil-Bott gives the cohomology of certain line bundles. How does (1) follow?

@Mohan OK, I'm no specialist in commutative algebra, I only have a working knowledge. I'll keep syzygy in mind.

@JasonStarr if there exists a surjection from a f.g. free module to $ M $ with kernel $ K $, I call $ K $ as $ tM $. Not once did I say that there is a unique surjection or that $ tM $ is well defined. Still the notation $ t(tM) $ makes sense, I hope I am clear now and the example should really convey what I've said before.