Ring homomorphism

Let us remark that all such systems M. (resp. M.) over a ring R (resp. a scheme S) form in a natural way a category (morphisms are isomorphisms). For any ring homomorphism R — R (resp. morphism of schemes f Sr S) we get a pullback functor Jlf. M. R (resp. 

A ring homomorphism f. k— R is called formally smooth, and R is called a formally smooth k-alaebra. if for every k-algebra C and ideal Ic C such that I2 = (0), the map... 

A ring D is called isomorphic to D if there exists a bijective ring homomorphism from D to D. We shall write D = D in order to indicate that D and D are isomorphic rings. 

Lemma 9.3.3 Let T be a normal closed subset of S. Then there exists a ring homomorphism

vs) = asTs sT-... 

This is primarily a technical chapter introducing another algebraic tool. We will use it at once to complete the proof of the smoothness theorem (11.6) and then draw on it throughout the rest of the book. To begin, we call a ring homomorphism A - B flat if, whenever M - JV is an injection of /1-modules, then MaB- N<8>aB is also an injection. For example, any localization A->S 1A is flat. Indeed, an element m a/s in M S = S lM is zero... 

PROOF A ring homomorphism f E ------- F contains the maximal ideal... 

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