Let A —> B be a finitely presented ring map. Then we can write B = A[x_1, …, x_n]\/I and we get a two term complex<\/p>\n
NL_{B\/A} : I\/I^2 —> ⨁ B dx_i<\/p>\n
given by differentiation. In the stacks project we call this the naive cotangent complex<\/em> of B over A, see Definition 07BN<\/a> and Lemma 00S1<\/a>. We say a ring map A —> B is smooth<\/em> if it is finitely presented and NL_{B\/A} is quasi-isomorphic to a projective module placed in degree 0, see Definition 00T2<\/a>.<\/p>\n A ring map A —> B is said to be formally smooth<\/em> if given a surjection C —> C’ of A-algebras with square zero kernel, any A-algebra map from B to C’ can be lifted to an A-algebra map from B to C, see Definition 00TI<\/a>. A first result on smooth ring maps is that given a finitely presented ring map A —> B we have<\/p>\n A —> B is formally smooth if and only if A —> B is smooth.<\/p><\/blockquote>\n See Proposition 00TN<\/a>. This equivalence means in particular that our definition of smooth ring maps agrees with everybody else’s definition.<\/p>\n A standard smooth<\/em> ring map is one of the form A —> B = A[x_1, …, x_n]\/(f_1, …, f_c) with the matrix of partial derivatives df_j \/ dx_i for i,j = 1, …, c invertible in B, seeDefinition 00T6<\/a>. A second result is that<\/p>\n a smooth ring map A —> B is Zariski locally on B standard smooth,<\/p><\/blockquote>\n see Lemma 00TA<\/a>.<\/p>\n A relative global complete intersection<\/em> is a ring map of the form A —> B = A[x_1, …, x_n]\/(f_1, …, f_c) such that the fibre rings have dimension n – c, see Definition 00SP<\/a>. A third result, see Lemma 00T7<\/a>, is that<\/p>\n a standard smooth ring map is a relative global complete intersection<\/p><\/blockquote>\n The proof of this requires a bit of dimension theory; it is essentially the “Jacobian criterion”.<\/p>\n Lemma 00SV<\/a> states that<\/p>\n a relative global complete intersection is flat.<\/p><\/blockquote>\n You prove this by reducing to the Noetherian case and using the local criterion of flatness. <\/p>\n So far, besides some basic commutative algebra there are only ingredients needed to proceed along the lines above are some dimension theory and the local criterion of flatness.<\/p>\n A final result is Lemma 00TF<\/a> which states that<\/p>\n a flat finitely presented ring map with smooth fibre rings is smooth.<\/p><\/blockquote>\n The current proof of this in the stacks project uses a large amount of technical material including limit techniques to reduce to the Noetherian case and Zariski’s main theorem to bound dimensions in nearby fibres.<\/p>\n","protected":false},"excerpt":{"rendered":" Let A —> B be a finitely presented ring map. Then we can write B = A[x_1, …, x_n]\/I and we get a two term complex NL_{B\/A} : I\/I^2 —> ⨁ B dx_i given by differentiation. In the stacks project … Continue reading