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By Avramov L.L. (ed.), Tchakerian K.B. (ed.)

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While the obvious surjection R2 −→ a has a splitting √ √ −1 + −5 2 − −5 √ x −→ x · , , 3 2 + −5 so that a is a direct summand of R2 . This is directly related to the fact that R is not a UFD. 6. Let R = R[x, y, z]/(x2 +y 2 +z 2 −1), the coordinate ring of the real 2-sphere S2 . The homomorphism R3 −→ R defined by the row vector ν = [x, y, z] is surjective, so the kernel P satisfies P ⊕ R ∼ = R3 . However, P 3 is not free. Every element of R gives a vector field on S2 , with ν defining the vector field pointing straight out from the origin.

To see that this sequence is exact, note that U is the disjoint union of open arcs that are diffeomorphic to the real line. On such an open arc, (i) the constants are the only functions that are annihilated by differentiation, and (ii) every smooth function can be integrated to a smooth function. On the other hand, as we will discuss later, the sequence is not exact on the right if U is S1 . For open sets V ⊆ U , sequences of the form above, along with appropriate restriction maps, fit together to give a commutative diagram d dt R(U ) −−−−→ D(U ) −−− −→ D(U )       d dt R(V ) −−−−→ D(V ) −−− −→ D(V ) .

Closely related is the Bass-Quillen conjecture, which asserts that for a regular ring R, every projective R[T ]-module is extended from R. Quillen’s and Suslin’s solutions of Serre’s conjecture proceed by proving this statement when R is a regular ring of dimension at most 1. Popescu’s celebrated theorem of general N´eron desingularization [128, 153], together with results of Lindel [100], proves the Bass-Quillen conjecture for regular local rings containing a field, unramified regular local rings of mixed characteristic, and also for excellent Henselian local rings.

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Algebra Some Current Trends by Avramov L.L. (ed.), Tchakerian K.B. (ed.)

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