SMOOTHING RING MAPS Contents
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چکیده
Our exposition follows Swan’s, but we first prove an intermediate result which let’s us work in a slightly simpler situation. Here is an overview. We first solve the following “lifting problem”: A flat infinitesimal deformation of a filtered colimit of smooth algebras is a filtered colimit of smooth algebras. This result essentially says that it suffices to prove the main theorem for maps between reduced Noetherian rings. Next we prove two very clever lemmas called the “lifting lemma” and the “desingularization lemma”. We show that these lemmas combined reduce the main theorem to proving a Noetherian, geometrically regular k-algebra Λ is a filtered limit of smooth k-algebras. Next, we discuss the necessary local tricks that go into the Popescu-Ogoma-Swan-André proof. Finally, in the last three sections we give the proof.
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Contents 1. Introduction 1 2. Advice for the reader 2 3. The cotangent complex of a ring map 2 4. Simplicial resolutions and derived lower shriek 3 5. Constructing a resolution 6 6. Functoriality 11 7. The fundamental triangle 13 8. Localization andétale ring maps 17 9. Smooth ring maps 19 10. Comparison with the naive cotangent complex 19 11. A spectral sequence of Quillen 21 12. Comparison wi...
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تاریخ انتشار 2012