Timeline for Simple motivation for mixed characteristic algebraic geometry?
Current License: CC BY-SA 4.0
6 events
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| Jun 25, 2023 at 16:30 | comment | added | Karl Schwede | Ok, but maybe you are asking where these Big Cohen-Macaulay algebras come from. How the ideas of Scholze, Bhatt-Scholze, etc are in the work of André, Gabbert, Bhatt-Scholze, etc on this stuff. The key thing is the perfectoid almost purity theorem and variants (also what Bhatt-Scholze call André’s flatness lemma is useful frequently). Now of course, Bhatt showed that the p-adic completion of $R^+$ is Cohen-Macaulay, via work of Bhatt-Lurie, via a different method. | |
| Jun 25, 2023 at 16:18 | comment | added | Karl Schwede | I think at this point the reduction to the unramified case isn't needed (for instance, if you have big Cohen-Macaulay algebras). For direct summand, reducing to the complete case is easy, if $R \subseteq S$ is finite, it remains finite after completion and completion is faithfully flat. From a big picture perspective, showing the existence of big Cohen-Macaulay algebras is (if graded) dual to saying you can dominate your scheme where something like Kodaira vanishing holds. Thus for things where the missing piece in mixed characteristic is vanishing theorems, this is useful. | |
| Jun 25, 2023 at 15:42 | history | edited | LSpice | CC BY-SA 4.0 |
Title of reference
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| Jun 25, 2023 at 14:09 | comment | added | Patrick Elliott | Yes, I would like to understand how/when/why these reductions work. | |
| Jun 25, 2023 at 14:08 | comment | added | Will Sawin | According to the abstract of that paper "M. Hochster conjectured that any finite extension of a regular commutative ring splits as a module. Building on his reduction to the case of an unramified complete regular local ring R of mixed characteristic" the conjecture was already reduced by the work of Hochster to the case of complete local mixed characteristic (i.e. p-adic) rings. So the fact that p-adic methods apply doesn't really have to do with the work of Bhatt and Scholze. Is the original reduction what you would like to understand? | |
| Jun 25, 2023 at 13:51 | history | asked | Patrick Elliott | CC BY-SA 4.0 |