![]() This is a functor, since if R –> S is a ring map, you get a map h X(R) –> h X(S) (e.g., you can reduce integer solutions mod a prime). (1) If you have a variety X (i.e., some equations to solve), you can cook up the functor h X, which take as input a ring R and produces an output h X(R) := the set of solutions to the equations defining X. (Added June 22, 2020.) On discord someone asked "What is a stack?". Then think about how much of the foundations you want to learn (2-categories, Grothendieck topologies, etc) and talk to your advisor about which are useful for you to learn. At the very least, look at the slides of Poonen, the MSRI lecture by Ravi, and the first chapter of Harris and Morrison. It was easier to answer the question of "how did I learn stacks" than "how should you learn stacks". That's quite a lot I obviously don't recommend doing all of that. This is really meant as more of a technical manual (I use it all the time), but there are numerous pieces of exposition that help (look through the table of contents to find them). ![]() It isn't identical to Anton's notes his include many examples and discussion that happened in class but didn't make it into the book.įinally, now, there is the motherload of all stacks resources Martin Olsson turned his notes from his course into a book These days, we have 2 additional great technical resources. (While there is a well known error about the lisse-etale site in chapter 13, this doesn't change the instructional value of the book, or the first 12 chapters.) ![]() I didn't read this cover to cover all at once, but did basically end up reading all of it. "Champs algebriques" Laumon, and Moret-Bailly is a great, EGA style treatment of stacks. At this point I've lost track of which expository articles I liked. There are some fantastic expository references, like "Stacks for Everybody", by Barbara Fantechi. Eventually I started writing papers about stacks too, which obviously helped.Īt Emory, my second year, I taught a topics course on stacks. There were several other things I read, like Vistoli's thesis but the content there has been distilled into other references by now.Įventually I started working on, then writing, my thesis, about generalizing rigid cohomology to stacks, and ended up learning a lot in the process. The first chapter has especially good exercies to get you started. This isn't about stacks per say, but I learned a lot by reading this. That part was overkill, but I enjoyed it.ĭuring a student seminar, we spent a lot of time reading Harris and Morrison's book "moduli of curves" I did a similar thing with the first book and a half of SGA IV the following semester (which is on Grothendieck topologies and topoi). One thing I did was to slowly read through Knutson's book on algebraic spaces.īasically, every morning for a semester, after breakfast and coffee, I would spend 45 minutes reading this book I would them make a few notes, and also usually have a few questions that I would ask people at tea. In preparation for that I started slowly learning even more of the background material. (There are videos too)Ī few semesters later, there was a large program at MSRI on algebraic geometryĪnd I learned quite a bit there. Here are the relevant links, but see especially Vakil's initial lecture. ![]() I was one of the TA's (though I was a TA for Bhargav Bhatt and Andrew Obus, so there was some imbalance.).Īnton also livetexed that. Then (also lucky) there was a stacks and deformation theory workshop at MSRI. That summer, we (about 5 of us) spent 1-2 hours a day re reading through the notes and fixing things we hadn't understood the first time. Then the following very lucky thing happened to me: Martin Olsson (stacks expert) moved to Berkeley, and his second semester (Spring of my 3rd year) he taught a well attended (40+ people) graduate course on stacks.Īnton Geraschenko lived texed the course, and we spent a lot of time improving the notes Which helped, but is really more focused on the foundations instead of actual stacks (which was still obviously important!). The summer after my first year, several of us worked through Vistoli's "Notes on Grothendieck topologies, fibered categories and descent theory" (Later I also learned about how stacks "explain" where the Frey curve in the proof of Fermat's Last Theorem come from, and lays out a blueprint of where to look for other Fermat equations in the simliar but non-stack language this is the content of Darmon's "Faltings plus epsilon" paper My history of learning stacks: the first substantial paper that I tried to read as a grad student wasĪnd Section 3 of that paper explains how stacks are used in the study of Fermat equations.Īround that time, Bjorn Poonen gave this talkĪnd I was hooked so I started asking lots of different people that I knew or that I met a conference to tell me about stacks. ![]()
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