# Pros and cons of Stacks Project as a reference compared with EGA/SGA

I would like to know pros and cons of Stacks Project compared with EGA and SGA and whether it serves as a nice alternative to them. Since I haven't read both of these texts, my attempt to compare the series in the following is based solely on the opinions previously posted by the users on MO.

The following are the pros and some personal opinions about Stacks Project:

• It was written by many notable algebraic geometers of 21st century, and its content is up to date, with more pages on stack.
• At this moment, the text contains more than 4500 pages, and its completeness ties with EGA and some (probably not all) volumes of SGA combined. (supported by comments of Prof. Vakil and Prof. Emerton in MO post The importance of EGA and SGA for “students of today”)
• It is written in English, so it is a bit easier to read for non-native French speaker and therefore less time-consuming. The fact that it is less time-consuming is important, considering the fact that both series are quite long. (I admit that one should be able to read math papers written in French. Yet, when there's a nice English alternative, it's more efficient to read English one if you're more fluent in English)
• According to Stack Project Blog, the generality of Stacks Project is the same as EGA/SGA, (so it's more general than Hartshorne's text).

But there should also be cons, especially content-wise. Some volumes of SGA is probably not covered well by Stacks Project because SGA has about 6000 pages. I understand that some topics covered by EGA/SGA are not covered by Stacks Project because they are no longer considered important. But I suppose SGA 3 is still the inescapable reference on group scheme, and SGA 4 and 4 1/2 have the same role on Etale Cohomology, though I'm not sure about Stacks Project's coverage of these topics. Also, it seems Stacks Project doesn't treat Etale fundamental groups (SGA 1) and Monodromy (SGA 7) according to its table of content. If I'm wrong, please tell me so. In order to know the cons of Stacks Project compared with EGA/SGA, I would like you to comment on the following points:

• whether Stacks Project really covers EGA well

• which volumes of SGA it covers well and which ones it does not

• whether there is any comprehensive, rigorous alternative to the volumes of EGA/SGA not covered well by Stacks Project (e.g. if Stacks Project doesn't cover Etale Cohomology as much as SGA does, does Etale Cohomology Theory by Lei Fu serve as a nice alternative?)

• The Stacks Project doesn't come close to subsuming any SGA (though some important technical topics in SGA1 and SGA4 do make an appearance in SP), so only the comparison with EGA is appropriate to consider. (The books of Frietag-Kiehl and Milne on etale cohomology, along with SGA 4.5, are collectively good to learn the core results and examples in that subject, but SGA4 remains an important reference for some refined topics.) There are a lot of important techniques and valuable results still only to be found in EGA. – grghxy May 23 '15 at 12:01
• Moreover, the table of contents of EGA scarcely begins to give a clear picture of the wealth of results and techniques given within. And the recent book FGA Explained is a valuable reference on some important highlights from the FGA Bourbaki talks by Grothendieck, covering some important topics (descent, some moduli schemes) which are not addressed in EGA (along with formal schemes as in EGA III$_1$). Best to be guided by learning specific topics in bite-sized portions. – grghxy May 23 '15 at 12:15
• Thanks for your remark that it is more appropriate to consider only the comparison with EGA, and I was definitely going to read FGA Explained, as many people here recommended. After covering a significant amount of SP and FGA Explained, I will cover the remaining important materials on EGA. Then I will read SGA for the topic I will be interested in. – Math.StackExchange May 23 '15 at 12:32

The first question you have to ask yourself is why do you think you have to read ALL of either set of sources.

In my limited experience in Algebraic Geometry, it pays to get the basic definitions under your belt, then to look at a theme, following that through several sources. When you are in some distance, pause that theme and take up something that has caught your eye along the way.

Why not look through Grothendieck's Esquisse d'un Programme (available on the net with commentary / translation in English), then follow up some themes from there. When you get, for instance, to fundamental groups (for the anabelian stuff in Esquisse) check back with BOTH SGA1, and Stacks plus any other surveys, books, etc. until you feel happy with that, then move on. Along the way, no doubt you will have met ideas that you do not yet know, so note them down and return.

Every so often check back on other ideas then use both EGA/SGA and Stacks project and n-Lab and .... Get to klnow where to look for the stuff, rather than thinking one source will fit all.

• Thanks for your suggestion of the paper. It will help me a lot to choose which volume of each series to read. – Math.StackExchange May 23 '15 at 12:42
• Although a very valuable source of reading, it is important to remember that the SGA are records of seminars, so there are some slips. The influence of more recent inputs can simplify some proofs or treatments. Grothendieck himself in Pursuing Stacks (and the other typescripts of the 1980s) wanted them made available AS HE HAD TYPED THEM so that beginners should see that even the well known names can have lots of false starts, made errors of judgement etc. (You do not learn to cook by eating cakes!) – Tim Porter May 23 '15 at 13:42
• There are two survey articles by Dieudonné in Advances in Mathematics vol 3 issue 3 which summarise both historical aspects and the EGA material. They may be useful as well. They sometimes say 'why' rather just 'what'! – Tim Porter May 23 '15 at 15:40