What math background for algebraic geometry?

[feuxfollets]

So I'm currently a sophomore math/CS major. I'm interesting in taking the algebraic geometry sequence next year; however I don't have the formal prereqs for the class, first year graduate algebra, which I would be taking simultaneously with the algebraic geometry class.

I'm wondering if this is doable? My current algebra background is all the standard undergraduate algebra, plus some module theory, some elementary homological algebra that I learned for the algebraic topology class I'm currently taking (like stuff in Dummit and Foote, a little bit of Chapter 6 from Rotman), plus some basic category theory. I also know a little about localization and spectrums, although not that much beyond basic definitions.



The course description for the algebraic geometry class is:


Algebraic geometry over algebraically closed fields, using ideas from commutative algebra. Topics include: Affine and projective algebraic varieties, morphisms and rational maps, singularities and blowing up, rings of functions, algebraic curves, Riemann Roch theorem, elliptic curves, Jacobian varieties, sheaves, schemes, divisors, line bundles, cohomology of varieties, classification of surfaces.


The course description for the first year grad algebra which is formally a prereq for that is:


Group theory: permutation groups, symmetry groups, linear algebraic groups, Jordan-Holder and Sylow theorems, finite abelian groups, solvable and nilpotent groups, p-groups, group extensions. Ring theory: Prime and maximal deals, localization, Hilbert basis theorem, integral extensions, Dedekind domains, primary decomposition, rings associated to affine varieties, semisimple rings, Wedderburn's theorem. Linear algebra: Diagonalization and canonical form of matrices, elementary representation theory, bilinear forms, quotient spaces, dual spaces, tensor products, exact sequences, exterior and symmetric algebras. Module theory: Tensor products, flat and projective modules, introduction to homological algebra, Nakayama's Lemma. Field theory: separable and normal extensions, cyclic extensions, fundamental theorem of Galois theory, solvability of equations.

 

[Number Nine]

Do you have any experience with commutative algebra?

 

[mathwonk]

In Dummit and Foote read the chapters on vector spaces and modules and commutative algebra and algebraic geometry.

I.e. learn about tensor products, localization, and integral extensions.

 

[feuxfollets]

I know about tensor products and localization already. if I finish the sections on commutative algebra/algebraic geometry, that would be sufficient background?

 

[micromass]

I know about tensor products and localization already. if I finish the sections on commutative algebra/algebraic geometry, that would be sufficient background?

No. I would absolutely not recommend you taking algebraic geometry before you took commutative algebra. You'll need most of that grad course algebra in your algebraic geometry class.

If you go in algebraic geometry underprepared then you will be absolutely destroyed.

On a related note, what book are they using in the algebra and the algebraic geometry course??

 

[feuxfollets]

Last year they used Lang's algebra book primarily for the algebra class, along with Artin/Dummit and Foote for supplements I think. Also one of the professors here, Shatz, wrote an algebra book that they sometimes use depending on who's teaching.

Hmmm the only website I can find for the algebraic geometry class was 2008-2009, and they used Mumford Algebraic Geometry II for the second semester. Not sure about the first.

 

[micromass]

Last year they used Lang's algebra book primarily for the algebra class, along with Artin/Dummit and Foote for supplements I think. Also one of the professors here, Shatz, wrote an algebra book that they sometimes use depending on who's teaching.

Hmmm the only website I can find for the algebraic geometry class was 2008-2009, and they used Mumford Algebraic Geometry II for the second semester. Not sure about the first.

An idea would be to mail the instructor and ask what text you'll be using. You might skim the text to see if you understand a lot or if you understand nothing.

But be warned: algebraic geometry is known to be very difficult.

 

[mathwonk]

as it happens i am a professional algebraic geometer. Of course that doesn't mean my advice is necessarily correct, but it has a better than average chance. Another plus is, if I could do it, how hard can it be?

You might take a look at Mumford's "red book" of algebraic geometry. The first chapter is called "some algebra". Basically noether's normalization lemma and consequences.

Everyone agrees that Atiyah - Macdonald's book on commutative algebra is very accessible.

The easiest alternative commutative algebra book may be the undergraduate one by Miles Reid.

I always liked the original by Zariski and Samuel for clarity.

When you say Mumford's alg geom II, do you mean the unpublished second volume, edited by Oda?

The most user friendly algebraic geometry books are probably the undergraduate one by Miles Reid, and Basic algebraic geometry, by Shafarevich. Especially the first edition of Shafarevich was rather self contained in terms of commutative algebra.

Joe Harris' algebraic geometry book has a lot of wonderful examples. And Fulton's book on algebraic curves is free nowadays I believe on his website.

 

[mathwonk]

Here are some notes from day 2 of my introductory course, last time i taught it. See if you can read these notes.

 

[mathwonk]

If you want good advice, go talk to Steve Shatz or Antonella Grassi or Ron Donagi or Ching Li Chai or whoever is teaching the course. In fact go see them anyway, it is worth meeting them.

 

[morphism]

I agree with mathwonk on both counts: 1) Go talk to the prof and see what they have to say; 2) If you read the above-mentioned sections in Dummit and Foote, you should be fine for the most part. You can always pick up additional commutative algebra as you need it. This is how most people learn algebraic geometry anyway.

 

[mathwonk]

There is perhaps not as much disagreement in the various advisements here as may seem. To summarize:
1. yes modern algebraic geometry uses commutative algebra.
2. it may be however that algebraic geometry is more useful for understanding commutative algebra than the other way around.
3. The commutative algebra prerequisites for a particular algebraic geometry course can only be learned from the professor, since one can teach in such a way as to include the needed commutative algebra within the course (as shafarevich did in his first edition), or at least to state without proof the needed algebraic results (as hartshorne does in his very algebraic treatment).
4. nonetheless it is prudent to acquire some few essential results related to integrality, such as noether's normalization lemma, cohen seidenberg's going up and down theorems, and hilbert's nullstellensatz.

Indeed those three results are provided in the first few pages of mumford's famous "red book" of algebraic geometry. Hence as a minimum I recommend reading those pages closely.

It is also recommended to have available supplementary sources for both algebra and geometry, such as miles reid's undergraduate commutative algebra (in particular his proof of noether's lemma may be more complete than mumford's), and shafarevich's BAG, preferably the first edition. Perhaps also reid's undergraduate algebraic geometry, and mumford's "red book" (originally harvard red, now springer yellow).But the question of whether one can understand algebraic geometry without knowing a lot of algebra is perhaps irrelevant to your personal question of whether you can survive the course at penn without it. for that you must speak to the professor. but even if the professor says "oh of course you can!" i still recommend hedging your bets by acquiring the supplementary sources mentioned above. Good luck!

 

[feuxfollets]

Okay, I will probably go talk to Antonella Grassi about it, as she's teaching it next semester. From what I see here it seems like it should be doable though. Thanks for the advice.

 

[mathwonk]

good plan. please let us know what she says. it will help us advise others.