In my work, I have to understand certain objects in commutative algebra (for example Gorenstein rings, Cohen–Macaulay rings e.t.c). I have a reasonable background in commutative algebra (I suppose!) and very basic knowledge of algebraic geometry. The problem is that most of the time I have tried to understand some properties about these objects from a paper, I found that it is written in the language of advanced algebraic geometry which is almost impossible for me to understand. Therefore my question is:

Are there notes, books, papers or survey articles which has parallel discussions about objects in commutative algebra and its counterpart in algebraic geometry?

Studying algebraic geometry is of course the answer but as both of these topics are not directly related to my work and I have already spent reasonable amount of time to understand one of them, this kind of literature would be extremely helpful for me.

PS1: Any suggestion is welcome. Thanks in advance.

PS2: I apologise if this question is not relevant in mathoverflow but I have asked people and searched but I could not find any advance literature like these.

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    $\begingroup$ Eisenbuds "Commutative Algebra with a view towards Algebraic Geometry" mentions the geometric meaning/view of the topics discussed. I'm not so sure it does so sufficiently to be able to read research articles in AG, but it might be a good start. Also, its very readable. $\endgroup$ – jorst Oct 28 '15 at 10:20
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    $\begingroup$ see "here" for geometric interpretation of Gorenstein rings, Complete intersections and regular rings. but for the general case I also wait for a good answer to this question. $\endgroup$ – user 1 Oct 28 '15 at 10:30
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    $\begingroup$ For a geometric interpretation of the CM property, see "here" $\endgroup$ – Sándor Kovács Oct 29 '15 at 18:55
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    $\begingroup$ For normality, you can see here and for S2 you can see here $\endgroup$ – Karl Schwede Nov 1 '15 at 16:04

I think this question is quite important. For a commutative algebraist, some knowledge of algebraic geometry is extremely useful (and vice versa). Not only that, and more important to me personally, it makes life more fun, like being able to talk in another language. You can converse meaningfully with more people, appreciate more talks and papers, and get more insights/motivations for your own research.

Unfortunately, with the explosion of knowledge and Balkanization of the areas, it is now virtually impossible for young people to learn deeply both subjects at the same time, unless if you are very motivated and is at a school with leading experts. So a lot of algebraists will go through their career without having to really learn what a "projective variety" or a "line bundle" is, and so are many geometers with "Gorenstein ring" or "integral closure of an ideal".

Below I will describe my own experience, hopefully it will help you. Unfortunately, I do not know any convenient text.

First, you don't have to become an expert. Even being able to translate simple phrases will serve you well. For example, in my thesis I had a problem about primes ideals over a local hypersurface. I translated it in to a question about subvarieties of a projective hypersurface, and realized that my question can be solved there with the "moving lemma". While that is not available in my local situation, I was able to find a weak version to make things work.

Second, you still need to invest a certain amount of time on learning some basic geometry, probably by selecting certain topics from Hartshorne's first 3 chapters and compare them with bits from Eisenbud. I would focus my attention to the following to start:

  • The connections between the geometry of a projective scheme $X$ and it's various embeddings. Even when $X$ is the projective space.

  • The connections between the graded coordinate ring with respect to an embedding and the local ring at the origin.

  • The connections between sheaf cohomology of a coherent sheaf and the local cohomology of the corresponding module.

Third, you can pick up more knowledge by occasionally looking at abstracts/papers, go to seminars/conferences, and try to match up what people are doing over there with what you know. For instance, you may learn that a smooth Calabi-Yau hypersurface is really a graded hypersurface of degree $n$ in $n$ variables with isolated singularity. So next time, when people talk about CY hypersurfaces, try to translate what they are interested in to your world.

Let me give a concrete example of the simplest Calabi-Yau hypersurface, namely an elliptic curve $E$ embedded in $\mathbb P^2$. Say $X= Proj(R)$ with $R=k[x,y,z]/(x^3+y^3+z^3)$. Then $H^1(E,\mathcal O_E)=k$, a fact you can relate to algebraically by realizing that this is the $0$-graded piece of local cohomology $H^2_m(R)_0$. You may also learn that the Frobenius action on this cohomology is $0$ or bijective depending on what reduction you make ("ordinary" and "supersingular" curves). It is not hard to compute this piece of local cohomology, it is generated by the class of $\frac{z^2}{xy}$, and the Frobenius takes this to $\frac{z^{2p}}{x^py^p}$. See for yourself why this class is $0$ or not depending on residue of $p$. Such an example would help you to see the connections more clearly and build up your confidence.

Last comment, make sure you read/watch people who are strong in both worlds and are good communicators. For instance, the papers by David Eisenbud and Karen Smith often have useful paragraphs that explain connections between local algebra and global geometry.


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