Survey articles in Commutative/Homological algebra

Question

I am a graduate student interested in Commutative algebra/Homological algebra.

I am comfortable with first eight chapters of Atiyah.

I am familiar with some algebraic geometry, first two chapters of Shaferevich.

I am looking for some papers in arxiv that a student with above background can read.

If I go on looking for each and every paper I would be wasting a lot of time.

Suggest some surcey articles/ conference proceedings (credits for these words to user Hoot) that I can read. I would be happy if you can suggest some that gives some motivation/historical background as well.

Answer 1

After these references it really depends on what type of commutative or homological algebra you intend to work in. However, some of the most widely useful general references are as follows.

Almost everything in the book Bruns and Herzog is lingua franca and can't be skipped.

Most of the chapters of Weibel's Homological algebra. Here its I guess okay to skip the group cohomology on the first pass, but the essentials about chain complexes, derived functors, spectral sequences, and the derived category are really great. Also, Gel'fand-Manin which bridges gaps that Weibel's text unfortunately misses.

The next set depends more on the types of problems you intend to work through.

For local cohomology, twenty four hours is a great start.

For problems related to Syzygies, Geometry of Syzygies and Graded Free Resolutions are essential.

One might want a more combinatorial approach to the subset and Miller-Sturmfels comes highly recommended.

Finally there are a few classics, like Serre's local algebra and Nagata that are useful.

I'm actually quite surprised no one mentioned these yet.

After these or in conjunction, it really comes down to reading many of the seminal papers/surveys and getting a feel for a specific type of problem. As likely you are being advised, it might make sense to ask your advisor for specific recommendations on where to start.

I hope this helps.

Answer 2

This is a very long comment, and not an answer!

I shall write only about the Grothendieck cohomology of sheaves of Abelian groups over a topological space!

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Let $X$ be a topological space; one can define \begin{equation*} \forall U\subseteq X\,\text{open},\,\mathcal{C}(U)=\{f:U\to\mathbb{R}\,\text{is continuous}\} \end{equation*} where over $\mathbb{R}$ (of course) one considers the natural topology. Easily, one can prove that $\mathcal{C}(U)$'s are commutative rings with unit; then:

• one associates to any open subset $U$ of $X$ a ring,
• for all $V\subseteq U\subseteq X$ open, $\mathcal{C}(U)$ is a subring of $\mathcal{C}(V)$.

How can we formalize these facts? One needs of category theory!

Via category theory, one defines a controvariant functor $\mathcal{C}$ from the category $\mathbf{Op}(X)$ of open subsets of $X$ and inclusions between subsets to the category $\mathbf{Ring}$ of commutative rings with unit; one calls $\mathcal{C}$ the presheaf of continuous functions on $X$. Moreover, $\mathcal{C}$ satisfies other two properties: the separatedness and the gluing; and one calls $\mathcal{C}$ sheaf.

This is the archetype of all (pre)sheaves!, even if, in general, one can consider a (pre)sheaf with values in a generic category.

For a very ample introduction to abstract sheaf theory: I suggest [S], the first five chapters.

One can repeat the same reasoning and find, for example, the sheaf $\mathcal{O}_X$ of holomorphic functions and the sheaf $\mathcal{O}_X^{\times}$ of invertible holomorphic functions on a complex manifold $X$; one can consider the sequence \begin{equation*} 0\to\mathbb{Z}\stackrel{i}{\to}\mathcal{O}_X(X)\stackrel{\exp}{\to}\mathcal{O}_X^{\times}(X) \end{equation*} where

• $i$ the inclusion of $\mathbb{Z}$ in $\mathcal{O}_X(X)$,
• $\exp:f\in\mathcal{O}_X(X)\to\exp(2\pi f)\in\mathcal{O}_X^{\times}(X)$;

easily one notes that $\mathbb{Z}$ is the kernel of $\exp$, but $\exp$ is not a surjective morphism of Abelian groups!, in other words: this sequence is only left exact!

This example is incredible, because the sequence of sheaves on $X$: \begin{equation*} 0\to\widetilde{\mathbb{Z}}\stackrel{i}{\to}\mathcal{O}_X\stackrel{\exp}{\to}\mathcal{O}_X^{\times}\to0 \end{equation*} is exact! (See [R], example 6.69)

What is the problem? The functor $H^0(X,\_)$ of global sections is not right exact!, where, for example, $H^0(X,\mathcal{O}_X)=\mathcal{O}_X(X)$.

Ideally, one woulds like to continue "the exactness on the right" via some other functor: the right derived functors of $H^0(X,\_)$!

In general, let $\mathcal{F}$ be a sheaf of Abelian groups on $X$, the $p$-th right derived $H^p(X,\mathcal{F})$ is called $p$-th Grothendieck cohomology group of $\mathcal{F}$. (See [R], chapter 6, sections 2.3, 2.4 and 3)

All this is completely and clearly exposed in the chapters 1 (except section 11), 2 (sections 1, 2 and 3) and 3 (sections 1 and 2) of [G].

At this point: what are the applications in classical algebraic geometry? Given an algebraic variety $X$ over a field $\mathbb{K}$, one defines the sheaf of regular functions $\mathcal{O}_X$ on $X$ and can study the Grothendieck cohomology of $(X,\mathcal{O}_X)$; but because $X$ (with the Zariski topology) is quasi-compact then it is paracompact, and it turns out that the Grothendieck cohomology of $\mathcal{O}_X$ coincides with the Čech cohomology: more easy to compute! (See [P], chapter 3)

In other words: in classical algebraic geometry, the Grothendieck cohomology is an overkilling!, but in the framework of schemes and ringed spaces (and more general spaces) it is a very powerful tool... and this is another subject.

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Bibliography

[G] Grothendieck A. - Sur quelques points d'algèbre homologique; Tohoku Math. J. (2) Volume 9, Number 2 (1957), 119-221 and Number 3 (1957), 119-221.

[P] Perrin D. - Algebraic Geometry. An Introduction (2008), Springer.

[R] Rotman J. J. - An Introduction to Homological Algebra (2009), Springer.

[S] Schapira P. - Algebra and Topology, lecture notes.

Answer 3

One source (which I have not read the whole article) might be the survey article "History of Homological Algebra" written by C. Weibel, http://www.math.uiuc.edu/K-theory/0245/survey.pdf.

Answer 4

When starting to learn commutative algebra, algebraic geometry, and homological algebra, I wish I would have known about Schneck's book

Computational Algebraic Geometry

because it goes into a lot of detail/examples required for doing more advanced algebraic geometry. For example, you will be much happier looking at hilbert schemes after reading this book because it gives a nice overview for how homological methods should be used to compute hilbert polynomials.