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So, I want to read about M sequence, or Koszul complexes, Cohen-macaulay modules. But I am realizing that to study these concepts I need to have some good background on homological algebra. So far I have finished atiyah MacDonald, so I have little background on this topic specially about tor and ext. But I want to know more on this topic. It will be great if you can recommend some introductory books for self reading mainly on homological algebra. And on this note, I know about commutative algebra, as per atiyah MacDonald and galois theory, so it will be great if you can inform me about some prerequisites I have to study in order to read those books.

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  • $\begingroup$ I think Aluffi's Algebra Chapter 0 might be good? Lots of exercises and it will allow you to go back and learn whatever you need from earlier chapters. $\endgroup$
    – user113102
    Apr 16, 2020 at 20:47

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Rotman's An Introduction to Homological Algebra, 2nd edition is a pretty solid intro to general homological algebra and is very accessible, although it doesn't deal with Koszul complexes.

Weibel's An Introduction to Homological Algebra is also accessible, but is more advanced in its delivery and topics than Rotman's book. It includes topics like Koszul complexes and local cohomology that are relevant to Cohen-Macaulay modules.

Relative Homological Algebra by Enochs and Jenda has a good look at topics in homological algebra, especially Gorenstein homological algebra and approximation theory. It is well contained, reasonable in length and has lots of good exercises. It includes regular sequences, Koszul complexes, local cohomology and has some discussions about CM modules.

There are two books on local cohomology that are accessible and are contain information about CM modules, namely Twenty-four hours of local cohomology by Iyengar et al, and Local Cohomology: an algebraic introduction with geometric applications by Brodmann and Sharp. Both contain more general information about related homological algebra and commutative algebra.

The first three chapters of Cohen-Macaulay rings by Bruns and Herzog has a lot of information on local rings, depth and related homological algebra, including local cohomology and Koszul complexes. It is quite good as a reference, but you could learn from it as well. I would also put the homological algebra sections of Eisenbud's Commutative Algebra with a view to Algebraic Geometry in with this.

I think you could build a good foundation from any of these books, and there are many more books on homological algebra that I haven't mentioned that are probably just as good.

More technical and specialist books about Cohen-Macaulay modules are Cohen-Macaulay Representations by Leuschke and Wiegand, and Cohen-Macaulay modules over Cohen-Macaulay rings by Yoshino. These are monographs rather than textbooks, but provide a good motivation and illustration of the representation theory of CM modules. The former is more heavy on commutative algebra, while the latter is more concerned with the categorical properties of the CM modules.

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