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I am a student who has completed master's but couldn't take admission to a PhD program due to some unfortunate reasons.

I have done 1 course in Commutative Algebra where I followed the book " Mostly Commutative Algebra" from Chambert-Loir.

I have not studied Homological Algebra but I know that Rotman's book is an excellent reference (Introduction to Homological Algebra).

I have decided to self study from a textbook topics equivalent to a 2nd course in Commutative Algebra in line with the current research lines. ( But I am more interested in commutative algebra than algebraic geometry as of now but I have no one in real life to whom I can ask how different is Algebraic Geometry from Commutative Algebra? There are certainly non-trivial links between them and the methods of Commutative Algebra are very much used in Algebraic Geometry.) I think Algebraic geometry is finding solutions to polynomial equations using algebraic methods but Commutative Algebra is not much concerned with geometric questions but with algebraic questions.

Can you please suggest some reference books?

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    $\begingroup$ I am not sure what research you want to pursue. If you want to study algebraic geometry, say, then maybe it is better to start with that, and consult the needed results in the main time. On the other hand, I would suggest Weibel's book for homological algebra. $\endgroup$
    – Z. M
    Commented May 26 at 12:02
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    $\begingroup$ There a bunch of books by Atiyah-Macdonald, Eisenbud, Matsumura and others, which might be suitable for you. $\endgroup$
    – Donu Arapura
    Commented May 26 at 14:24
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    $\begingroup$ If you're interested in algebraic geometry, then Eisenbud is a great reference. But it's very complete, and I don't recommend going through it linearly (the table of contents spans 10 pages!). I think it really shines in the homological methods (chapters 17–21), and it's one of the best sources I know for Gröbner bases (chapter 15), but of course both assume a fairly solid background already. If that's too advanced, you could try Atiyah–MacDonald first. (I think I agree with this question that Matsumura is hard to read.) $\endgroup$
    – R. van Dobben de Bruyn
    Commented May 26 at 15:16

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I would suggest Altman and Kleiman's "A Term of Commutative Algebra". As far as I know, the most recent version can be found at https://dspace.mit.edu/handle/1721.1/116075.2.

Quoting from the preface: "There is no shortage of books on Commutative Algebra, but the present book is different. Most books are monographs, with extensive coverage. But there is one notable exception: Atiyah and Macdonald’s 1969 classic [4]. It is a clear, concise, and efficient textbook, aimed at beginners, with a good selection of topics. So it has remained popular. However, its age and flaws do show. So there is need for an updated and improved version, which the present book aims to be."

It's not only based on Atiyah-Macdonald, but it's a substantially expanded and arguably better structured version of it, IMHO with better balance between theory and exercises (a substantial amount of theory in Atiyah and Macdonald's text is contained in the exercises), with several additional chapters and exercises. Moreover, the second part of the book contains solution proposals to many of the exercises, which is helpful if you get stuck (since you are talking about self-studying the subject).

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  • $\begingroup$ Honestly, I don't quite understand the objection to my suggestion. It's in line with the other suggestions in the comments, and the textbook is very comprehensive (and also freely available). $\endgroup$
    – M.G.
    Commented May 27 at 14:09
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Ferretti, Andrea, Commutative algebra, Graduate Studies in Mathematics 233. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-7127-9/hbk; 978-1-4704-7434-8/pbk; 978-1-4704-7433-1/ebook). xvii, 373 p. (2023). ZBL1529.13001.

Ferretti, Andrea, Homological methods in commutative algebra, ZBL07725932.

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