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I've read Cox, Little & O'Shea's Ideals, Varieties, and Algorithms up to Section $7$ where Buchberger's algorithm for finding a Groebner basis is proved. At this point I'm feeling very comfortable with Groebner basis for ideals in $k[x_1,\ldots,x_n]$. I'd like to learn about the generalization to module theory, which is essentially Chapter $15$ in Eisenbud's Commutative Algebra (with a view toward Algebraic Geometry). However, possibly unsurprisingly, I am finding Eisenbud to be a little bit of a stiff read. I'm familiar with the basics of module theory, but I want a source that is a little more comprehensive and easier to read. I've looked at Graded Syzygies by Irena Peeva, but it lacks many proofs, which makes it tricky to comprehend.

I am looking for a (possibly several) references regarding free resolutions of modules and preferably, methods of computing them (like Schreyer's algorithm). I'd like something that is more approachable than Eisenbud, containing more detail than Peeva, yet still on the introductory side of things.

Any books, PDFs, etc. that you may know of would be appreciated.

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A Singular introduction to Commutative Algebra by Greuel and Pfister is probably worth a look. It provides a good introduction to commutative and homological algebra and emphasises computational and algorithmic methods with an abundance of examples in Singular. The content is not as advanced as Eisenbud's book.

Another book (also focused on Singular) that is more advanced is Dekker and Lossen's Computing in Algebraic Geometry. This is much more concise than Greuel and Pfister's book and the level varies from introductory to quite advanced. Again, there is a strong influence from computational methods and practical sessions in Singular.

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