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I am an Applied Math student. I have concluded my BSc and will be moving to Pure Math this September.
I am writing to have some advice about what Algebra I should study to start tackling Algebraic Geometry and (advanced) Algebraic Topology.

My background consists in Linear Algebra and a 'fundational' course covering the basics of Groups, Rings and Fields.
So I think I can learn about this topics, having all the necessary prerequisites, but do not know which are the most relevant and fundamental topics to the stated area. Any advice and reference to some compact text apt to self study would be great!

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    $\begingroup$ I may come across as a bit opinionated but for algebraic geometry it is absolutely necessary that you have a solid understanding of basic commutative algebra. Everything else comes second! For commutative algebra I suggest you read Andreas Gathmann's online lecture notes: (mathematik.uni-kl.de/~gathmann/class/commalg-2013/…). Go through chapters 1-6. Gathmann explicitly goes through the geometric meaning behind every concept in commutative algebra. Some people would also suggest Atiyah and MacDonald's book. It's decent but come on we are living in 2019. $\endgroup$
    – Parthiv Basu
    Jul 23, 2019 at 9:11
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    $\begingroup$ I can't speak for algebraic geometry, but for studying algebraic topology, you can get away with very little knowledge of algebra. I took an algebraic topology course last year (from Hatcher's book) and haven't had a graduate level algebra course (or self-studied any algebra). Some exercises that require a bit more algebra may stump you, but for the most part, based on your stated prerequisites you can pick up what you need along the way. $\endgroup$
    – Aweygan
    Jul 23, 2019 at 14:12

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For algebraic topology, it really depends on what areas of you'll be studying. If you'll be looking into homology and cohomology, you should read Allen Hatcher's Algebraic Topology (chapters 2 and 3). If you're going to be spending your time on homotopy theory, then I'd recommend Arkowitz' book on Homotopy theory. Hatcher also treats homotopy theory, but I don't particularly like how he goes about that. Arkowitz is better for this, in my opinion.

For algebraic geometry, I don't know that there's a standard 'compact' text which will be of that much use in a first course. Perhaps Miles Reid's Undergraduate Algebraic Geometry.

For both things, I'd recommend learning some category theory. Learn about functors. In both algebraic geometry and algebraic topology we use functors to go from the "geometric" (or topological) world of spaces to the algebraic world of rings and groups (among other things). When you start your algebraic topology course you should aim to understand (for example) how the fundamental group is actually a functor from Top to Grp. When you start doing algebraic geometry, try to understand the functors which you're using there too. It helps understand the bigger picture.

In both cases, an in depth understanding of the algebraic objects at hand is absolutely necessary. In the case of a first course in algebraic topology, group theory is important. In the case of algebraic geometry, ring theory. Particularly a good understanding of polynomial rings is important - but then again, algebraic geometry is basically designed to help us understand those :)

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    $\begingroup$ I think OP is looking for prerequisite texts in algebra, and not introductory texts to algebraic geometry/topology. $\endgroup$
    – Santana Afton
    Jul 23, 2019 at 13:32
  • $\begingroup$ I see. Upon re-reading the question I think you're probably right. I'll leave this answer to stand, since I think the two references above are at least a little useful (if for no other reason than being freely available online). $\endgroup$
    – Matt
    Jul 23, 2019 at 19:14

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