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!
ormaybesaladqh
Answered question
2022-10-23
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!
Answer & Explanation
Reese Hobbs
Beginner2022-10-24Added 13 answers
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 each matters, i'd endorse studying some class concept. find out about functors. In each algebraic geometry and algebraic topology we use functors to go from the "geometric" (or topological) international of areas to the algebraic world of earrings and businesses (among other matters). whilst you start your algebraic topology path you should aim to apprehend (as an example) how the essential institution is honestly a functor from top to Grp. while you start doing algebraic geometry, try to understand the functors which you're using there too. It allows understand the larger photograph.
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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