But what if there isn't? How can you prove that? Another success of homological algebra is the standard proof of the First Brauer-Thrall conjecture:. Roiter A finite dimensional algebra over an algebraically closed field is either representation-finite or admits indecomposable modules of arbitrary dimension. This statement is, in principle, quite independent of homological algebra, yet we now prove it using what's called Auslander-Reiten theory, which is at its heart homological algebra I don't know what Roiter's argument was.
Koszul duality is a nice topic where classical constructions and homological algebra intertwine themselves.
Semi-infinite homological algebra
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I am just starting homological algebra and so notions of global dimension and projective dimension are a little beyond me at this point although, I shall learn them soon enough. That said, even with my minimal commutative algebra background Atiyah-Macdonald I can definitely understand your example.
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Thanks again! I was well-aware of the applications of homological algebra to topology, but was unaware of the result of Roiter--it's a nice result!
Gorenstein Homological Algebra - CRC Press Book
Thanks also for this post I'm commenting on, it definitely starts to make me realize the similarity of homological algebra to representation theory. Qiaochu Yuan Qiaochu Yuan k 33 33 gold badges silver badges bronze badges. Avitus Avitus 12k 1 1 gold badge 19 19 silver badges 41 41 bronze badges. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password.
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Q Quasi-isomorphism. R Resolution algebra. S Schanuel's lemma Schur multiplier Secondary calculus and cohomological physics Section category theory Severi—Brauer variety Shapiro's lemma Sheaf cohomology Short five lemma Six operations Snake lemma Splitting lemma Standard complex.
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T T-structure Tate cohomology group Tensor product of modules Topological half-exact functor Tor functor Torsion algebra Transgression map Triangulated category. U Universal coefficient theorem. V Verdier duality. W Weak dimension Weak equivalence homotopy theory Weil conjectures. Z Zig-zag lemma. Categories : Abstract algebra Algebraic topology Category theory. Hidden categories: Commons category link is on Wikidata.