Semi-infinite homological algebra
I want to understand better why studying what we study in homological algebra gives us valuable information about the ring over which we are concerned--what information does it exactly give us? Fine, I can see why this obviously gives us motivation to study some of the things we study in homological algebra, but the only problem is I have no intuition for why Morita's theorem should be true.
Can anyone elucidate this, in the most simple terms possible?
While this very well may be equivalent to what I have asked in the above paragraph if so, feel free to concatenate answers I was wondering if someone could more fully explain Eisenbud's analogy that homological algebra is to ring theory as representation theory is to group theory in your own opinion, I know you don't know what he was thinking. The classic example being that Burnside's theorem has nothing directly to do with representation theory there is no use of character theoretic language in its statement yet the only "simple" proof uses character theory.
Unfortunately, in the realm of pure algebra I have been able to find very few examples of such uses of homological algebra--the only exception being the Schur-Zassenhaus theorem. So, any as elementary possible applications of homological algebra to problems in more elementary algebra group theory, module theory, ring theory, and to some extent [but preferably less so] commutative algebra where the statements would seem to suggest that the proof could be self-contained, yet realistically requires homological algebra would be great.
Thank you very much friends, help with any of these questions would go a LONG way to helping a very excited, and eager learner of homological algebra. I certainly won't try to give a general philosophical answer to your question, but I'll mention a success story that persuaded specialists that homological algebra was an amazingly powerful tool in commutative algebra. Auslander, Buchsbaum and Serre proved that a local noetherian ring is regular if and only if it has finite global dimension.
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From this it is easy to deduce that the localization at a prime ideal of a regular local ring is still regular. The statement of that result has nothing to do with homological algebra but since nobody had managed to prove it before, without homological algebra, this duly impressed algebraists.
Optional technicalities Let me give some relevant definitions here. This definition, due to Zariski, is a purely algebraic way of ensuring that an algebraic variety has no singularities. Rings appear in nature, most often, through their representations, that is, their modules. Modules are what you see of a ring experimentally. If two rings have equivalent categories of modules, then they are indistinguishable in that you cannot tell them apart by looking at their modules.
Most interesting things that you want to know about rings are things that you can learn about it from looking at their modules: it is therefore quite natural that two Morita equivalent rings share, for the most part, lots of properties.
In particular, you can compute the center of a ring from the knowledge of its modules alone and the maps between them and one can easily check then that two Morita equivalent rings have isomorphic center. In the special case in which the two rings are commutative, so that they coincide with their centers, they are then isomorphic, as you say. I have no intuition for why Morita's theorem should be true. Summarizing, the center of a category consists of natural families of endomorphisms of its objects, and naturality implies in particular that each endomorphism is central.
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Homological algebra - Wikipedia
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Homology and Rings. Homology and Groups. Spectral Sequences. Back Matter Pages About this book Introduction With a wealth of examples as well as abundant applications to Algebra, this is a must-read work: a clearly written, easy-to-follow guide to Homological Algebra. Cech cohomology Cohomology Homological algebra Sheaf cohomology algebra.