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Results tagged with homological-algebra
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user 450
(Co)chain complexes, abelian Categories, (pre)sheaves, (co)homology in various (possibly highly generalized) settings, spectra, derived functors, resolutions, spectral sequences, homotopy categories. Chain complexes in an abelian category form the heart of homological algebra.
72
votes
3
answers
8k
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Where do all these projection formulas come from?
I have been intrigued for a long time by the formal similarity of results from different areas of mathematics. Here are some examples.
Set theory Given a map $f:X\to Y$ and subsets $X' \subset X, Y'\ …
- 3,793
21
votes
Motivating the category of chain complexes
"...utinam intelligere possim rationacinationes pulcherrimas quae e propositione concisa DE QUADRATUM NIHILO EXAEQUARI fluunt."
Henri CARTAN
[...if I could only understand the beautiful consequences …
- 47.5k
21
votes
Accepted
Flat module and torsion-free module
Dear liu,
1) If $A$ is a domain in which every finitely generated ideal is principal, then a module over $A$ is flat iff it is torsion free (Bourbaki, Comm.Alg.,I,§2, 4, Prop.3). Of course a PID has …
- 47.5k
11
votes
0
answers
869
views
Who proved the exactness of Amitsur's complex ?
A foundational result in Grothendieck's descent theory and in his étale cohomology is the exactness of Amitsur's complex. More precisely, suppose we have an $A$-algebra
$A\to B$; then there is a cosim …
- 47.5k