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Results tagged with homological-algebra
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user 65
(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.
13
votes
1
answer
1k
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When do six operations work?
This question comes (heavily edited) from my notes, thus slightly unusual structure.
We know that algebraic maps have very strict structure, and in many settings the operations f_*, f_!, their adjo …
- 32.5k
73
votes
3
answers
18k
views
What is Koszul duality?
Okay, let's make sure I'm on the same page with those who know homological algebra.
What is Koszul duality in general?
What does it mean that categories are Koszul dual (I guess representations of K …
CommunityBot
- 1
6
votes
Some intuition behind the five lemma?
One example would be a map induced by a morphism $f: X \to Y$ in the long homology sequence.
E.g. suppose the top row is a cohomology of pair $(X, A)$ and the bottom row is the cohomology of pair $( …
1
vote
Is there an analogue of the Lefschetz fixed point theorem for discrete dynamical systems?
I think this is how you can construct the required homology theory:
$n$-chains = maps of chain graphs $[n] = 0 \to 1 \to \dots\to n$ into your graph (where, perhaps, one edge of $[n]$ maps to many e …
- 15.1k
36
votes
6
answers
5k
views
How to think about model categories?
I've read about model categories from an Appendix to one of Lurie's papers.
What are the examples of model categories? What should be my intuition about them?
E.g. I understand the typical examples …
- 15.1k
10
votes
7
votes
3
answers
3k
views
Beilinson-Bernstein and Koszul duality
For geometric representation theorists down here.
Consider the Beilinson-Bernstein theorem:
Functor of global sections establishes
the correspondence between twisted
D-modules with fixed twi …
- 24.1k