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Results tagged with homological-algebra
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user 2106
(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.
65
votes
What is Koszul duality?
I've spent many years researching Koszul duality in its various versions. To me, Koszul duality is a fundamental homological phenomenon which has many manifestations, e.g.
the relation between the h …
- 15.6k
34
votes
Accepted
Does the derived category remember the homological dimension?
Let $V$ be a finite-dimensional vector space, $\mathcal{A}$ be the abelian category of finitely generated graded modules over the symmetric algebra $S(V)$, and $\mathcal{B}$ be the abelian category of …
- 15.6k
28
votes
Accepted
Is there an additive functor between abelian categories which isn't exact in the middle?
Consider the abelian category of morphisms of vector spaces, i.e., the objects are linear maps $f:U\to V$, and the morphisms are commutative squares. Let the functor $Im$ assign to a morphism $f$ its …
- 15.6k
26
votes
"Pick up a homological algebra book and prove all of the theorems yourself" (exercise from L...
In the Russian 1968 translation of Lang's Algebra (which is done from the Addison-Wesley 1965 edition), this exercise is there on page 126, exactly as quoted in Jonas' answer. It is complemented by t …
24
votes
1
answer
2k
views
Sums of injective modules, products of projective modules?
Under what assumptions on a noncommutative ring R does a countable direct sum of injective left R-modules necessarily have a finite injective dimension?
Analogously, under what assumptions on R does …
- 15.6k
21
votes
The composition of derived functors - commutation fails hazardly?
My favorite (counter)example is this: let $A$, $B$, $C$ be the categories of left modules over some rings $R$, $S$, $T$ (respectively), and let $F$ and $G$ be the functors of tensor product with some …
- 15.6k
21
votes
0
answers
1k
views
Homotopy flat DG-modules
A right DG-module $F$ over an associative DG-algebra (or DG-category) $A$ is said to be homotopy flat (h-flat for brevity) if for any acyclic left DG-module $M$ over $A$ the complex of abelian groups …
- 15.6k
19
votes
Accepted
Existence of projective resolutions in abelian categories
Among the standard examples of abelian categories without enough projectives, there are
the categories of sheaves of abelian groups on a topological space (as VA said), or sheaves of modules over a …
- 15.6k
17
votes
Accepted
When is bar-cobar duality an equivalence?
What the references are saying is correct, and you are right. Yes, $\Omega BA \to A$ is always a quasi-isomorphism. No, $\Omega$ does not in general take quasi-isomorphisms to quasi-isomorphisms.
A …
- 15.6k
17
votes
Comodule exercises desired
My favorite exercise is: prove that a comodule (over a coalgebra over a field) is coflat if and only if it is injective. This presumes that you already know that any coalgebra is the union of its fin …
- 15.6k
16
votes
Accepted
Splitting of exact triangles in derived category
In any triangulated category, the necessary and sufficient condition for a distinguished triangle $A\to B\to C\to A[1]$ to split is that the morphism $C\to A[1]$ in this distinguished triangle vanishe …
- 15.6k
15
votes
Accepted
Tensored Over Abelian Groups?
Given an object $X$ in an additive category $C$ and an abelian group $A$, define the object $A\otimes X$ in $C$ by the rule $Hom_C(A\otimes X,\:Y) = Hom_{Ab}(A,Hom_C(X,Y))$, where $Ab$ denotes the cat …
- 15.6k
14
votes
Does homology detect chain homotopy equivalence?
Yes, this is true, and it does not matter whether the complexes are bounded from any side (nor of course does it matter whether the homology is finitely generated). This is so because:
The homotopy …
- 15.6k
14
votes
Accepted
Is there an adjoint to the inclusion of I-adically complete modules to all modules?
Contrary to the skepticism expressed in the question, for a finitely generated ideal $I$ in a commutative ring $R$, the completion functor $\Lambda_I\colon M\longmapsto \varprojlim_n M/I^nM$ is, in fa …
- 15.6k
13
votes
Accepted
koszul duality and algebras over operads
The situation for graded modules over a pair of Koszul dual algebras is more complicated, actually. What the question says is true for Koszul algebras $A$ and $A^!$ provided that $A$ is Noetherian an …
- 15.6k