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Mar
20
comment A quasi-isomorphism between the total complex of a Cartan-Eilenberg resolution and the complex per se.
@AndreaGagna I just skimmed G&M and found that they assume that the double complex is bounded. It's a different story. But the spectral sequence with filtration by rows works (assuming AB5), since it's upper half-plane therefore the spectral sequence converges to $\operatorname{Tot}^\oplus$.
Mar
13
comment The geometric interpretation for extension of ideals?
Further remarks to complete this answer: first, instead of fiber, I prefer to consider the whole morphism: $\operatorname{Spec}(A/f(I)A)\to\operatorname{Spec}(B/I)$, just like the restriction of fiber bundle into a subset. Second, I want to figure out that $A\otimes_BB/I\cong A/f(I)A$ is a corollary of the right exactness of $A\otimes_B-$, though could be checked directly.
Mar
13
revised The hyper-derived functors $\mathbb L_\bullet F$ are just derived functors of $H_0F$?
added 248 characters in body
Mar
13
answered The hyper-derived functors $\mathbb L_\bullet F$ are just derived functors of $H_0F$?
Mar
11
comment A quasi-isomorphism between the total complex of a Cartan-Eilenberg resolution and the complex per se.
@AndreaGagna Well, so only the condition that $H_n(P,d^h)$ is exact is used here? It is claimed in the text that AB5 should be assumed in the classical convergence theorem, though it seems to me that the construction and proof for everything is for modules, not generally for abelian categories. If you are right, I hope you can post an answer and I'll read it and accept it in weekends.
Mar
8
comment Long exact sequence for a triple follows from long exact sequence for a pair?
In fact, my initial idea is that the diagram chasing of the homology exact sequence of a pair could be systematized by the construction of derived couples. I guess that it's a corollary of usage of the exactness of repeated derived couples. Well, I will read your proof in the coming weekend, and sorry for my retardation.
Mar
8
comment Poincare Duality Reference
The fact that the star complexes constitutes a cellular filtration only depends on the homology of the manifold, just as Serfert & Threlfall shows. I cannot see how to determine the topology, not just homology of that, and how to take advantage of the tubular neighborhood you've mentioned. In addition, sorry for my ignorance, I don't know the tubular neighborhood theorem for triangulated manifolds, but not for differentiable manifolds.
Mar
8
comment Poincare Duality Reference
Sorry, I cannot follow your idea. I want to go back to the language introduced in Seifert and Threlfall. The dual cell decomposition is constructed as follows: there's a point, the barycenter, associated to each $n$ dim simplex, then a $1$ dim star complex associated to each $n-1$ dim simplex, whose center is the barycenter of the complex and whose outer boundary is the totality of star complexes associated to each incident $n$ dim simplex, and so forth. Maybe it's easy to prove that such a star complex is a cellular decomposition, i.e. $H_m(X_{n+1},X_n)=0$ for $m\neq n$.
Mar
8
revised The hyper-derived functors $\mathbb L_\bullet F$ are just derived functors of $H_0F$?
added 2 characters in body
Mar
7
comment Four Color Theorems: Graphs vs. Maps
In fact, it's just a matter of duality, therefore the conditions of duality theorems should be satisfied. For more details, see Alexander duality, or Poincaré duality, etc. I'm not familiar with these stuff.
Mar
7
asked The hyper-derived functors $\mathbb L_\bullet F$ are just derived functors of $H_0F$?
Mar
7
comment Four Color Theorems: Graphs vs. Maps
Since the graph is embedded as a subset of $\mathbb R^2$, therefore the complement is just the set-theoretic complement. For example, if the graph is $S^1$ embedded in $\mathbb R^2$, then the complement is of two connected components (Jordan curve theorem).
Mar
7
comment Four Color Theorems: Graphs vs. Maps
It seems to me that a (finite) planar graph is a (finite) graph embedded in $\mathbb R^2$. The regions are the connected components of the complement of the embedded graph.
Mar
7
revised A quasi-isomorphism between the total complex of a Cartan-Eilenberg resolution and the complex per se.
added 32 characters in body
Mar
7
comment A quasi-isomorphism between the total complex of a Cartan-Eilenberg resolution and the complex per se.
@ZhenLin Yes, since for fixed $n$, in order to compute $H_n$ etc, we only need a finite portion.
Mar
7
asked A quasi-isomorphism between the total complex of a Cartan-Eilenberg resolution and the complex per se.
Mar
7
comment Homology and (co)Limits
@Exterior In general, chain homology functor commutes with exact functors, a consequence of the FHHF theorem (cf. Ravi Vakil's notes, FOAGjan2915, 1.6.H). If, instead of exact functors, consider arbitrary right exact functor, then maybe a spectral sequence intervenes.
Mar
7
comment Poincare Duality Reference
@AndrewMarshall Well, with these abstract nonsense (say five lemma or even spectral sequence argument), the combinatorial meaning is hidden. I think that original approach is still valuable.
Mar
7
comment Poincare Duality Reference
Is there any reference for the proof that the dual decomposition is really a CW-decomposition? Apparently, it depends on the fact that the original simplicial complex is a manifold. I don't know how to take advantage of this homogeneity explicitly.
Mar
7
comment Poincare Duality Reference
In which section of Schubert's book? I didn't find that.