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Results tagged with homological-algebra
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user 51164
(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.
5
votes
Accepted
A conservative, non faithful functor between triangulated categories
Here is also a simple example where the restriction of $F$ to the heart is faithful but $F$ itself is not. Let $Vec$ be the abelian category of complex vector spaces and let $Rep(\mathbb{Z})$ be the a …
- 9,491
5
votes
0
answers
111
views
Homological characterization of perfect resolutions
Suppose that $R$ is a left Noetherian associative ring with unit and $M$ a finitely generated left $R$-module. It is a standard fact that if the $\mathrm{Ext}$-groups $\mathrm{Ext}^i_R(M,N)$ vanishe f …
- 9,491
3
votes
Pro-representability of deformation functor associated to a DG Lie algebra
Edit: The answer below only works for the case where all the $L^i$'s are finite dimensional.
The statement is true. In fact, under the assumptions of the question, it is also true that ${\scr C}(L)^* …
- 9,491
15
votes
Accepted
Example of an additive functor admitting no right derived functor
Let ${\cal C}$ be the category of finite dimensional ${\bf Z}/2$-vector spaces equipped with a ${\bf Z}/2$ action, let ${\cal C'}$ be the category of finite dimensional ${\bf Z}/2$-vector spaces and l …
- 9,491
2
votes
Homological vs. cohomological dimension of a group/space
If $X$ is a finite CW complex then $hd(X) = cd(X)$. To prove this it is clearly enough to assume that $X$ is connected. Let $n$ be the dimension of $X$, let $G$ be the fundamental group of $X$ and let …
- 9,491
6
votes
Accepted
Spelling out explicitly the data of a two step filtration in terms of pieces and gluing data
Technically speaking the answer to your question is no, in the sense that the data of $(\alpha,\beta,\gamma,\delta)$ alone does not determine the filtered object $V_0 \subseteq V_1 \subseteq V_2$. How …
- 9,491
10
votes
Accepted
Different definitions of derived functors
The total right derived functor ${\bf R}F(-)$ contains a bit more information than just its individual cohomologies ${\bf R}^iF(-) = H^i({\bf R}F(-))$. This information can indeed be described as a ki …
- 9,491