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Results tagged with homological-algebra
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user 750
(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.
6
votes
1
answer
334
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A potential resolution of $R/r$
The DGA
For $k$ some field, let $R$ be a $k$-algebra, and let $r\in R$.
Define a differential graded algebra $\mathbf{R}_r$ as follows. As a graded algebra, it is isomorphic to $R\langle t\rangle$, t …
- 13k
17
votes
Can a module be an extension in two really different ways?
I believe this is a counter example. Let $R=\mathbb{C}[x]$, and consider finite-dimensional modules (ie, f.d. vector spaces equipped with a distinguished endomorphism). For convenience, I will iden …
- 13k
9
votes
Free resolution dimension?
When requiring finitely-generatedness of the resolution, then the free dimension of a projective module can be infinite.
As a simple example, take the ring $R=k\oplus k$, and let $e_1=(1,0)$ and $e_2= …
- 13k
14
votes
Elementary $\mathrm{Ext}^1$ intuition
It seems like you already can see this, but $Ext^1(M,N)$ is measuring all the ways to form distinct short exact sequences $0\to N\to ?\to M\to 0$
If you are looking for intuition, what you should do i …
- 13k
3
votes
Graded or stacky Serre duality
Theres graded local duality which works just like local duality; however, it requires that $A_0$ is a field. I've had some luck making things work when A_0 is not a field, but then the local duality …
- 13k
17
votes
3
answers
1k
views
Freyd-Mitchell for triangulated categories?
Is there a nice analog of the Freyd-Mitchell theorem for triangulated categories (potentially with some requirements)? Freyd-Mitchell is the theorem which says that any small abelian category is a fu …
- 13k