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Results tagged with homological-algebra
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user 4194
(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
Heuristic behind $A_{\infty}$ - algebras
There is one aspect of $A_\infty$-algebras I haven't seen mentioned yet: Massey products and related structures.
Suppose you have a differential graded algebra $(A,d)$: a chain complex with a product …
- 4,254
1
vote
Accepted
Boundary operator in the colimit of complexes.
For each integer $n$, the $n$th term $C[n]$ in the colimit complex $C$ is the colimit of the $n$th terms $C_i[n]$. For each $i \in \mathcal{I}$ and each integer $n$, the $n$th boundary map in $C_i$ i …
- 4,254
2
votes
Accepted
Why do we use the diagonal for diagonal approximations ?
The diagonal map $\Delta$ is "coassociative": the two maps $(\Delta \otimes 1) \circ \Delta$ and $(1 \otimes \Delta) \circ \Delta$ from $\mathbb{Z}G$ to $(\mathbb{Z}G)^{\otimes 3}$ are equal. Therefor …
- 4,254
2
votes
$\inf\{i\in \mathbb N \cup \{0\}\cup\infty\mid Ext^i_R(R/I,R)\neq 0\}=0 ?$
I think the answers to both questions are yes.
Put a grading on $R$ so that it is connected (zero in negative degrees, $k$ in degree 0): for example, put $x_n$ in degree $n$. (It also seems safest to …
- 4,254
3
votes
0
answers
39
views
Filtering a pre-Koszul algebra to get a homogeneous associated graded algebra
In Priddy's paper "Koszul resolutions", on p. 42 he defines an algebra $A$ to be pre-Koszul if it can be written as a quotient of a free algebra $F = F\langle x_i \rangle$ with generators $\{x_i\}$ by …
- 4,254
4
votes
Accepted
Adams spectral sequence and short exact sequences. Some clarifications
The red dot in (3,0) comes from a map $\Sigma^3 D \to \mathbb{F}_2$, and this map is the image of a map $\mathbb{R}P^\infty \to \mathbb{F}_2$, so it goes to zero under the coboundary map. This agrees …
- 4,254
15
votes
Accepted
Is homology finitely generated as an algebra?
Another counterexample: let $A$ be the algebra $\mathbb{Q}[y,z]/(y^2) \otimes \bigwedge(x)$ with $x$ in degree 1, $y$ and $z$ in degree 2. Put a differential on this by $z \mapsto xy$. This is a commu …
- 4,254