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4 votes
1 answer
184 views

FI-homology of a spectral sequence of rational FI-modules

Let $(E^r_{p,q})$ be a spectral sequence of rational $\mathsf{FI}$-modules. Call $H^{\mathsf{FI}}$ the $\mathsf{FI}$-homology (see here) and $t_k X= \deg H^{\text{FI}}_k X$ the $k$-th generation ...
8 votes
2 answers
960 views

Is the derived category of local systems equivalent to the derived category of sheaves of vector spaces with local system cohomology?

Let $k$ be a field and $X$ a topological space. Write $\mathrm{Sh}(X)$ for the category of sheaves of vector spaces on $X$, and $\mathrm{Loc}(X)$ for the subcategory of local systems of finite ...
4 votes
0 answers
192 views

Extended double 2-cocycle conditions: Mathematical structure behind?

Note: For experts, to save your time, you can just read the highlighted texts and Eqs directly. The ordinary group 2-cocycle condition: Let us remind the usual so-called homogeneous group 2-cocycle $...
10 votes
1 answer
920 views

Is the homotopy category of an abelian model category abelian?

A model structure on an abelian category $A$ is called an abelian model structure if the cofibrations are precisely the monomorphisms with cofibrant cokernel, and if the fibrations are precisely the ...
1 vote
0 answers
120 views

Computing $\text{Tor}_*^{R_G} (\mathbb{Z}, \mathbb{Z}) $ for a compact Lie group $G$

Let $R_G$ be the representation ring of $G$ a connected, simply connected Lie group, $I_G$ the augmentation ideal and $\mathbb{Z}=R_G/I_G$. $R_G$ acts on $\mathbb{Z}$ via $V \cdot n = (\dim V ) n$. I ...