10
votes
Formality of classifying spaces
This is an old question. But sometimes old questions get answered!
Benson, Greenlees, Formality of cochains on BG
Here is the abstract:
Let $G$ be a compact Lie group with maximal torus $T$. If $|N_G(...
- 6,162
10
votes
Accepted
Explicit example of an equivariant embedding of $U(n)/( U(k) \times U(n-k))$ into a finite dimensional $U(n)$-representation
You can just use the embedding $f\colon G/H=U(n)/(U(k)\times U(n-k))\to M_n(\mathbb{C})$ given by $f(gH)=gpg^{-1}$, where $p=1_k\oplus 0_{n-k}$. This gives a homeomorphism from $G/H$ to the space
$$ ...
- 56.9k
10
votes
Vector bundles on $\mathbb{A}^n / G$
In the paper Affine varieties dominated by $\mathbf{C}^2$ Gurjar considers a slightly more general situation, namely an affine normal variety $\mathrm{X}$ with a proper surjective morphism $\mathbf{A}^...
- 2,808
8
votes
Accepted
effective descent of coherent sheaves
The answer depends on the action:
If you take an action which is free, then the morphism $\pi$ is étale and, by Etale descent (cf. Bosch-Lutkebohmer-Raynaud "Neron Models page. 139) and you have ...
- 378
7
votes
Terminology about G- simplicial complexes
I've never stumbled across a standard one in all of the literature, though I've definitely appreciated a remark by Ken Brown in his group cohomology bible: "The hypothesis that $G_\sigma$ fixes $\...
- 17.5k
6
votes
Deequivariantisation of indecomposable sheaves
Take $X=pt$ and $G=G_m$ and $k$ to have characteristic zero. The equivariant derived category in this case is equivalent to modules for the homology of the circle, ie exterior algebra on a generator ...
- 24k
6
votes
Equivariant resolution of singularities
To any variety $X$ over a field of characteristic zero, say $k$, one can attach a resolution of singularities, say $X'\to X$, with the following properties:
$X'\to X$ is an isomorphism over $X_{\...
- 657
6
votes
Formality of classifying spaces
I've only just seen this rather old thread. I've recently been computing with cochains on $BG$ for $G$ a finite group in characteristic $p$, and have some rather surprising conclusions. If $G$ has ...
- 16.2k
5
votes
Accepted
Resolution by locally free $G$-equivariant sheaves on varieties
Suppose $L$ on $X$ is a (very) ample line bundle. Then $\oplus_{g \in G} g^* L$ is a $G$ equivariant vector bundle. Its determinant is also $G$ equivariant and isomorphic to $\otimes_{g \in G} {g^*...
- 3,948
5
votes
Accepted
Approximation of $C^1$-smooth equivariant maps by infinitely smooth ones
One option is to use the harmonic map flow developed by Eels and Sampson [1]. In a certain sense this is a (non-linear) analog of the heat equation for maps $M \to N$.
Endow the manifolds $M$ and $N$ ...
- 5,038
4
votes
Deequivariantisation of indecomposable sheaves
Take $G_m$ acting on itself via $z\mapsto z^2$, and take $k$ to be of characteristic $2$. The equivariant derived category here is the derived category of $\mathbb{Z}/2\mathbb{Z}$-modules. Now take ...
- 563
4
votes
Interactions (functors) between equivariant sheaves for different groups?
In light of Marc Hoyois's observation that $Sh_G(X)$ is the same as presheaves on some easy category, I will answer the more general question:
Given a functor $f \colon \mathcal{C} \to \mathcal{D}$,...
- 5,898
4
votes
Accepted
Classification of (complex algebraic) vector bundles on punctured affine space
I spoke to my colleague Song Sun, and he reminded me of a discussion that he and I had about Question 2 some time ago. For $n\geq 2$, there are many examples of locally free sheaves on $X_{n} = \...
4
votes
Accepted
${\rm SL}_2(\mathbb C)$-equivariant K-theory of $\mathbb C P^1$
I am just posting the comments as one answer (I am happy for somebody else to answer). For a maximal torus $\mathbb{G}_m \hookrightarrow \mathbf{SL}_2$, the $\mathbb{G}_m$-equivariant K-theory of a ...
4
votes
What does the Serre functor of equivariant category of fractional CY category look like?
I don't think you can understand $\kappa$ abstractly. In some sense, the computation of $\kappa$ is a more precise version (taking into account the group action) of computation of the Serre functor.
...
- 39.3k
3
votes
Accepted
Equivariant sheaves on $\mathbb P^1$
Let me explain why the line bundle $\mathcal{O}(1)$ does not admit a $\mathrm{PGL}(2)$-equivariant structure. Indeed, if it does, then the vector space
$$
\mathrm{Hom}(\mathcal{O}, \mathcal{O}(1))
$$
...
- 39.3k
3
votes
Accepted
Short exact sequence of equivariant line bundles on $\mathbb P^1$
This is equivariant. The map $\Lambda \to \mathcal O$ explicitly on sections sends a section $(e_m, f_1) $ valued in $ L \subseteq \mathbb C^2$ to the section $f_1$ of the trivial line bundle. This ...
- 148k
3
votes
Deequivariantisation of indecomposable sheaves
This is essentially a "down to earth" version of what David Ben-Zvi is saying. The object/example produced below essentially matches with what David is suggesting. I am just producing it &...
- 563
3
votes
Accepted
Graded commutativity of the $n$th Browder bracket
Both papers choose a normalization that is different from yours: they define
$$
[a,b] = (-1)^{na+1} s(a \otimes b).
$$
This is definition 5.7 in Cohen's paper that you mentioned.
The reason for this ...
- 52.6k
3
votes
Classification of (complex algebraic) vector bundles on punctured affine space
Jean-Pierre Serre, Prolongement de faisceaux analytiques coh ́erents, Ann. Inst. Fourier (Grenoble) 16 (1966), no. fasc. 1, 363–374. MR MR0212214 (35 #3088) p. 372 proves that there are infinitely ...
- 26.3k
3
votes
Accepted
What is the pointed Borel construction of the $0$-sphere?
Let's apply your definition (which I think has typos on the RHS - the two "+" subscripts on the EG should not be there I think). Let's model everything as topological spaces and do the ...
- 2,052
2
votes
What is equivariant chains on a representation sphere?
Exercise 10 of section 1 of Chapter II of tom Dieck's book Transformation Groups gives you one answer to your question. It reads:
Let $S(V)$ be the representation sphere of a finite group $G$. Show ...
- 11.1k
1
vote
Accepted
Representation of equivariant maps
Consider
$$
(|\det(X)| g(x_i))_{i=1}^{n, \top}.
$$
General theorem in this direction can be found e.g. in section 7 of Michor's Topics in differential geometry. The space of equivariant maps is a ...
- 8,597
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