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comment Cohomology of geometric realization of a simplicial topological space
I believe the answer is "yes" (I'm more confident for the corresponding statement about homology, but I think this is ok too). Non-elementary proof: C_* commutes with homotopy colimits. Elementary proof: maybe use that there is a map between both objects and a spectral sequence computing their homology that has the same E_2 term?
Feb
1
awarded  Nice Answer
Jan
24
comment Why is symmetric group action needed for symmetric spectra?
We know that connective spectra are spaces with an action of the E_infty operad, and you can explicitly write down deloopings. I wonder if you can reinvent this formula by writing down the formula for the deloopings of a smash product of E_infty spaces? The symmetric group appears because it acts on the spaces of the operad
Jan
24
comment Fibration and induced mapping
And the definition of Serre fibration via the homotopy lifting property
Jan
24
comment Fibration and induced mapping
You can do it without that theorem, straight from the definition of fibration... Use the adjunction between maps and products/smash products
Jan
18
comment $d$ operator for Mayer Vietoris sequence in De Rahm cohomology
Use exactness- in the case in question the term after the target of delta is zero.
Jan
14
comment Surjective Homomorphisms of Isomorphic Abelian Groups
Hint: use the classification of finitely generated abelian groups. Try taking a quotient and seeing whether it's possible to end up with what you started with...
Jan
6
comment If s is a (global) section of a quasi coherent sheaf F on a scheme X, is there a reasonable scheme structure to gives its vanishing locus?
If you had a global section of the dual of a quasicoherent sheaf then that would give a morphism from X to $\mathbf{V}(\mathcal{E})$ (relative spec of sym on your bundle). You also have the zero section, always, and then you can form the fiber product of schemes to put a scheme structure on the zero locus.
Jan
6
comment Does the invertible sheaf to line bundle correspondence commute with taking duals?
EGA II.1.7. In particular, EGA II.1.7.9 and II.1.7.10 contain the statement you want (which says sections of the associated vector bundle are sections of the dual of your qcoh sheaf).
Jan
4
comment A categorical perspective on the equivalence of sheaf cohomology and Cech cohomology?
Should've read Qiaochu's answer more closely- mine is basically the same... Sorry!
Jan
4
answered A categorical perspective on the equivalence of sheaf cohomology and Cech cohomology?
Jan
4
comment Chinese remainder theorem as sheaf condition?
It would be weird for that diagram to be an equalizer since the first map is already an isomorphism! And the geometric way of thinking about this is that the vanishing sets of comaximal ideals are disjoint in Spec R, and disconnected Spec's are the Spec of the products of the coordinate rings of the components.
Nov
7
accepted Is every CW complex homotopic to a Delta-Complex?
Jul
29
awarded  Yearling
Apr
26
comment Show $\mathbb{CP^2/CP^1}$ is not a retract of $\mathbb{CP^4/CP^1}$.
I'm confused- why doesn't Olivier's answer just answer the question. There is no nonzero element in degree 4 that squares to zero, so there's not even a nonzero map $H^*(\mathbb{C}P^2/\mathbb{C}P^1) \rightarrow H^*(\mathbb{C}P^4/\mathbb{C}P^1)$ let alone a retract.
Mar
20
awarded  Enlightened
Mar
20
awarded  Nice Answer
Feb
8
comment How can I show this is a homotopy.
If I break up my domain into two closed pieces and define continuous functions on each piece that agree on the intersection of the pieces, the result is a continuous function.
Jan
31
comment A Way to make the following “proof” of the Hairy Ball Theorem rigorous?
no. but someone thinks i am: I tried to take down the picture and they put it back up.
Dec
11
awarded  Good Answer