Comment by Zhaoting Wei on Does a fully faithful functor always preserve...
The problem is that not all cones in $D$ are of the form $Fc$.
View ArticleComment by Zhaoting Wei on Does flatness morphisms between ringed spaces...
@assaferan There are two problems: first, I am interested in not just schemes but general ringed spaces; second, even if we consider schemes, the fibers of $f$ need not to be affine.
View ArticleComment by Zhaoting Wei on How to compute the genus of the (singular)...
@Sasha Yes the only possibilities are $x_1=\pm x_3$ and $x_2=\mp x_4$ so there are two components. I made some changes on the problem.
View ArticleComment by Zhaoting Wei on Does the Hochschild cohomology of an...
@A.S. The Deligne conjecture says that the Hochschild cochain has the structure of an $E_2$-algebra, but my question is whether the Hochschild cohomology has the structure of an algebra.
View ArticleComment by Zhaoting Wei on Can we compute the Hochschild cohomology for...
@PedroTamaroff I am trying to deform $k[x]$ to a curved dg-algebra with trivial differential but non-trivial curvature and then compute its Hochschild cohomology. In this case I'm afraid that the...
View ArticleComment by Zhaoting Wei on How to complete $i^*i_*\mathcal{F}\to \mathcal{F}$...
Is it well-known? How to see it?
View ArticleComment by Zhaoting Wei on Do we have an equivariant Newlander-Nirenberg...
@MichaelAlbanese You are right. I made the edit according to your comments.
View ArticleComment by Zhaoting Wei on Does homotopy equivalence between deformations of...
@JonPridham If we tensor with the maximal ideal of an Artin local ring, is these two definitions equivalent? I have edited my question and could you have a look again?
View ArticleDoes the Mukai's lemma hold for non-algebraic $K3$ surfaces?
In Huybrechts' book Fourier-Mukai Transforms in Algebraic Geometry I found the following result due to Mukai (Page 232, Lemma 10.6)Let $X$ and $Y$ be two $K3$ surfaces. Then the Mukai vector of any...
View ArticleDoes $X\times Y$ have the resolution property if both $X$ and $Y$ have?
We say a complex manifold $X$ has the resolution property if every coherent sheaf $\mathcal{M}$ on $X$ admits a surjection $\mathcal{E}\twoheadrightarrow \mathcal{M}$ by some finite rank locally free...
View ArticleWhat is the significance that the Springer resolution is a moment map?
Let $\mathcal{B}$ be the flag variety and $\mathcal{N} \subset \mathfrak{g}$ is the nilpotent cone. We know that the Springer resolution$$\mu: T^*\mathcal{B}\rightarrow \mathcal{N}$$is the moment map,...
View ArticleWhat is the cohomology of the $C^{\infty}$-Koszul complex on $\mathbb{C}^2$?
First we consider the holomorphic Koszul complex on $\mathbb{C}^2$:$$0\to \mathcal{O}(\mathbb{C}^2)\overset{\begin{pmatrix}-z_2\\z_1\end{pmatrix}}{\to} \mathcal{O}(\mathbb{C}^2)^{\oplus...
View ArticleDo we have the Oka coherence theorem for finite group actions?
We first consider the sheaf of holomorphic functions $\mathcal{O}(\mathbb{C}^n)$ on $\mathbb{C}^n$. By Oka coherence theorem, $\mathcal{O}(\mathbb{C}^n)$ is coherent over itself.Now we consider a...
View ArticleWhy do we need cofiltered condition on the index category in the definition...
Let $\mathcal{C}$ be a category. The pro-category pro-$\mathcal{C}$ is defined as (see this nLab page) follows: its objects are diagrams $F: D\to \mathcal{C}$ where $D$ is a small cofiltered category....
View ArticleEquivariant K-theory of $S^1$-action on $S^2$
Is there any references for the structure of the equivariant K-theory $K_{S^1}(S^2)$ where the action of $S^1$ on $S^2$ is defined to be rotation about the $z$-axis? What is the ring structore of...
View ArticleKasparov's Dirac element and the index map
In Kasparov's 1988 paper Equivariant KK-theory and the Novikov conjecture section 4 he defined the Dirac element for a (non-spin) $G$- Riemanian manifold $X$ as an element in the $K$-homology...
View ArticleCan we classify two dimensional complex vector bundles over $\mathbb{R}P^2$?
I know it is easy to classify line bundles over $\mathbb{R}P^2$. But do we have a classification of two dimensional complex vector bundles over $\mathbb{R}P^2$?
View ArticleWhat is the cone of $\mathcal{F}\to i_*i^*\mathcal{F}$ for a divisor $i:...
Let $X$ be a smooth projective variety of dimension $n$ and $i: D\hookrightarrow X$ be a smooth, anti-canonical divisor in $X$. In other words,$[D]+[\omega_X]=[0]$ where $\omega_X$ is the canonical...
View ArticleDo we have a left adjoint of $i^*$ for a closed immersion $i$?
Let $i: X\hookrightarrow Y$ be a closed immersion of varieties. We have the derived pullback functor $i^*: D^b_{coh}(Y)\to D^b_{coh}(X)$.My questions is: can we construct a left adjoint of $i^*$ in...
View ArticleHow to complete $i^*i_*\mathcal{F}\to \mathcal{F}$ into an exact triangle for...
Let $Y$ be a smooth algebraic variety and $i: X\hookrightarrow Y$ be a smooth divisor. We consider the derived functors $i^*: D^b_{coh}(Y)\to D^b_{coh}(X)$ and $i_*: D^b_{coh}(X)\to D^b_{coh}(Y)$. By...
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