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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 $Fc$.

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Comment 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.

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Comment 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.

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Comment 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.

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Comment 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...

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Comment by Zhaoting Wei on How to complete $i^*i_*\mathcal{F}\to \mathcal{F}$...

Is it well-known? How to see it?

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Comment by Zhaoting Wei on Do we have an equivariant Newlander-Nirenberg...

@MichaelAlbanese You are right. I made the edit according to your comments.

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Comment 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?

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What 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...

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Do 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...

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Why 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....

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Equivariant 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...

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Kasparov'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...

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Can 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 $\mathbb{R}P^2$?

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What 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...

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Do 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...

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How 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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Does a compact Lie group action on a family of compact manifolds have...

Let $\pi: M\to B$ be a fiber bundle of smooth manifolds with $B$ connected and each fiber of $\pi$ is a compact manifold. Let $G$ be a compact Lie group acting smoothly on $M$ such that$\pi(g\cdot...

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How to see $\delta_2(\hat{\chi}(V))=\chi(V)$ in differential cohomology?

I'm reading the paper "Differential Characters and Geometric Invariants" by Cheeger and Simons. In Page 62 the authors defined the differential Euler character $\hat{\chi}(V)\in \hat{H}^{2n-1}(M,...

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Do we have an equivariant version of integrability theorem of flat connections?

I am reading Donaldson and Kronheimer's book The Geometry of Four-Manifolds. In page 48, I found Theorem 2.2.1:Let $H$ be the hypercube $H=\{\mathbf{x}\in \mathbb{R}^d|~|x_i|<1\}$. If $E$ is a...

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Can we define $\partial\bar{\partial}(\log|z_1|^2)\wedge...

In complex analysis, by Poincare-Lelong theorem, we have$$\frac{\sqrt{-1}}{\pi}\partial\bar{\partial}(\log|z|^2)=T_{z=0}$$as currents, where$$T_{z=0}(\eta)=\int_{z=0}\eta.$$Now suppose we have two...

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Do we have $d(P_{+}(\omega\wedge \theta))=d_{+}\omega\wedge...

Let $(X, g)$ be a smooth, oriented, Riemannian 4-diemnsional manifold. Let $\Lambda^2$ denote the bundle of 2-forms on $X$. Then the Hodge-star operator decomposes $\Lambda^2$ into the space of...

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Is there any "deep" relation between the localization theorem of equivariant...

First let's consider equivariant cohomology: if a compact Lie group $G$ acts on a compact manifold $M$. We have the equivariant cohomology $ H_G(M)$ defined as the cohomology of the cochain complex...

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Is any deformation of an acyclic complex gauge equivalent to a trivial one?

This question is related to this question. Let $(C^{\bullet},d)$ be a cochain complex over a field $k$ with char$k$=0. We fix an Artin local $k$-algebra $(A,m)$. A deformation of $d$ is a new operator...

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Is an isomorphism between holomorphic vector bundles still holomorphic with...

Let $X$ be a compact complex manifold and $E$ be a finite dimensional holomorphic vector bundle on $X$ with a fixed $\bar{\partial}$-connection $\bar{\partial}_E$.Now we consider a small neighborhood...

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Do we have a Grauert-Fischer theorem for non-trivial families?

This question is related to my previous question. Let $X$ be a compact complex manifolds and $\Delta\in \mathbb{C}^n$ be a small neighborhood of $0$. A family of deformations of $X$ over $\Delta$ is a...

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Is an infinite direct sum of quasi-coherent $\mathcal{O}_X$-modules...

On any ringed space $(X,\mathcal{O}_X)$ we can define quasi-coherent $\mathcal{O}_X$-modules: A sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ is quasi-coherent if for every point $x\in X$ there exists...

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Has anyone studied the derived category of Higgs sheaves?

Let $X$ be a complex manifold and $\Omega^1_X$ be the sheaf of holomorphic $1$-forms on $X$. A Higgs bundle on $X$ is a holomorphic vector bundle $E$ together with a morphism of $\mathcal{O}_X$-modules...

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Answer by Zhaoting Wei for Proof of the Hirzebruch-Riemann-Roch theorem using...

I think the top of page 148 computes the case where $W$ is trivial, whose second term $\frac{1}{2}\sum_k(Rw_k,\bar{w}_k)$ is exactly $\frac{1}{2}\text{Tr}_{T^{1,0}M}(R^+)$. The general case then follows.

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Answer by Zhaoting Wei for Does there exist a GRR-like generalization of the...

I'm sorry for the self-citation. But your question is largely answered in the monograph Coherent Sheaves, Superconnections, and Riemann-Roch-Grothendieck, or the arxiv version, joint work of...

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What are the norms of the generators of the standard Podleś sphere?

Fix a real number $0<q<1$. We consider the standard Podles sphere $A_q$ as the universal unit $C^*$-algebra generated by $a$ and $b$ with relations\begin{equation*}\begin{split}&a=a^*,~...

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How obtain the right definition of smooth elements in a $C^*$-algebra?

In Alain Connes'$C^*$-algèbres et géométrie différentielle (an English translation is here,), for a $C^*$-algebra $A$, we consider a $C^*$-dynamic system $(A,G,\alpha)$, where $G$ is a Lie group and...

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Can we have a nontrivial division of a irreducible root system as the union...

The question is related to this MO question. Let $(\Phi, E)$ be a irreducible crystallographic root system where $\Phi$ is the set of all roots and $E$ is the $\mathbb{R}$-span of $\Phi$. As in the...

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Comment by Zhaoting Wei on Is the deformation of a $C^{\infty}$-manifold over...

@Z.M It is just a intuitive definition. We should use the ringed space definition as the true definition.

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