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pull-back

ˈpull-ˌback [f. phr. to pull back, pull v. 24.] 1. a. The action or an act of pulling back.1668 Dryden Evening's Love Epil. 14 In the French stoop, and the pull-back o' the arm. 1900 G. Swift Somerley 146 An occasional wrench and pull-back of the arms gave him considerable pain. 1903 A. Maclaren Las... Oxford English Dictionary

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Pull back (disambiguation)

category theory Pullback attractor, an aspect of a random dynamical system Pullback bundle, the fiber bundle induced by a map of its base space Other Pull-back wikipedia.org

en.wikipedia.org

For Kurds, US pull-back feels like being abandoned once more

Iraq's Kurds rose up again in the 1980s with Iranian backing during the Iran-Iraq war. Iraqi leader Saddam Hussein's army waged a brutal scorched-earth campaign, using poison gas and forcibly resettling up to 100,000 Kurds in the southern desert. The second event came in 1991, after the U.S.-led Gulf War that liberated Kuwait from Iraqi forces.

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Definition of pull-back analogous to push-forward Let $U\subseteq\mathbb{R}^n, V\subseteq\mathbb{R}^m$ be open subsets, and $f:U\rightarrow V$ a $C^1$ map. Let $u,v$ be vector fields on $U,V$ respectively. Then the **...

Since there is a natural Riemannian metric on $\mathbb R^n$ and $\mathbb R^m$, a notion of pull back for vector fields could be defined as follows: let $v$ be a vector field on $V$. Then $f^*v$ is a vector field on $U$ so that for all vector fields $w$ on $U$ we have $\langle f^* v,w\rangle = \langl...

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Section and pull-back bundle I find in general that the pull-back of a section of a vector bundle is a section of the pull-back bundle, but this seems to be false for the cotangent bundle. Let $\phi:M\to N$ be a smoo...

You are confused by two different notions of pullback. The pullback of $f:M\rightarrow N$ by $g:P\rightarrow N$ is $(x,y)\in M\times P$ such that $f(x)=g(x)$. A $1$-form defined on $N$ is a morphism $\omega:N\rightarrow T^*N$, if you have $\phi:M\rightarrow N$, you cannot defined the pullback of $\o...

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does a commutative diagram implies pull-back? Let $\xi=(E,p,B),\xi'=(E',p',B')$ be fibre bundles. Let $f: B\to B'$, $\bar f: E\to E'$ be maps such that the diagram commutes $\require{AMScd}$ \begin{CD} E @>\displaysty...

If you add the hypothesis that $\bar f$ restricts to a homeomorphism on each fiber, then it is true that $\xi$ is isomorphic to the pullback bundle. To see this note that the total space of the pullback bundle is $$f^*E' = \\{(b,e')\in B\times E': p'(e') = f(b)\\}. $$ Define a bundle map $\phi\colon...

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Pull-back of push-out under the diagonal embedding Let $X$ be a variety, $\Delta:X\to X\times X$ be a diagonal embedding. How to compute $\Delta^\Delta_\mathcal{O}_X$?

Recall the defintion of pullback: Let $f:X\rightarrow Y$ and $G$ be a sheaf on $Y$. Then $f^*G:$$=f^{-1}{G} \otimes_{f^{-1}O_Y}O_X$. Let $F$ be a sheaf on $X$. If $f$ is injective then we have $$f^{-1}f_*F(U)=\lim_{\rightarrow_{f(U)\subset V}}f_*F(V)=\lim_{\rightarrow_{U\subset f^{-1}(V)}}F(f^{-1}(V...

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How do you compute the pull-back of a complex differential (1,1)-form given its potential? Let $F: X \to Y$ be a holomorphic map. Let $\omega$ be a complex differential $(1,1)$-form on $Y$, $\omega=\partial \bar \part...

Suppose the charts on $Y$ follow coordinates $w \in \mathbb{C}^n$. Then we may write $\omega$ locally in a chart as $$ \omega(w) = \frac{\partial^2 f}{\partial w \partial \bar w }(w) dw \wedge d\bar w $$ Now if we have a nice holo map $F: X \to Y$ (i.e. structure preserving), the pull back is just a...

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One-Form pullback equation Let $U \subseteq \mathbb R^n$ and $V \subseteq \mathbb R^m,g: U \to \mathbb V$ continuably differentiable and let be $g$ One-Form on $V$. We define the pull-back $g^*w$. Let $w=\sum_{i=1}^m ...

Let $x\in R^n$, $u\in T_xR^n$, $(g^*w)_x(u)=w_{g(x)}dg.u=(f_1dy_1+..+f_mdy_m)_{g(x)}(dg_1(u),..,dg_m(u))$ $=f_1(g(x))dg_1(u)+...+f_m(g(x))dg_m(u)$.

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Pull-back of sections of vector bundles I'm sure this is a silly question but I'm stuck at the concept of pulling back sections of a vector bundle. Let $\pi:E\to X$ be a vector bundle on a variety $X$ and $f:Y\to X$ a...

This is the pull-back of $\sigma$ to $f^*E$.

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smooth map between manifolds which induces a pull-back of tangent bundles Let $M,N$ be two differentiable manifolds and $f: M\to N$ be a smooth map. Then the tangent map $Tf: TM\to TN$ is a homomorphism of vector bund...

Well, in general $Tf$ induces a push forward between $TM$ and $TN$. Now, if you take $T^*N$ and $T^*M$, then $(Tf)^* : T^*N \to T^*M$ is the natural pull back. If $\dim TM < \dim TN$ then I don't know how you can induce a pull back as $TM = (Tf)^* TN$ for vectors that lie outside the image of $Tf$. ...

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Pull-back of injective morphism of locally free sheaves is injective? Let $i:X \hookrightarrow \mathbb{P}^n$ be a smooth projective variety. Let $f:\mathcal{F}_1 \to \mathcal{F}_2$ be an injective morphism of $\mathca...

No, this is not true. Denote by $S=k[X_0,...,X_n]$, $X$ a hypersurface in $\mathbb{P}^n$ defined by an equation $F$ of degree $d$. Then there is an injective morphism $S(-d) \to S$ given by multiplication by $F$. But the induced morphism $(S/(F))(-d) \to S/(F)$ is not injective. Applying the associa...

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A line bundle is torsion iff its pull-back is trivial? Let $f\colon X\to Y$ be a finite flat morphism of varieties. Probably some assumptions on $f$ are required, but I have often seen the following claim being used: ...

If $f$ is a finite flat map, there exists maps $f^*:\mathrm{Pic}\, Y\to \mathrm{Pic}\, X$ and a map $f_*$ in the reverse direction, with the composite $f_*f^*$ multiplication by $\deg f$. $f_*$ is defined as follows. For a line bundle $M$ on $X$, $f_*(M)\in \mathrm{Pic}\, Y$ is $\Lambda^df_*M\otimes...

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$X \cong \operatorname{Proj} \oplus_n H^0(X, O(n)) $ Let $ X \subset \mathbb{P}^n$ be a smooth closed subvariety, and $O(1)$ is the pull-back of the line bundle of $O(1)$ on $\mathbb{P}^n$. Then it is claimed: > $$X ...

This follows from (EGA, II, 4.5.1, (b)) or (Stacks, 23.24.11), since $O(1)$ is ample on $X$.

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Simple question about the definition of divisor Let $C$ a complex, compact riemann surface and $\pi:C^{'} \rightarrow C$ a generic cover of $C$. If $\pi^{*}$ is the pull-back and $E$ a divisor on $C$, how can i define...

It is essentially the preimage of $E$ but with multiplicities. If $p\in C$ is a point, then $$\pi^*(p):=\sum_{x\in\pi^{-1}(p)}(\mbox{mult}_x\pi)x$$ where $\mbox{mult}_x\pi$ is the local multiplicity of $\pi$ at $x$ (i.e., $\pi$ looks like $z\mapsto z^{\mbox{mult}_x\pi}$ at $x$). You can then extend ...

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pull-back

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pull-back

Pull back (disambiguation)

For Kurds, US pull-back feels like being abandoned once more

Definition of pull-back analogous to push-forward Let $U\subseteq\mathbb{R}^n, V\subseteq\mathbb{R}^m$ be open subsets, and $f:U\rightarrow V$ a $C^1$ map. Let $u,v$ be vector fields on $U,V$ respectively. Then the **...

Section and pull-back bundle I find in general that the pull-back of a section of a vector bundle is a section of the pull-back bundle, but this seems to be false for the cotangent bundle. Let $\phi:M\to N$ be a smoo...

does a commutative diagram implies pull-back? Let $\xi=(E,p,B),\xi'=(E',p',B')$ be fibre bundles. Let $f: B\to B'$, $\bar f: E\to E'$ be maps such that the diagram commutes $\require{AMScd}$ \begin{CD} E @>\displaysty...

Pull-back of push-out under the diagonal embedding Let $X$ be a variety, $\Delta:X\to X\times X$ be a diagonal embedding. How to compute $\Delta^\Delta_\mathcal{O}_X$?

How do you compute the pull-back of a complex differential (1,1)-form given its potential? Let $F: X \to Y$ be a holomorphic map. Let $\omega$ be a complex differential $(1,1)$-form on $Y$, $\omega=\partial \bar \part...

One-Form pullback equation Let $U \subseteq \mathbb R^n$ and $V \subseteq \mathbb R^m,g: U \to \mathbb V$ continuably differentiable and let be $g$ One-Form on $V$. We define the pull-back $g^*w$. Let $w=\sum_{i=1}^m ...

Pull-back of sections of vector bundles I'm sure this is a silly question but I'm stuck at the concept of pulling back sections of a vector bundle. Let $\pi:E\to X$ be a vector bundle on a variety $X$ and $f:Y\to X$ a...

smooth map between manifolds which induces a pull-back of tangent bundles Let $M,N$ be two differentiable manifolds and $f: M\to N$ be a smooth map. Then the tangent map $Tf: TM\to TN$ is a homomorphism of vector bund...

Pull-back of injective morphism of locally free sheaves is injective? Let $i:X \hookrightarrow \mathbb{P}^n$ be a smooth projective variety. Let $f:\mathcal{F}_1 \to \mathcal{F}_2$ be an injective morphism of $\mathca...

A line bundle is torsion iff its pull-back is trivial? Let $f\colon X\to Y$ be a finite flat morphism of varieties. Probably some assumptions on $f$ are required, but I have often seen the following claim being used: ...

$X \cong \operatorname{Proj} \oplus_n H^0(X, O(n)) $ Let $ X \subset \mathbb{P}^n$ be a smooth closed subvariety, and $O(1)$ is the pull-back of the line bundle of $O(1)$ on $\mathbb{P}^n$. Then it is claimed: > $$X ...

Simple question about the definition of divisor Let $C$ a complex, compact riemann surface and $\pi:C^{'} \rightarrow C$ a generic cover of $C$. If $\pi^{*}$ is the pull-back and $E$ a divisor on $C$, how can i define...