Questions tagged [pullback]
The pullback tag has no usage guidance.
2
votes
1answer
172 views
Restriction of Ext sheaves on closed subschemes
Let $f:X\rightarrow C$ be a morphism, where $C$ is a smooth curve. For $t\in C$ let $i_t:X_t = f^{-1}(t)\rightarrow X$ be the inclusion of the fiber of $f$ over $t$, and let $\mathcal{F}$ a coherent ...
2
votes
0answers
104 views
Restriction of the sheaf of relative differentials
Let $f:X\rightarrow C$ be a morphism, where $C$ is a smooth curve, and let $\Omega_f$ be the sheaf of relative differentials.
For $t\in C$ let $i_t:X_t = f^{-1}(t)\rightarrow X$ be the inclusion of ...
0
votes
0answers
73 views
$H$ very ample, $f$ finite, is there uniform $C=C(\mathrm{deg}(f))$ for $C f^* H$ very ample?
Let $X$ and $Y$ be normal projective varieties over $\mathbb{C}$ of dimension $n$. Let $f: X \rightarrow Y$ be a finite morphism. Also, let $H$ be a very ample divisor on $Y$.
Is there a constant $C=...
4
votes
0answers
131 views
Exercise in the book “Lectures on Kähler geometry”
I am currently studying the book "Lectures on Kähler geometry" by Andrei Moroianu and am looking for help concerning Exercise 5.8 (3) which is to prove the following Lemma 5.11
Let $f: M \...
5
votes
0answers
80 views
Pull back group cohomology onto handle decomposition
A construction encountered in the Dijkgraaf-Witten invariant uses the following ingredients:
An oriented, (assumed here to be smooth) manifold $M^n$
A finite group $G$ (and a field, chosen to be $\...
1
vote
1answer
108 views
Properties of codimension under pull back
If I pull back a cycle of codimension $c$ along a morphism of schemes I can easily see that the codimension can stay the same or the codimension can drop to all the way to $0$. But intuitively it ...
10
votes
1answer
590 views
History of the pullback corner notation
Where/when did the convention originate of marking pullback (and/or pushout) squares by that little right-angle symbol in the corner?
The earliest instance I’ve been able to find is in Paul Taylor’s ...
1
vote
0answers
124 views
Kernel of Gysin map
Let $X$ be a variety, let $D_1$, $D_2$ be linearly equivalent effective Cartier divisors on $X$ and let $\varphi_i\colon A_{k+1}(X)\to A_{k}(D_i)$ be the Gysin homomorphism. Is $\ker(\varphi_1) = \ker(...
0
votes
0answers
198 views
Isomorphism of probability spaces
Consider a surjective map $f:(X, \sigma(f)) \to (Y, \mathcal{Y})$, if a measure $\nu$ is given in $(Y, \mathcal{Y})$ the pullback $\nu(f(\cdot))$ is a measure on $(X, \sigma(f))$, similarly if $\mu$ ...
1
vote
2answers
275 views
Does “symmetry” of a pullback connection should be obvious?
$\newcommand{\M}{M}$
$\newcommand{\N}{N}$
$\newcommand{\TM}{TM}$
$\newcommand{\TN}{TN}$
$\newcommand{\TstarM}{T^*M}$
$\newcommand{\Ga}{\Gamma}$
Let $\M,\N$ be smooth manifolds, $\phi:\M \to \N$ be a ...
2
votes
0answers
81 views
Smooth submersions - smallest universal subclass of regular epimorphisms?
The smooth category has many problems. One is that pullbacks of regular epimorphisms need not exist.
However, pullbacks along submersions always exist. It also seems that submersions are universal (i....
3
votes
1answer
92 views
Fiber product of nilmanifolds
Let $M_1$ and $M_2$ be nilmanifolds. We can see them as total spaces of torus bundles $\pi_i:M_i \to B_i\ \ i=1,2$. Suppose that $B_1=B_2$ and that the fibers are torus of the same dimension and ...
4
votes
1answer
459 views
Left adjoint of pullback
In Awodey's Category Theory, p233 of 2nd ed. (or p205 of 1st ed.), he states:
Indeed, the UMP of pullbacks essentially states that composition along
any function α is left adjoint to pullback ...
11
votes
2answers
912 views
Pull-back of a fibration along a homotopy equivalence and homotopy classes of sections
I previously asked this on Math.SE but didn't receive a satisfactory answer.
Let $p:E\rightarrow B$ be a fibration (i.e. have the homotopy lifting property with respect to all spaces), and $f: B'\...
1
vote
1answer
197 views
finite generation of a certain type of subring
Let $k$ be a field, and let $R$ be a finitely generated $k$-algebra. (If it helps, you may assume $R$ is an integral domain.) Let $I$ be an ideal of finite colength. Note that $A:=k+I$ is a subring ...
2
votes
0answers
205 views
Is continuity of a functor stable under pullback?
Let $p:C\rightarrow D$, $i:F\rightarrow D$ be functors of 2-categories, and we form the lax pullback of $p$ along $i$ $$
\bar{p}:C\times_D^{lax} F\rightarrow F$$
Q1: Is it true that if $p$ ...
0
votes
2answers
830 views
Pullback of a constant sheaf
Let $\varphi:X\to Y$ be a surjective morphism of schemes which are connected and of finite type.
Let $A$ be an abelian group, $\mathscr{F}$ be the constant sheaf on $X$ with fibers $A$ and $\mathscr{...
1
vote
2answers
699 views
Definition of subobject classifier in presheaves
I am reading Awodey (Category Theory, 1st edition), p 175, and I have difficulties to understand the paragraph about the subobject classifier of $\mathbf{Sets}^{\mathbf{C^{op}}}$.
First let me quote ...
17
votes
3answers
5k views
Pullback measures
Why do all measure theory textbooks present the concept of push-forward measure, but never the concept of pull-back measure? Doesn't the latter exist?
It's true that the naive treatment of such a ...
