Explicit sections after sheafification
$begingroup$
Hartshorne (in his book on algebraic geometry) constructs, for a presheaf $mathscr{F}$ over a topological space $X$, the sheafification of $mathscr{F}$, $mathscr{F}^+$, as
$$ mathscr{F}^+(U) = { s!: Uto bigsqcup_{pin U}mathscr{F}_p text{ such that }(*) } $$
with $(*)$ being the following conditions:
- for each $pin U$, $s(p)inmathscr{F}_p$.
- for each $pin U$, there is a neighbourhood $Vsubset U$ of $p$, and some $tinmathscr{F}(V)$, such that for any $qin V$, the germ $t_q = s(q)$.
My question essentially boils down to the following (and I'll expand on this):
What, explicitly, are the elements of $mathscr{F}^+(U)$?
What I mean is that, while the description above is in some ways explicit, it leaves too many details out (at least for me), and so I want some "classification" of the various kinds of sections we can get.
For example, it is clear that any element $sinmathscr{F}^+(U)$ is of the form $s!: pmapsto t_p$, with $t$ not necessarily being fixed. The case where it is fixed, of course, is the image of the natural morphism $theta!:mathscr{F}tomathscr{F}^+$, since this is given by $theta(s)mapsto (pmapsto s_p)$. This gives some of the elements of $mathscr{F}^+(U)$, but not all of them.
As far as I've been able to tell, any section $ainmathscr{F}^+(U)$ will be determined by considering an open cover ${U_i}_{iin I}$ of $U$, with classes of sections $[t^i]subsetmathscr{F}(U_i)$, such that all $sin[t^i]$ agree on germs (i.e. for any $s,vin[t^i]$ and $pin U_i$ we have $s_p = v_p$) and, similarly, for any $sin[t^i]$, $vin[t^j]$, and $pin U_icap U_j$ we have $s_p = v_p$ (that is, the germs agree on intersections).
So, a secondary question is:
Is the above characterization of the sections of $mathscr{F}^+$ correct?
sheaf-theory
$endgroup$
add a comment |
$begingroup$
Hartshorne (in his book on algebraic geometry) constructs, for a presheaf $mathscr{F}$ over a topological space $X$, the sheafification of $mathscr{F}$, $mathscr{F}^+$, as
$$ mathscr{F}^+(U) = { s!: Uto bigsqcup_{pin U}mathscr{F}_p text{ such that }(*) } $$
with $(*)$ being the following conditions:
- for each $pin U$, $s(p)inmathscr{F}_p$.
- for each $pin U$, there is a neighbourhood $Vsubset U$ of $p$, and some $tinmathscr{F}(V)$, such that for any $qin V$, the germ $t_q = s(q)$.
My question essentially boils down to the following (and I'll expand on this):
What, explicitly, are the elements of $mathscr{F}^+(U)$?
What I mean is that, while the description above is in some ways explicit, it leaves too many details out (at least for me), and so I want some "classification" of the various kinds of sections we can get.
For example, it is clear that any element $sinmathscr{F}^+(U)$ is of the form $s!: pmapsto t_p$, with $t$ not necessarily being fixed. The case where it is fixed, of course, is the image of the natural morphism $theta!:mathscr{F}tomathscr{F}^+$, since this is given by $theta(s)mapsto (pmapsto s_p)$. This gives some of the elements of $mathscr{F}^+(U)$, but not all of them.
As far as I've been able to tell, any section $ainmathscr{F}^+(U)$ will be determined by considering an open cover ${U_i}_{iin I}$ of $U$, with classes of sections $[t^i]subsetmathscr{F}(U_i)$, such that all $sin[t^i]$ agree on germs (i.e. for any $s,vin[t^i]$ and $pin U_i$ we have $s_p = v_p$) and, similarly, for any $sin[t^i]$, $vin[t^j]$, and $pin U_icap U_j$ we have $s_p = v_p$ (that is, the germs agree on intersections).
So, a secondary question is:
Is the above characterization of the sections of $mathscr{F}^+$ correct?
sheaf-theory
$endgroup$
$begingroup$
Well, you would need to mod out by the equivalence relation that equates the "gluing" of $s_i in mathscr{F}(U_i)$ and of $t_j in mathscr{F}(V_j)$ if and only if for each $i,j$, there is an open cover $W_k$ of $U_i cap V_j$ such that $s_i |_{W_k} = t_j |_{W_k}$ for all $k$. Otherwise, that does form an alternate description of the sheafification.
$endgroup$
– Daniel Schepler
Dec 4 '18 at 23:01
$begingroup$
Yeah, I can see why that would be required to make $mathscr{F}^+(U)$ and $mathscr{F}^+(V)$ actually "comparable" when $Ucap Vnot=emptyset$. Thanks!
$endgroup$
– Carl-Fredrik Lidgren
Dec 4 '18 at 23:07
add a comment |
$begingroup$
Hartshorne (in his book on algebraic geometry) constructs, for a presheaf $mathscr{F}$ over a topological space $X$, the sheafification of $mathscr{F}$, $mathscr{F}^+$, as
$$ mathscr{F}^+(U) = { s!: Uto bigsqcup_{pin U}mathscr{F}_p text{ such that }(*) } $$
with $(*)$ being the following conditions:
- for each $pin U$, $s(p)inmathscr{F}_p$.
- for each $pin U$, there is a neighbourhood $Vsubset U$ of $p$, and some $tinmathscr{F}(V)$, such that for any $qin V$, the germ $t_q = s(q)$.
My question essentially boils down to the following (and I'll expand on this):
What, explicitly, are the elements of $mathscr{F}^+(U)$?
What I mean is that, while the description above is in some ways explicit, it leaves too many details out (at least for me), and so I want some "classification" of the various kinds of sections we can get.
For example, it is clear that any element $sinmathscr{F}^+(U)$ is of the form $s!: pmapsto t_p$, with $t$ not necessarily being fixed. The case where it is fixed, of course, is the image of the natural morphism $theta!:mathscr{F}tomathscr{F}^+$, since this is given by $theta(s)mapsto (pmapsto s_p)$. This gives some of the elements of $mathscr{F}^+(U)$, but not all of them.
As far as I've been able to tell, any section $ainmathscr{F}^+(U)$ will be determined by considering an open cover ${U_i}_{iin I}$ of $U$, with classes of sections $[t^i]subsetmathscr{F}(U_i)$, such that all $sin[t^i]$ agree on germs (i.e. for any $s,vin[t^i]$ and $pin U_i$ we have $s_p = v_p$) and, similarly, for any $sin[t^i]$, $vin[t^j]$, and $pin U_icap U_j$ we have $s_p = v_p$ (that is, the germs agree on intersections).
So, a secondary question is:
Is the above characterization of the sections of $mathscr{F}^+$ correct?
sheaf-theory
$endgroup$
Hartshorne (in his book on algebraic geometry) constructs, for a presheaf $mathscr{F}$ over a topological space $X$, the sheafification of $mathscr{F}$, $mathscr{F}^+$, as
$$ mathscr{F}^+(U) = { s!: Uto bigsqcup_{pin U}mathscr{F}_p text{ such that }(*) } $$
with $(*)$ being the following conditions:
- for each $pin U$, $s(p)inmathscr{F}_p$.
- for each $pin U$, there is a neighbourhood $Vsubset U$ of $p$, and some $tinmathscr{F}(V)$, such that for any $qin V$, the germ $t_q = s(q)$.
My question essentially boils down to the following (and I'll expand on this):
What, explicitly, are the elements of $mathscr{F}^+(U)$?
What I mean is that, while the description above is in some ways explicit, it leaves too many details out (at least for me), and so I want some "classification" of the various kinds of sections we can get.
For example, it is clear that any element $sinmathscr{F}^+(U)$ is of the form $s!: pmapsto t_p$, with $t$ not necessarily being fixed. The case where it is fixed, of course, is the image of the natural morphism $theta!:mathscr{F}tomathscr{F}^+$, since this is given by $theta(s)mapsto (pmapsto s_p)$. This gives some of the elements of $mathscr{F}^+(U)$, but not all of them.
As far as I've been able to tell, any section $ainmathscr{F}^+(U)$ will be determined by considering an open cover ${U_i}_{iin I}$ of $U$, with classes of sections $[t^i]subsetmathscr{F}(U_i)$, such that all $sin[t^i]$ agree on germs (i.e. for any $s,vin[t^i]$ and $pin U_i$ we have $s_p = v_p$) and, similarly, for any $sin[t^i]$, $vin[t^j]$, and $pin U_icap U_j$ we have $s_p = v_p$ (that is, the germs agree on intersections).
So, a secondary question is:
Is the above characterization of the sections of $mathscr{F}^+$ correct?
sheaf-theory
sheaf-theory
asked Dec 4 '18 at 22:51
Carl-Fredrik LidgrenCarl-Fredrik Lidgren
106
106
$begingroup$
Well, you would need to mod out by the equivalence relation that equates the "gluing" of $s_i in mathscr{F}(U_i)$ and of $t_j in mathscr{F}(V_j)$ if and only if for each $i,j$, there is an open cover $W_k$ of $U_i cap V_j$ such that $s_i |_{W_k} = t_j |_{W_k}$ for all $k$. Otherwise, that does form an alternate description of the sheafification.
$endgroup$
– Daniel Schepler
Dec 4 '18 at 23:01
$begingroup$
Yeah, I can see why that would be required to make $mathscr{F}^+(U)$ and $mathscr{F}^+(V)$ actually "comparable" when $Ucap Vnot=emptyset$. Thanks!
$endgroup$
– Carl-Fredrik Lidgren
Dec 4 '18 at 23:07
add a comment |
$begingroup$
Well, you would need to mod out by the equivalence relation that equates the "gluing" of $s_i in mathscr{F}(U_i)$ and of $t_j in mathscr{F}(V_j)$ if and only if for each $i,j$, there is an open cover $W_k$ of $U_i cap V_j$ such that $s_i |_{W_k} = t_j |_{W_k}$ for all $k$. Otherwise, that does form an alternate description of the sheafification.
$endgroup$
– Daniel Schepler
Dec 4 '18 at 23:01
$begingroup$
Yeah, I can see why that would be required to make $mathscr{F}^+(U)$ and $mathscr{F}^+(V)$ actually "comparable" when $Ucap Vnot=emptyset$. Thanks!
$endgroup$
– Carl-Fredrik Lidgren
Dec 4 '18 at 23:07
$begingroup$
Well, you would need to mod out by the equivalence relation that equates the "gluing" of $s_i in mathscr{F}(U_i)$ and of $t_j in mathscr{F}(V_j)$ if and only if for each $i,j$, there is an open cover $W_k$ of $U_i cap V_j$ such that $s_i |_{W_k} = t_j |_{W_k}$ for all $k$. Otherwise, that does form an alternate description of the sheafification.
$endgroup$
– Daniel Schepler
Dec 4 '18 at 23:01
$begingroup$
Well, you would need to mod out by the equivalence relation that equates the "gluing" of $s_i in mathscr{F}(U_i)$ and of $t_j in mathscr{F}(V_j)$ if and only if for each $i,j$, there is an open cover $W_k$ of $U_i cap V_j$ such that $s_i |_{W_k} = t_j |_{W_k}$ for all $k$. Otherwise, that does form an alternate description of the sheafification.
$endgroup$
– Daniel Schepler
Dec 4 '18 at 23:01
$begingroup$
Yeah, I can see why that would be required to make $mathscr{F}^+(U)$ and $mathscr{F}^+(V)$ actually "comparable" when $Ucap Vnot=emptyset$. Thanks!
$endgroup$
– Carl-Fredrik Lidgren
Dec 4 '18 at 23:07
$begingroup$
Yeah, I can see why that would be required to make $mathscr{F}^+(U)$ and $mathscr{F}^+(V)$ actually "comparable" when $Ucap Vnot=emptyset$. Thanks!
$endgroup$
– Carl-Fredrik Lidgren
Dec 4 '18 at 23:07
add a comment |
1 Answer
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active
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$begingroup$
So, basically the sections of the associated sheaf are functions which are locally sections of the presheaf. Maybe a clear way to see this is the following example: take a smooth manifold $M$ and define a presheaf $mathscr{F}$ by $mathscr{F}(U)={f:Uto mathbb{R}:f:text{is constant}}$. If $M$ is not connected, then write $M_1sqcup M_2$ for instance, both clopen. We can write a function $f_1:M_1to mathbb{R}$ by $f_1(x)=1$ and $f_2:M_2to mathbb{R}$ by $f_2(x)=2$. Because $M_1cap M_2=varnothing$, if $mathscr{F}$ were a sheaf, then the piecewise function $f:Mto mathbb{R}$ given by
$$f(x)= begin{cases}
f_1(x)&xin M_1\
f_2(x)& xin M_2.
end{cases}$$
would have to be in the sheaf. However, it is not because $f$ is not constant globally, only locally. Once we pass to the sheafification, $mathscr{F}^+$, our sections on $Usubseteq M$ are
$$ mathscr{F}^+(U)={f:Uto mathbb{R}:(1):&:(2)}.$$
$(1)$ is the property that $f(P)in mathscr{F}_P$. $mathscr{F}_P=mathbb{R}$ so this is fine.
$(2)$ is the property that around each $Pin U$, there exists a neighborhood $Vsubseteq U$ with $f|_V(P)=s_P$ for $sin mathscr{F}(U)$. That is, $f$ is locally a member of $mathscr{F}$. That is, $f$ is locally constant.
So, the sections of $mathscr{F}^+(U)$ are locally constant functions valued in $mathbb{R}$. Morally, a sheaf is supposed to be an object that locally assigns functions on the space, and the sheafification formalism says that indeed this is the case.
$endgroup$
add a comment |
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$begingroup$
So, basically the sections of the associated sheaf are functions which are locally sections of the presheaf. Maybe a clear way to see this is the following example: take a smooth manifold $M$ and define a presheaf $mathscr{F}$ by $mathscr{F}(U)={f:Uto mathbb{R}:f:text{is constant}}$. If $M$ is not connected, then write $M_1sqcup M_2$ for instance, both clopen. We can write a function $f_1:M_1to mathbb{R}$ by $f_1(x)=1$ and $f_2:M_2to mathbb{R}$ by $f_2(x)=2$. Because $M_1cap M_2=varnothing$, if $mathscr{F}$ were a sheaf, then the piecewise function $f:Mto mathbb{R}$ given by
$$f(x)= begin{cases}
f_1(x)&xin M_1\
f_2(x)& xin M_2.
end{cases}$$
would have to be in the sheaf. However, it is not because $f$ is not constant globally, only locally. Once we pass to the sheafification, $mathscr{F}^+$, our sections on $Usubseteq M$ are
$$ mathscr{F}^+(U)={f:Uto mathbb{R}:(1):&:(2)}.$$
$(1)$ is the property that $f(P)in mathscr{F}_P$. $mathscr{F}_P=mathbb{R}$ so this is fine.
$(2)$ is the property that around each $Pin U$, there exists a neighborhood $Vsubseteq U$ with $f|_V(P)=s_P$ for $sin mathscr{F}(U)$. That is, $f$ is locally a member of $mathscr{F}$. That is, $f$ is locally constant.
So, the sections of $mathscr{F}^+(U)$ are locally constant functions valued in $mathbb{R}$. Morally, a sheaf is supposed to be an object that locally assigns functions on the space, and the sheafification formalism says that indeed this is the case.
$endgroup$
add a comment |
$begingroup$
So, basically the sections of the associated sheaf are functions which are locally sections of the presheaf. Maybe a clear way to see this is the following example: take a smooth manifold $M$ and define a presheaf $mathscr{F}$ by $mathscr{F}(U)={f:Uto mathbb{R}:f:text{is constant}}$. If $M$ is not connected, then write $M_1sqcup M_2$ for instance, both clopen. We can write a function $f_1:M_1to mathbb{R}$ by $f_1(x)=1$ and $f_2:M_2to mathbb{R}$ by $f_2(x)=2$. Because $M_1cap M_2=varnothing$, if $mathscr{F}$ were a sheaf, then the piecewise function $f:Mto mathbb{R}$ given by
$$f(x)= begin{cases}
f_1(x)&xin M_1\
f_2(x)& xin M_2.
end{cases}$$
would have to be in the sheaf. However, it is not because $f$ is not constant globally, only locally. Once we pass to the sheafification, $mathscr{F}^+$, our sections on $Usubseteq M$ are
$$ mathscr{F}^+(U)={f:Uto mathbb{R}:(1):&:(2)}.$$
$(1)$ is the property that $f(P)in mathscr{F}_P$. $mathscr{F}_P=mathbb{R}$ so this is fine.
$(2)$ is the property that around each $Pin U$, there exists a neighborhood $Vsubseteq U$ with $f|_V(P)=s_P$ for $sin mathscr{F}(U)$. That is, $f$ is locally a member of $mathscr{F}$. That is, $f$ is locally constant.
So, the sections of $mathscr{F}^+(U)$ are locally constant functions valued in $mathbb{R}$. Morally, a sheaf is supposed to be an object that locally assigns functions on the space, and the sheafification formalism says that indeed this is the case.
$endgroup$
add a comment |
$begingroup$
So, basically the sections of the associated sheaf are functions which are locally sections of the presheaf. Maybe a clear way to see this is the following example: take a smooth manifold $M$ and define a presheaf $mathscr{F}$ by $mathscr{F}(U)={f:Uto mathbb{R}:f:text{is constant}}$. If $M$ is not connected, then write $M_1sqcup M_2$ for instance, both clopen. We can write a function $f_1:M_1to mathbb{R}$ by $f_1(x)=1$ and $f_2:M_2to mathbb{R}$ by $f_2(x)=2$. Because $M_1cap M_2=varnothing$, if $mathscr{F}$ were a sheaf, then the piecewise function $f:Mto mathbb{R}$ given by
$$f(x)= begin{cases}
f_1(x)&xin M_1\
f_2(x)& xin M_2.
end{cases}$$
would have to be in the sheaf. However, it is not because $f$ is not constant globally, only locally. Once we pass to the sheafification, $mathscr{F}^+$, our sections on $Usubseteq M$ are
$$ mathscr{F}^+(U)={f:Uto mathbb{R}:(1):&:(2)}.$$
$(1)$ is the property that $f(P)in mathscr{F}_P$. $mathscr{F}_P=mathbb{R}$ so this is fine.
$(2)$ is the property that around each $Pin U$, there exists a neighborhood $Vsubseteq U$ with $f|_V(P)=s_P$ for $sin mathscr{F}(U)$. That is, $f$ is locally a member of $mathscr{F}$. That is, $f$ is locally constant.
So, the sections of $mathscr{F}^+(U)$ are locally constant functions valued in $mathbb{R}$. Morally, a sheaf is supposed to be an object that locally assigns functions on the space, and the sheafification formalism says that indeed this is the case.
$endgroup$
So, basically the sections of the associated sheaf are functions which are locally sections of the presheaf. Maybe a clear way to see this is the following example: take a smooth manifold $M$ and define a presheaf $mathscr{F}$ by $mathscr{F}(U)={f:Uto mathbb{R}:f:text{is constant}}$. If $M$ is not connected, then write $M_1sqcup M_2$ for instance, both clopen. We can write a function $f_1:M_1to mathbb{R}$ by $f_1(x)=1$ and $f_2:M_2to mathbb{R}$ by $f_2(x)=2$. Because $M_1cap M_2=varnothing$, if $mathscr{F}$ were a sheaf, then the piecewise function $f:Mto mathbb{R}$ given by
$$f(x)= begin{cases}
f_1(x)&xin M_1\
f_2(x)& xin M_2.
end{cases}$$
would have to be in the sheaf. However, it is not because $f$ is not constant globally, only locally. Once we pass to the sheafification, $mathscr{F}^+$, our sections on $Usubseteq M$ are
$$ mathscr{F}^+(U)={f:Uto mathbb{R}:(1):&:(2)}.$$
$(1)$ is the property that $f(P)in mathscr{F}_P$. $mathscr{F}_P=mathbb{R}$ so this is fine.
$(2)$ is the property that around each $Pin U$, there exists a neighborhood $Vsubseteq U$ with $f|_V(P)=s_P$ for $sin mathscr{F}(U)$. That is, $f$ is locally a member of $mathscr{F}$. That is, $f$ is locally constant.
So, the sections of $mathscr{F}^+(U)$ are locally constant functions valued in $mathbb{R}$. Morally, a sheaf is supposed to be an object that locally assigns functions on the space, and the sheafification formalism says that indeed this is the case.
answered Dec 5 '18 at 0:25
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
10.5k41641
10.5k41641
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$begingroup$
Well, you would need to mod out by the equivalence relation that equates the "gluing" of $s_i in mathscr{F}(U_i)$ and of $t_j in mathscr{F}(V_j)$ if and only if for each $i,j$, there is an open cover $W_k$ of $U_i cap V_j$ such that $s_i |_{W_k} = t_j |_{W_k}$ for all $k$. Otherwise, that does form an alternate description of the sheafification.
$endgroup$
– Daniel Schepler
Dec 4 '18 at 23:01
$begingroup$
Yeah, I can see why that would be required to make $mathscr{F}^+(U)$ and $mathscr{F}^+(V)$ actually "comparable" when $Ucap Vnot=emptyset$. Thanks!
$endgroup$
– Carl-Fredrik Lidgren
Dec 4 '18 at 23:07