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xiao_shi_tou_
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Dabei seit: 12.08.2014
Mitteilungen: 1250
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 Beitrag No.6, eingetragen 2019-12-11 12:08    [Diesen Beitrag zitieren]
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2019-11-29 01:18 - Triceratops in Beitrag No. 5 schreibt:
2019-11-28 23:24 - xiao_shi_tou_ in Beitrag No. 4 schreibt:
Ich dachte an eine Eigenschaft der Art:
Ein morphismus von stacks $\c{F}\to \c{G}$ über einem Situs $\c{C}$ ist eine $\Gm$-Gerbe (oder allgemeiner für eine andere Gruppe), genau dann, wenn die Universelle Eigenschaft... erfüllt ist.

Kennst du irgendein Beispiel dieser Art? Meinetwegen:

Eine Gruppe ist zyklisch, wenn [universelle Eigenschaft] gilt?
Ein Ring ist kommutativ, wenn [universelle Eigenschaft] gilt?

Siehst du, worauf ich hinauswill?

Universelle Eigenschaften beziehen sich auf Konstruktionen (Objekte / Funktoren), nicht auf Eigenschaften.

Ich habe jetzt aber ein anderes interessantes Resultat in einem Paper von V.Lafforge gefunden:
"Every quotient of an algebraic variety by a finite étale group scheme is a Deligne-Mumford stack."

Ist das nicht trivial?
Hallo Triceratops.

Ich habe einen Fehler gemacht.
Die Aussage ist nicht, dass $\cPic_{X/S}$ ein DM-stack ist.
Man bekommt nur einen glatten Atlas $U\to \cPic_{X/S}$, aber keinen $\etale$n.

Meine Fragen hierzu haben sich jetzt weitgehend geklärt :). Vielen Dank für Deine Hilfe!
Viele Grüße


\(\endgroup\)

Triceratops
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Dabei seit: 28.04.2016
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 Beitrag No.5, eingetragen 2019-11-29 01:18    [Diesen Beitrag zitieren]
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2019-11-28 23:24 - xiao_shi_tou_ in Beitrag No. 4 schreibt:
Ich dachte an eine Eigenschaft der Art:
Ein morphismus von stacks $\c{F}\to \c{G}$ über einem Situs $\c{C}$ ist eine $\Gm$-Gerbe (oder allgemeiner für eine andere Gruppe), genau dann, wenn die Universelle Eigenschaft... erfüllt ist.

Kennst du irgendein Beispiel dieser Art? Meinetwegen:

Eine Gruppe ist zyklisch, wenn [universelle Eigenschaft] gilt?
Ein Ring ist kommutativ, wenn [universelle Eigenschaft] gilt?

Siehst du, worauf ich hinauswill?

Universelle Eigenschaften beziehen sich auf Konstruktionen (Objekte / Funktoren), nicht auf Eigenschaften.

Ich habe jetzt aber ein anderes interessantes Resultat in einem Paper von V.Lafforge gefunden:
"Every quotient of an algebraic variety by a finite étale group scheme is a Deligne-Mumford stack."

Ist das nicht trivial?
\(\endgroup\)

xiao_shi_tou_
Senior
Dabei seit: 12.08.2014
Mitteilungen: 1250
Herkunft: Bonn
 Beitrag No.4, eingetragen 2019-11-28 23:24    [Diesen Beitrag zitieren]
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2019-11-28 19:09 - Triceratops in Beitrag No. 3 schreibt:
2019-11-28 15:44 - xiao_shi_tou_ in Beitrag No. 2 schreibt:
Mich würde noch interessieren, ob man Gerben durch universelle Eigenschaften Charakterisieren kann!?

Kannst du diese Frage präzisieren?

Gerben sind ja Stacks mit einer gewissen Eigenschaft, siehe stacks.math.columbia.edu/tag/06NY . Diese Definition ist aber ganz offensichtlich keine universelle Eigenschaft.
Hi Triceratops.
Ich dachte an eine Eigenschaft der Art:
Ein morphismus von stacks $\c{F}\to \c{G}$ über einem Situs $\c{C}$ ist eine $\Gm$-Gerbe (oder allgemeiner für eine andere Gruppe), genau dann, wenn die Universelle Eigenschaft... erfüllt ist.

Vielleicht meine ich auch etwas wie einen Quotienten stack, aber damit bin ich noch nicht so vertraut. Ich müsste mich noch ein bisschen mehr mit den Begriffen beschäftigen, vielleicht klärt sich meine Frage dann von selbst.

Ich habe jetzt aber ein anderes interessantes Resultat in einem Paper von V.Lafforge gefunden:
"Every quotient of an algebraic variety by a finite étale group scheme is a Deligne-Mumford stack."

Ich glaube, dass aus dieser Aussage zusammen mit der Darstellbarkeit des Picard Funktors folgen könnte, dass der Picard stack ein DM-stack ist. Dem will ich jetzt mal auf den Grund gehen.

Viele Grüße
XST
 

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Triceratops
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 Beitrag No.3, eingetragen 2019-11-28 19:09    [Diesen Beitrag zitieren]

2019-11-28 15:44 - xiao_shi_tou_ in Beitrag No. 2 schreibt:
Mich würde noch interessieren, ob man Gerben durch universelle Eigenschaften Charakterisieren kann!?

Kannst du diese Frage präzisieren?

Gerben sind ja Stacks mit einer gewissen Eigenschaft, siehe stacks.math.columbia.edu/tag/06NY . Diese Definition ist aber ganz offensichtlich keine universelle Eigenschaft.


xiao_shi_tou_
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 Beitrag No.2, eingetragen 2019-11-28 15:44    [Diesen Beitrag zitieren]
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2019-11-27 20:44 - Triceratops in Beitrag No. 1 schreibt:
Ich kenne mich mit dem Thema nicht wirklich gut aus, aber ich schätze, dass
1) die beiden Aussagen nicht inhaltlich voneinander abhängen,
2) die Aussage, dass $\mathcal{Pic}$ ein DM-Stack ist, viel einfacher als die Darstellbarkeit von $\mathrm{Pic}$ zu beweisen ist. Hast du dir die Beweise einmal angesehen?

Hallo Triceratops.
Ja, ich habe mir die Beweise angesehen, aber noch nicht sehr sorgfältig. Der Beweis zur Darstellbarkeit des Picard Funktors ist wirklich nicht so einfach (ich habe den Beweis im Buch über Neron Modelle von Bosch vorliegen). Wenn überhaupt würde also eher aus der Darstellbarkeit des Picard Funktors folgen, dass $\cPic_{X/S}$ ein DM-stack ist, aber ich sehe nicht, wie. Wenn es nicht möglich ist, dann würde ich eben den Beweis in stacks Projekt nehmen.

Ich dachte, dass eine Verbindung bestehen könnte, weil die Voraussetzungen an den Strukturmorphismus $X\to S$ sehr ähnlich sind.
Mich würde noch interessieren, ob man Gerben durch universelle Eigenschaften Charakterisieren kann!? Ich habe bisher nur die übliche Definition wie in stacks project gefunden.

Vielen Dank für die Hilfsbereitschaft
Viele Grüße
XST


 
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Triceratops
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 Beitrag No.1, eingetragen 2019-11-27 20:44    [Diesen Beitrag zitieren]

Ich kenne mich mit dem Thema nicht wirklich gut aus, aber ich schätze, dass
1) die beiden Aussagen nicht inhaltlich voneinander abhängen,
2) die Aussage, dass $\mathcal{Pic}$ ein DM-Stack ist, viel einfacher als die Darstellbarkeit von $\mathrm{Pic}$ zu beweisen ist. Hast du dir die Beweise einmal angesehen?


xiao_shi_tou_
Senior
Dabei seit: 12.08.2014
Mitteilungen: 1250
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 Themenstart: 2019-11-27 18:33    [Diesen Beitrag zitieren]
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Hallo zusammen.
Vor kurzem habe ich die Begriffe "Gerbe" und "Algebraischer Stack (DM-Stack)" kennengelernt und habe nun eine Frage zum Thema.
Es gibt ja den Picard stack $\cPic_{X/S}\colon (\mathbf{Sch}/S)_{fppf}^{op}\to \mathbf{Grpd}$ und den relativen Picard Funktor $\Pic_{X/S}\colon (\mathbf{Sch}/S)_{fppf}^{op}\to \mathbf{Sets}$.

Man kann dann zeigen, dass der Morphismus
$$\cPic_{X/S}\to \Pic_{X/S}$$,
welcher ein Geradenbündel $\lineb$ auf seine Isomorphie Klasse $[\lineb]$ schickt eine $\mathbb{G}_m$-Gerbe ist. Objekte in $\cPic_{X/S}$ haben $\Gm$ als Automorphismengruppe.

Man sieht auch ein, dass der Picard Funktor als $fppf$-Garbifizierung von
$T\mapsto \cPic_{X/S}\mid_T/{\cong}$ gegeben ist.

Der Picard stack selbst kann nicht als Schema darstellbar sein, da es nicht-triviale Automorphismen gibt, doch der relative Picard Funktor ist unter gewissen Voraussetzungen an $X\to S$ bekanntlich durch ein Schema darstellbar (Grothendieck).

Unter passenden Voraussetzungen an $X\to S$ kann man aber zeigen, dass $\cPic_{X/S}$ ein DM-Stack ist, also insbesondere bedeutet das ja, dass jeder Morphismus $T\to \cPic_{X/S}$ von einem Schema $T$ darstellbar ist und dass es einen $\etale n$ surjektiven Atlas $U\to \cPic_{X/S}$ gibt.

Ich verstehe das so, dass eine DM-stack das beste ist was man als Ersatz für ein Schema bekommen kann, wenn der stack nicht darstellbar ist und das der Atlas sozusagen die beste Antwort auf die Frage ist, wie man aus dem Stack ein Schema machen könnte, wobei mir nicht klar ist, ob man in der Praxis mit dem Stack selbst arbeitet (denn die Theorie der algebraischen stacks ist ja sehr ausgereift) oder ob man immer mit dem Atlas arbeitet, beziehungsweise, welche Rolle dieser Atlas spielt?

Mich interessiert nun hauptsächlich, wie die beiden Aussagen:
$\bul$ $\Pic_{X/S}$ ist durch ein Schema darstellbar.
$\bul$ $\cPic_{X/S}$ ist ein DM-stack
miteinander zusammenhängen.
Sind sie äquivalent, oder kann man wenigstens eins aus dem anderen folgern?

Wenn ich zum Beispiel die Darstellbarkeit von $\Pic_{X/S}$ annehme, dann wäre ja eine Frage, was denn dann der Atlas ist.
Klar ist ja schonmal, dass es einen Morphismus $U\to \cPic_{X/S}\to \Pic_{X/S}=h_{\underline{\Pic}_{X/S}}$ geben müsste, also einen Morphismus $U\to \underline{\Pic}_{X/S}$.
Ich habe versucht $\hom(-,\cPic)$ mit $\hom_S(-,\underline{\Pic}_{X/S})$ in Verbindung zu bringen, denn dann könnte man zum Beispiel die Identität in $\Pic_{X/S}(\underline{\Pic}_{X/S})=\hom_S(\underline{\Pic}_{X/S},\underline{\Pic}_{X/S})$ auf einen Morphismus $U\colon=\underline{\Pic}_{X/S}\to \cPic_{X/S}$ schicken, der dann als Atlas fungieren soll.

Das Problem ist, dass mir nicht klar ist, wie $\hom(-,\cPic_{X/S})$ mit $\hom(-,\Pic_{X/S})$ zusammenhängt. Insbesondere ist mir nicht klar in welchen Kategorien ich die Morphismen betrachten sollte, denn man kann ja den Funktor $\Pic_{X/S}$ auch als Stack auffassen. (Gibt es so etwas wie einen linksadjungierten Funktor zum Vergissfunktor (stacks)$\to$ ($fppf$-Garben)?).

Für die kovarianten Funktoren $\hom(\cPic_{X/S},-)$ und $\hom(\underline{\Pic}_{X/S},-)$ würde ich hingegen etwas ähnliches wie $\hom(X/\sim,-)\cong \set{\psi\in \hom(X,-)}{x\sim x'\implies \psi(x)=\psi(x')}$ erwarten, da ja $\Pic_{X/S}$ aus dem stack entsteht indem man die Isomorphismen rausteilt. Aber das würde mich bei meiner Frage glaube ich nicht weiter bringen, falls es überhaupt stimmt.

Einen Beweis, dass $\cPic_{X/S}$ ein algebraischer Stack ist gibt es ja im stacks projekt, aber mir geht es hauptsächlich darum, ob und wie das mit der Darstellbarkeit des Picard Funktors zusammenhängt.

Ich hoffe ich konnte meine Frage verständlich erklären und entschuldige mich dafür, dass sie so lang geworden ist.

Viele Grüße
XST
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