# Math 193a: Algebraic Stacks, Fall 2011

The class meets 9–10am MWF in 257 Sloan. Office hours are Wednesday 2–4pm in 374 Sloan (but moving to the common room for tea at 3:30).

# Course Description

Algebraic stacks arise naturally as solutions to classification (moduli) problems, so it is desirable to understand their geometry. In this course, we will assume a working knowledge of the geometry of schemes. We will extend the definitions and techniques used to study schemes to algebraic spaces and algebraic stacks. Topics will include Grothendieck topologies, descent, algebraic spaces, fibered categories, and algebraic stacks.

Specific topics will be included based on feedback from students.

# Exercises

Try to do the following problems. If you get stuck, come to my office hours. If you're taking the class for a grade, make sure you hand in a substantial fraction of the exercises (or talk to me about doing a project of some sort). I know some of them are very tedious to write up, so you don't need to hand in everything, but please do attempt all of the problems. Also, please don't hand in solutions to problems that were assigned several weeks ago; for concreteness, let's not hand in solutions to problems in Set $n$ any later than week $n+2$.

## Set 1 (due Oct. 14)

1. If you have never done so before, prove Yoneda's Lemma: for any category $\C$, taking an object $X\in \C$ to the functor $h_X\colon \C^{op}\to (Set)$ (defined by $h_X(T)=Hom_\C(T,X)$) defines a fully faithful functor $\C\to Func(\C^{op},(Set))$. [Edit: Actually, I'd like you to show a bit more. Show that for any functor $F:\C^{op}\to (Set)$, we have $Hom(h_X,F)\cong F(X)$.]

2. Play the game "find the representing object" whenever you get the chance. Determine if the following functors are representable. If they are, find the representing object.

• The functor $(Top)\to(Set)$ taking a topological space $X$ to the set of open subsets of $X$.
• The functor $(Top)\to(Set)$ taking a topological space $X$ to the set of closed subsets of $X$.
• The functor $(Top)\to(Set)$ taking a topological space $X$ to the open subsets of $X$ whose complement is also open.
• The functor $GL_n:(CommRing)\to(Set)$ taking a commutative ring $A$ to the set of invertible $n\times n$ matrices with entries in $A$. [Edit: In this case, try to find a ring $R$ so that $Hom(R,A)=GL_n(A)$, rather than $Hom(A,R)$.]
• The functor $Nil:(CommRing)\to(Set)$ taking a commutative ring $A$ to $\{x\in A| x^n=0$ for some integer $n\}$. [Edit: In this case, try to find a ring $R$ so that $Hom(R,A)=Nil(A)$, rather than $Hom(A,R)$.]
• The functor $\AA^n-\{0\}:(Sch)\to(Set)$ taking a scheme $T$ to $\{(f_1,\dots, f_n)\in \O_T(T)^n|$the $f_i$ do not all simultaneously vanish$\}$.
• The functor $(\AA^n-\{0\})/\GG_m:(Sch)\to(Set)$ taking a scheme $T$ to $(\AA^n-\{0\})(T)/\sim$, where $\sim$ is the equivalence relation $(f_1,\dots, f_n)\sim (f_1',\dots, f_n')$ if there is a unit $u\in \O_T(T)$ such that $f_i'=uf_i$ for each $i$.

3. Let $A:\C\to \D$ and $B:\D\to \C$ be functors. Show that an adjunction $Hom(A-,-)\cong \hom(-,B-)$ is equivalent to a choice of natural transformations $\epsilon:id_\D\to BA$ (a unit) and $\eta:AB\to id_\C$ (a counit) such that the compositions $A\xrightarrow{A\epsilon} ABA\xrightarrow{\eta A}A$ and $B\xrightarrow{\epsilon B} BAB\xrightarrow{B\eta} B$ are $id_A$ and $id_B$, respectively.

4. With the notation in the previous problem, show that $A$ is fully faithful (i.e. $Hom(-,-)\to Hom(A-,A-)$ is an isomorphism) if and only if the unit of adjunction $\epsilon$ is an isomorphism. Similarly, show that $B$ is fully faithful if and only if $\eta$ is an isomorphism. (Hint: use Yoneda's Lemma)

## Set 2 (due Oct. 21)

1. Suppose $X$ and $X'$ are hausdorff topological spaces. Let $T$ and $T'$ denote the topoi of $X$ and $X'$, respectively, using the classical topology. Show that every morphism of topoi $T\to T'$ is induced by a continuous map $X\to X'$. (I'm pretty sure this is true, but I haven't done this exercise) [Edit: This follows from statement 4.2.3 in SGA4 Expose IV, but I can't find the proof of that statement ... perhaps your French is better than mine.]

2. (How to pull back representable sheaves) Let $f:\C'\to \C$ be a continuous morphism of sites. Let $Y\in \C'$ be an object, and suppose the functor $h_Y$ is a sheaf. Show that $f^{-1}h_Y\cong h_{f(Y)}$. (Hint: use Yoneda's lemma.)

3. (Non-functoriality of the lisse-étale topos) For a scheme $X$, the lisse-étale site on $X$ is the category of smooth schemes over $X$, where a collection of morphisms over $X$ $\{f_i:U_i\to Y\}$ is said to be a covering if each $f_i$ is étale and the $f_i$ are jointly surjective. We donote the lisse-étale topos of $X$ by $\def\liset{\text{lis-et}}X_\liset$.

Let $\O$ in $\AA^1_\liset$ be given by sending any object $U\to \AA^1$ to $\Gamma(U,\O_U)$ (we will see in class that this is a sheaf). Define a morphism $t\cdot -:\O\to \O$ by multiplication by the coordinate on $\AA^1$ (what does this do on each $U$?). Show that $t\cdot -$ has no kernel.

Let $f:Spec(k)\to \AA^1$ be the inclusion of the origin. We get a continuous morphism of lisse-étale sites $f:\liset(\AA^1)\to\liset(Spec(k))$ given by sending $U\to \AA^1$ to $U\times_{\AA^1}Spec(k)\to Spec(k)$. Show that $f^{-1}:\AA^1_\liset\to Spec(k)_\liset$ takes $t\cdot -$ to a morphism with a non-trivial kernel (Hint: use the previous exercise to compute $f^{-1}\O$). Conclude that $f^{-1}$ does not commute with finite limits.

en français, «lisse»

Bonus: The above problem ($f^{-1}$ not commuting with finite limits) does not occur in the big étale topology. Where does the argument break down? We will see later that the lisse-étale topology has the advantage that $f_*$ usually respects quasi-coherence, a property not enjoyed by the big étale site.

4. Let $\C$ be a site, and let $X$ be an object in $\C$. Recall that the comma category $\C/X$ inherits the structure of a site.

(a) Show that there is an equivalence of categories between $Sh(\C/X)$ and $Sh(\C)/h_X$.
(b) Show that $j^*:Sh(\C)\to Sh(\C)/h_X$, given by $F\mapsto (F\times h_X\xrightarrow{p_2}h_X)$ commutes with finite projective limits and has a right adjoint $j_*$. Therefore, we have a morphism of topoi $Sh(\C)/h_X\to Sh(\C)$.

5. (Facts about representability) Recall that a morphism of sheaves $\phi:F\to G$ is representable if for every object $T\in \C$ and every morphism $T\to G$, the fiber product $T\times_G F$ is in $\C$.

(a) Show that representability is stable under base change.
(b) Show that a composition of representable morphisms is representable.
(c) Suppose $F\xrightarrow\phi G\xrightarrow\psi H$ are morphisms of sheaves, where $\psi$ has representable diagonal. Show that if $\psi\circ\phi$ and $\psi$ are representable, then so is $\phi$. (Hint: use the "property P argument")

## Set 3 (due Oct. 28)

1. Suppose $f:X\to Y$ is a morphism of schemes. If $f$ is surjective as a morphism of schemes, must it be surjective as a morphism of (zariski, étale, smooth, fppf, or fpqc) sheaves? Conversely, if $f$ is surjective as a morphism of sheaves (in one of our topologies), must it be surjective as a morphism of schemes?

2. Suppose $p:\D\to \C$ is a fibered category fibered in groupoids (i.e. for any object $X$ of $\C$, every morphism in $\D(X)$ is an isomorphism). Show that every arrow of $\D$ is cartesian.

3. The "real" definition of a quasi-coherent sheaf on a site is as follows.

Let $\O$ be a sheaf of rings on a site $\C$, and let $F$ be an $\O$-module. We say $F$ is quasi-coherent if for every object $Y$ of $\C$, there is a cover $X\to Y$ so that $F|_{\C/X}$ has a presentation (i.e. $F|_{\C/X}$ is the cokernel of a module morphism $\O^J|_{\C/X}\to \O^I|_{\C/X}$ for some (possibly infinite) sets $I$ and $J$).
• (a) If you have never done so, show that this definition agrees with the other notion of quasi-coherence for the small Zariski topology ($F$ is quasi-coherent if for any open affine $U=Spec(A)$ and any regular function $f\in A$, $F(Spec(A_f))$ is the localization $F(U)_f$).
• (b) Show that the definition of a quasi-coherent big sheaf given in class is the same as the above notion of a quasi-coherent sheaf in the fpqc topology on $Sch$.

4. (Descent for affine morphisms) Suppose $f:X\to Y$ is an fpqc morphism of schemes. Suppose $Z\to X$ is an affine morphism, and there is an isomorphism $Z\times_{X,p_2}(X\times_Y X)\cong Z\times_{X,p_1}(X\times_Y X)$ satisfying the natural cocycle condition. Show that there is an affine morphism $Z_Y\to Y$ so that $Z\cong Z_Y\times_Y X$.

5. (Descent for immersions) Suppose $f:X\to Y$ is an fpqc morphism of schemes. Suppose $U\to X$ is an open immersion such that $U\times_{X,p_1}(X\times_Y X)=U\times_{X,p_2}(X\times_Y X)$. Show that $U$ is the pullback of an open immersion to $Y$. (Hint: consider the closed complement of $U$) Conclude descent for all immersions. (Hint: we showed descent for closed immersions in class)

[Edit: Bonus. (Descent for quasi-affine morphisms) Suppose $f:X\to Y$ is an fpqc morphism of schemes. Suppose $Z\to X$ is a quasi-affine morphism, and there is an isomorphism $Z\times_{X,p_2}(X\times_Y X)\cong Z\times_{X,p_1}(X\times_Y X)$ satisfying the natural cocycle condition. Show that there is a quasi-affine morphism $Z_Y\to Y$ so that $Z\cong Z_Y\times_Y X$. (Hint: use the fact that quasi-affine morphisms have canonical factorizations as open immersions followed by affine morphisms, and that these factorizations commute with flat base change. Specifically, any quasi-affine morphism $f:Z\to X$ factors as $Z\to Spec_X(f_*\O_Z)\to X$, with the first morphism an open immersion.)]

## Set 4 (due Nov. 4)

1. Suppose $\C$ is a site, and $\D$ is a full subcategory of $\C$ which has all finite limits. Let $\D$ have the induced topology (i.e. $\{U_i\to X\}$ is a cover in $\D$ if and only if it is a cover in $\C$). Suppose that every object of $\C$ has a cover by objects in $\D$. Show that the morphism of topoi $Sh(\C)\to Sh(\D)$ induced by the inclusion $\D\to \C$ is an equivalence.

2. Suppose $Spec(A_1)\rightrightarrows Spec(A_0)$ is a finite flat equivalence relation. Let $A=Eq(A_0\rightrightarrows A_1)$. Show that $A_0$ is a finite $A$-module. This was a sticky point in lecture when we were proving that $Spec(A)$ is the quotient of $Spec(A_0)$ by $Spec(A_1)$ in the category of fpqc sheaves.

3. Let $Y$ is a noetherian algebraic space. Let $P$ be a property of algebraic spaces. Suppose that for any closed subspace $Z\subseteq Y$, if every proper closed subspace of $Z$ has $P$, then $Z$ has $P$. Prove that $Y$ has $P$.

## Set 5 (due Nov. 11)

1. Let $Y$ be a locally noetherian algebraic space, and $F$ a quasi-coherent $\O_Y$-module. Suppose there is an étale cover by a scheme $U\to Y$ so that $F|_{zar(U)}$ is coherent. Show that for every étale morphism from a scheme $X\to Y$, $F|_{zar(X)}$ is coherent.

2. Show that for algebraic spaces, formation of pushforward of a quasi-coherent sheaf commutes with flat base change. That is, suppose $f:X\to Y$ is a quasi-compact quasi-separated morphism of algebraic spaces, $F$ is a quasi-coherent sheaf on $X$, $g:Y'\to Y$ is a flat morphism of algebraic spaces, and the following diagram is cartesian: $\begin{matrix} X' & \xrightarrow{g'} & X\\ f'\downarrow\qquad & & \quad\downarrow f\\ Y'& \xrightarrow{g} & Y \end{matrix}$ Show that the natural morphism $g^*f_*F\to f'_*g'^*F$ is an isomorphism. (You may assume this result is true for schemes.)

## Set 6 (due Nov. 18)

1. Let $Y$ be an algebraic space. Show that $\def\spec{\mathcal{Spec}}\spec$ defines an anti-equivalence of categories between the category of quasi-coherent $\O_Y$-algebras and the category of algebraic spaces affine over $Y$.

2. Show that scheme-theoretic closure commutes with etale base change. That is, suppose $f:X\to Y$ is a morphism of algebraic spaces and $Y'\to Y$ is an etale morphism. Let $Z\subseteq Y$ be the scheme-theoretic image of $X$ in $Y$, and let $Z'$ be the scheme-theoretic image of $X\times_Y Y'$ in $Y'$. Show that $Z'\cong Z\times_Y Y'$.

## Set 7 (due Nov. 28)

1. Check the things I asked you to check in lecture (you don't need to write these up, but you should do them):

• For a presheaf $F$ on $\C$, show that $F^{fib}$ is a fibered category.
• Show that $F$ is a separated presheaf (resp. a sheaf) if and only if $F^{fib}$ is a prestack (resp. a stack).
• For a presheaf $F$, check that $F=(F^{fib})^{sh}$. For a category fibered in sets $\D\to \C$, check that $\D\to (\D^{sh})^{fib}$ is an equivalence.
• For good measure, review the proof of the 2-Yoneda Lemma.

2. Show that the fiber product of two stacks (as fibered categories) is a stack.

3. Suppose $\X$ is an algebraic stack with quasi-affine diagonal (e.g. any stack with separated and quasi-finite diagonal). Show that for any two schemes $U$ and $V$ and any morphisms $U\to \X$, $V\to \X$, the fiber product $U\times_\X V$ is a scheme.

## Set 8 (due Dec. 10)

1. Show that there is a correspondence between $GL_r$-torsors on a scheme $X$ and rank $r$ vector bundles on $X$. Specifically, given a $GL_r$-torsor $P$ on $X$, show that the sheaf of sections of $P\times^{GL_r}\O_X^r$ is a rank $r$ vector bundle. Given a rank $r$ vector bundle $\E$, show that $Isom(\E,\O^r)$ is a $GL_r$-torsor. Finally, show that these operations are inverse.

2. (Stackification). Suppose $\C$ is a site. Suppose $\D\to \C$ is a prestack (i.e. for any covering $X\to Y$ in $\C$, the functor $\D(Y)\to \D(X\to Y)$ is fully faithful). Define $\D^+\to \C$ to be the fibered category whose objects are triples $(Y\in \C, \{X\to Y\}\in Cov(Y), F\in \D(X\to Y))$, and whose morphisms consist of a morphism $f:Y\to Y'$, a common refinement $Z\to Y$ of the covers $X\to Y$ and $X\times_{Y'}Y\to Y$, and a descent-data-respecting isomorphism between the restrictions of $F$ and $f^{-1}F'$ to $Z$. Show that if $\E$ is any stack on $\C$, there is an equivalence $HOM_\C(\D,\E)\to HOM_\C(\D^+,\E)$.

## Other exercises

• Suppose $T$ is the topos associated to a site $\C$. Define a collection of morphisms $\{F_i\to F\}$ in $T$ to be a covering if $\coprod F_i\to F$ is an epimorphism. Show that this defines a topology on $T$, and that the category of sheaves on this topology is equivalent to $T$.

# Resources

Jay Daigle is live-TeXing notes for the course. He's posted here here. Thanks Jay!

The stacks project by Johan de Jong et. al.
Algebraic spaces by Donald Knutson
Champs algébriques by Gérard Laumon and Laurent Moret-Bailly
Notes on Grothendieck topologies, fibered categories and descent theory by Angelo Vistoli
my notes from Martin Olsson's course at Berkeley (source available in an svn repo)
Algebraization of Formal Moduli I by Michael Artin

## Bjorn's table

The following table is taken from page 179 of Bjorn Poonen's Rational points on varieties. I copied it because it's slightly more accessible to me as a web page, and so that I can add information to it.

 Definition Composition Base Change fpqc Descent Spreading Out affine EGA II, 1.6.1 EGA II, 1.6.2(ii) EGA II, 1.6.2(iii) EGA IV2, 2.7.1(xiii) EGA IV3, 8.10.5(viii) bijective YES NO EGA IV2, 2.6.1(iv) NO closed EGA I, 2.2.6 EGA I, 2.2.7(i) NO EGA IV2, 2.6.2(ii) closed immersion EGA I, 4.2.1 EGA I, 4.2.5 EGA I, 4.3.2 EGA IV2, 2.7.1(xii) EGA IV3, 8.10.5(iv) dominant EGA I, 2.2.6 EGA I, 2.2.7(i) NO etale EGA IV4, 17.3.1 EGA IV4, 17.3.3(ii) EGA IV4, 17.3.3(ii) EGA IV4, 17.7.4(vi) EGA IV4, 17.7.8(ii) faithfully flat EGA I, 0:6.7.8 YES YES YES finite EGA II, 6.1.1 EGA II, 6.1.5(ii) EGA II, 6.1.5(iii) EGA IV2, 2.7.1(xv) EGA IV3, 8.10.5(x) finite presentation EGA IV1, 1.6.1 EGA IV1, 1.6.2(ii) EGA IV1, 1.6.2(iii) EGA IV2, 2.7.1(vi) finite type EGA I, 6.3.1 EGA I, 6.3.4(ii) EGA I, 6.3.4(iv) EGA IV2, 2.7.1(v) flat EGA I, 0:6.7.1 EGA IV2, 2.1.6 EGA IV2, 2.1.4 EGA IV2, 2.2.11(iv) formally etale EGA IV4, 17.1.1 EGA IV4, 17.1.3(ii) EGA IV4, 17.1.3(iii) formally smooth EGA IV4, 17.1.1 EGA IV4, 17.1.3(ii) EGA IV4, 17.1.3(iii) formally unram. EGA IV4, 17.1.1 EGA IV4, 17.1.3(ii) EGA IV4, 17.1.3(iii) fppf Definition 3.4.1 YES YES YES fpqc Vis05, 2.34 Vis05, 2.35(i) Vis05, 2.35(v) good moduli space Alp08 4.1 YES Alp08 4.7(i) Alp08 4.7(ii) homeomorphism YES NO EGA IV2, 2.6.2(iv) immersion EGA I, 4.2.1 EGA I, 4.2.5 EGA I, 4.3.2 EGA IV3, 8.10.5(iii) injective EGA I, 3.5.11 YES NO EGA IV2, 2.6.1(ii) NO isomorphism EGA I, 2.2.2 YES YES EGA IV2, 2.7.1(viii) EGA IV3, 8.10.5(i) loc. immersion EGA I, 4.5.1 EGA I, 4.5.5(i) EGA I, 4.5.5(iii) YES loc. isomorphism EGA I, 4.5.2 EGA I, 4.5.5(i) EGA I, 4.5.5(iii) loc. of finite pres. EGA IV1, 1.4.2 EGA IV1, 1.4.2(ii) EGA IV1, 1.4.2(iii) EGA IV2, 2.7.1(iv) loc. of finite type EGA I, 6.6.2 EGA I, 6.6.6(ii) EGA I, 6.6.6(iii) EGA IV2, 2.7.1(iii) monomorphism EGA I, 0:4.1.1 YES YES EGA IV2, 2.7.1(ix) EGA IV3, 8.10.5(ii) open EGA I, 2.2.6 EGA I, 2.2.7(i) NO EGA IV2, 2.6.2(i) open immersion EGA I, 4.2.1 EGA I, 4.2.5 EGA I, 4.3.2 EGA IV2, 2.7.1(x) EGA IV3, 8.10.5(iii) projective EGA II, 5.5.2 EGA II, 5.5.5(ii). EGA II, 5.5.5(iii) NO. EGA IV3, 8.10.5(xiii) proper EGA II, 5.4.1 EGA II, 5.4.2(ii) EGA II, 5.4.2(iii) EGA IV2, 2.7.1(vii) EGA IV3, 8.10.5(xii) quasi-affine EGA II, 5.1.1 EGA II, 5.1.10(ii) EGA II, 5.1.10(iii) EGA IV2, 2.7.1(xiv) EGA IV3, 8.10.5(ix) quasi-compact EGA I, 6.6.1 EGA I, 6.6.4(ii) EGA I, 6.6.4(iii) EGA IV2, 2.6.4(v) quasi-finite EGA II, 6.2.3 EGA II, 6.2.4(ii) EGA II, 6.2.4(iii) EGA IV2, 2.7.1(xvi) EGA IV3, 8.10.5(xi) quasi-projective EGA II, 5.3.1 EGA II, 5.3.4(ii) EGA II, 5.3.4(iii) NO EGA IV3, 8.10.5(xiv) quasi-separated EGA IV1, 1.2.1 EGA IV1, 1.2.2(ii) EGA IV1, 1.2.2(iii) EGA IV2, 2.7.1(ii) radicial EGA I, 3.5.4 EGA I, 3.5.6(i) EGA I, 3.5.7(ii) EGA IV2, 2.6.1(v) EGA IV3, 8.10.5(vii) sch.-th. dominant EGA IV3, 11.10.2 YES NO EGA IV3, 11.10.5(i) separated EGA I, 5.4.1 EGA I, 5.5.1(ii) EGA I, 5.5.1(iv) EGA IV2, 2.7.1(i) EGA IV3, 8.10.5(v) smooth EGA IV4, 17.3.1 EGA IV4, 17.3.3(ii) EGA IV4, 17.3.3(iii) EGA IV4, 17.7.4(v) EGA IV4, 17.7.8(ii) surjective EGA I, 2.2.6 EGA I, 2.2.7(i) EGA I, 3.5.2(ii) EGA IV2, 2.6.1(i) EGA IV3, 8.10.5(vi) univ. bicontinuous EGA IV2, 2.4.2 EGA IV2, 2.4.3(ii) EGA IV2, 2.4.3(iii) univ. closed EGA II, 5.4.9 EGA IV2, 2.4.3(ii) EGA IV2, 2.4.3(iii) EGA IV2, 2.6.4(ii) univ. homeom. EGA IV2, 2.4.2 EGA IV2, 2.4.3(ii) EGA IV2, 2.4.3(iii) EGA IV2, 2.6.4(iv) univ. open EGA IV2, 2.4.2 EGA IV2, 2.4.3(ii) EGA IV2, 2.4.3(iii) EGA IV2, 2.6.4(i) unramified EGA IV4, 17.3.1 EGA IV4, 17.3.3(ii) EGA IV4, 17.3.3(iii) EGA IV4, 17.7.4(iv) EGA IV4, 17.7.8(ii)