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<title>Generation in prime characteristic/a GUT for flops and derived categories</title>
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<div class="slides">
<section>
<section>
<h4>Generation in prime characteristic</h4>
<p>
<small><a href="mattrobball.com">Matthew Robert Ballard</a> <br>
Derived Categories, Arithmetic, and
Reconstruction in Algebraic Geometry <br>
Banff International Research Station <br>
July 7 2022
</small>
</p>
</section>
<section>
<img src="https://patlank.com/assets/profile.jpg" style="height:250px">
<p>Joint with <a href="patlank.com">Pat Lank</a><br>
Supported by the <a href="https://www.simonsfoundation.org">Simons Foundation</a></p>
</section>
<section>
<p>
Given an algebraic structure, one of the first steps to understanding
it is to find generators.
</p>
</section>
<section>
<p>
For a group $G$, we look for a set $ \lbrace g_a \mid a \in A \rbrace $
of elements such that for any $ x \in G $ we have
\[
x = g_{a_1}^{n_1} \cdots g_{a_l}^{n_l}
\]
for some $ n_i \in \mathbb{Z}$.
</p>
</section>
<section>
<p>
For a module $ M $ over a ring $ R $, we seek a surjection
\[
R^{\oplus A} \to M
\]
</p>
</section>
<section>
<p>
For a triangulated category $ T $, our generative operation is forming
cones
\[
X \overset{f}{\to} Y \to C(f) \to X[1]
\]
</p>
</section>
<section>
<p>
<b> Definition </b>. We say that a (full) subcategory $ S $ generates
$ T $ if the smallest subcategory of $ T $
<ul>
<li>closed under finite sums, shifts, and summands </li>
<li>closed under triangles and </li>
<li>containing $ S $ </li>
</ul> <br>
is $ T $ itself.
</p>
</section>
<section>
<p> We write $ \langle S \rangle $ for this smallest subcategory in
general and say
\[
\langle S \rangle = T
\]
if $ S $ generates.
</p>
</section>
<section>
<p> We have
$ \langle R \rangle = \operatorname{perf} R
$,
the subcategory of bounded complexes with finitely-generated projective
components.
</p>
<p>
If we are Noetherian (and commutative), $ R $ generates $ D^b(\operatorname{mod} R) $
if and only if $ R $ is regular by Auslander-Buchsbaum.
</p>
</section>
<section>
<p><b>Challenge</b>: I hand you $ R $. You hand back a generator for
$ D^b(\operatorname{mod} R) $.
</p>
</section>
<section>
<p>Some examples to meditate upon:
<ul>
<li> $ k[x]/(x^2) $ </li>
<li> $ k[x,y,z]/(xy,xz) $ </li>
</p>
</section>
<section>
<p> Assume that $ R $ lives over a field $ k $ with $\operatorname{char} k > 0 $. </p>
<p><b>Corollary</b>: If $ R $ is Noetherian and $ F_\ast R $ is
finitely-generated, then for $ e \gg 0 $, $F_\ast^e R$ generates
$ D^b(\operatorname{mod} R )$.
</p>
</section>
<section>
<p> This doesn't work globally. <br>
If $ C $ is smooth and proper with $ g(C) > 0 $, then $F_\ast^e \mathcal O_C $
does not generate for any $ e$.
</p>
</section>
<section>
<p> Two natural questions:
<ol>
<li> For what $ X $ does $ F_\ast^e \mathcal O_X $ generate? </li>
<li> Does $ F_\ast^e \operatorname{perf} X $ generate? </li>
</ol>
</p>
</section>
<section>
<p> Regarding Question 1, we have a guess.
</p>
<p><b>Conjecture</b>: For $ X $ smooth and projective, $F_\ast^e \mathcal O_X $
generates if and only if $ D^b(\operatorname{coh} X) $ possesses a full
exceptional collection.
</p>
</section>
<section>
<p> Question 2 has an answer.
</p>
<p><b>Theorem</b>: (MRB-Lank) For $ e \gg 0 $, $ F_\ast^e \operatorname{perf} X $
generates $D^b(\operatorname{coh} X) $.
</p>
</section>
<section>
<p> This has the marvelous by-product that $ F_\ast^e $ converts weak generators into
strong ones. <br>
If $ X $ is not regular, then any perfect generator $ G $ takes an unbounded number of
cones to generate all of $ \operatorname{perf} X $. <br>
However, $ F_\ast^e G $ will only need a uniformly bounded number of cones to generate
$ D^b(\operatorname{coh} X )$.
</p>
</section>
<section>
<p><b>Corollary</b>: If $ \mathcal L $ is a very ample line bundle on $ X $, then
\[
\bigoplus_{i=0}^{\dim X} F_\ast^e \mathcal L^i
\]
is a strong generator for $ D^b(\operatorname{coh} X )$.
</p>
</section>
<section>
<p>Relations to other works:
<ul>
<li>One can find generators of $D^b(\operatorname{coh} X) $ pretty generally:
Rouquier, Iyengar-Takahashi, Neeman,... but not as constructively.</li>
<li> Independently Iyengar-Pollitz-Mukhopadhyay recently announced the affine case.
They also say that you can take $e =1 $ for complete intersections. </li>
</ul>
</p>
</section>
</section>
<section>
<section>
<h4>A GUT for flops and derived categories</h4>
<p>
<small><a href="mattrobball.com">Matthew Robert Ballard</a> <br>
Derived Categories, Arithmetic, and Reconstruction in Algebraic Geometry<br>
Banff International Research Station <br>
July 7 2022
</small>
</p>
</section>
<section>
<img src="https://guests.mpim-bonn.mpg.de/kcnitin/img/mockup/pic.jpg" style="height:250px"
alt="Picture of Nitin Chidambaram">    
<img src="https://sites.ualberta.ca/~favero/images/Home%20Page%20Photo.png"
style="height:250px" alt="Picture of David Favero">
<p> Joint with <a href="https://guests.mpim-bonn.mpg.de/kcnitin/">Nitin Chidambaram</a>
and <a href="https://sites.ualberta.ca/~favero/">David Favero</a>.<br>
Supported by the <a href="https://www.simonsfoundation.org">Simons Foundation</a>
and <a href="https://www.nserc-crsng.gc.ca">NSERC</a>.</p>
</p>
</section>
<section>
<p>Recall that a flip relative to a divisor $ D $ is a diagram
\[
\begin{CD}
@. X \\
@. @VV f V \\
X^+ @>> f^+ > Y
\end{CD}
\]
inducing an isomorphisms in codimension $\geq 1$ and such that $-D$ is $f$-ample
and $D$ is $f^+$-ample.
</p>
<p> A flip is a flop if $f^\ast K_X = (f^+)^\ast K_{X^+}$.
</section>
<section>
<p>
Geometric Invariant Theory provides a natural source of flips. For example, given
a $\mathbb{Z}$-graded ring $R$ we get a flip (under some mild assumptions) where
\[
X = \operatorname{Proj}~ \bigoplus_{n \leq 0} R_n \\
X^+ = \operatorname{Proj}~ \bigoplus_{n \geq 0} R_n \\
Y = \operatorname{Spec} R_0
\]
</p>
</section>
<section>
<p> Groups with higher rank often provide more intricate examples. For example,
let $k$ be a field and $V,W$ be finite-dimensional vector spaces over $k$ and
assume that $\dim V < \dim W$.</p>
</section>
<section>
<p>
Set $ G = \operatorname{Aut}(V) $.
For $A \in \operatorname{Hom}(W,V)$ and $B \in \operatorname{Hom}(V,W)$ set
\[
Z = \lbrace (A,B) \mid AB = 0 \rbrace
\]
We get two GIT quotients $Z^{ss}(+)/G$ and $Z^{ss}(-)/G$ related by a flop.
</p>
</section>
<section>
<p>
For $2 \leq \dim V \leq \dim W -2 $, the two GIT quotients of $Z$ give a
stratified Mukai flop.
</p>
</section>
<section>
<p> <b>Conjecture</b>: (Bondal-Orlov) Two smooth projective varieties related by a
flop are derived equivalent.
</p>
<p> This was amplified by Kawamata to a pair of K-equivalent varieties.
</p>
</section>
<section>
<p> For a brief moment in time, it was hoped that the structure of sheaf of the
fiber product
\[
\mathcal O_{X \times_Y X^+}
\]
furnished a Fourier-Mukai kernel.
</p>
</section>
<section>
<p> Kawamata/Namikawa: For stratified Mukai flops with $\dim V=2$, it is not. 💔
</p>
</section>
<section>
<p> However, Cautis-Kamnitzer-Licata produced Fourier-Mukai kernels for stratified
Mukai flops.
</p>
</section>
<section>
<p> MRB-Diemer-Favero described a procedure to associate an integral kernel to
any affine GIT problem given a linear algebraic monoid $M$ with $M^\times = G $.
</p>
</section>
<section>
<p> If $G$ is acting on $Z$ via $\sigma : G \times Z \to Z $, then let
\[
Q = \operatorname{im}\left(k[Z \times Z \times M] \overset{\phi}{\to} k[Z \times G]\right)
\]
where
\[
r \otimes r^\prime \otimes f \overset{\phi}{\mapsto} r \sigma^\ast(r') f
\]
</p>
</section>
<section>
<p> By restricting $Q$ to $Z^{ss}(+) \times Z^{ss}(-)$, we get a integral kernel
\[
P \in D(\operatorname{Qcoh} Z^{ss}(+)/G \times Z^{ss}(-)/G)
\]
</p>
</section>
<section>
<p> <b>Question</b>: When is $\Phi_P$ fully-faithful? </p>
</section>
<section>
<p> For example, if $\dim V = 1$ and $\dim W = n+1$, then we get the graded ring
\[
R = \frac{k[x_0,\ldots,x_n,y_0,\ldots,y_n]}{(x_0y_0+\cdots+x_ny_n)}
\]
the associate flop is called a Mukai flop and $P$ is graph of the induced birational
map -- which is not a FM kernel.
</p>
</section>
<section>
<p> Fix: one needs to derive $R$ </p>
<p> Then we get $\mathcal O_{X \times_Y X^+}$ which is FM here.
</section>
<section>
<p> For general stratified Mukai flop, $R = k[A,B]/(AB) $, we couldn't get our
hands on the representation theory of $Q_{der}$ so we looked at its relative Koszul dual.
</p>
<p>
\[
S = k[A,B,T]
\]
for $T \in \operatorname{End}(V)$.
</p>
</section>
<section>
<p>
An explicit presentation for $Q$ is </p>
<p>
\[
\scriptsize Q = \frac{k[A_L,B_L,T_L,A_R,B_R,T_R,C]}
{(CA_L-A_R,B_L-B_RC,CT_L-T_RC,c_i(T_L)-c_i(T_R))}
\]</p>
<p>
where $ C \in \operatorname{End}(V) $ and $c_i$ are the coefficients of the
characteristic polynomial.
</p>
</section>
<section>
<p> <b>Theorem</b>:(MRB-Chidambaram-Favero) <br> For $\dim V = 2$, $P$ is FM for the
flop associated to $S$.
</p>
</section>
<section>
<p> <b>Corollary</b>: The Koszul dual to $P$ is an FM kernel for the stratified
Mukai flop.
</p>
<p> We know that $\mathcal H^0P^!$ is a CKL kernel. </p>
</section>
<section data-background-image="assets/images/Dock_with_Trees_0.jpg">
<p> AMS Mathematical Research Community <br>
<a href="https://www.ams.org/programs/research-communities/2023MRC-DerivedCategories">
Derived Categories, Arithmetic, and Geometry</a> <br>
June 4-10 2023
</p>
</section>
<section data-background-image="assets/images/Dock_with_Trees_0.jpg">
<h4 style="color:black"> Organizers </h4>
<a href="http://mattrobball.com">
<img src="https://www.matthewrobertballard.com/assets/img/prof_pic.jpg"
style="height:200px"></a>
<a href="https://www.sfu.ca/~khonigs/">
<img src="https://www.sfu.ca/content/sfu/math/people/faculty/khonigs.img.-838256229.png"
style="height:200px"></a>
<a href="http://dkrashen.org">
<img src="https://github.com/dkrashen/dkrashen.github.io/blob/master/images/maxpicofme.jpg?raw=true"
style="height:200px" ></a>
<a href="http://alicialamarche.com">
<img src="http://alicialamarche.com/assets/img/face.JPG" style="height:200px"></a>
<a href="https://www.imo.universite-paris-saclay.fr/~macri">
<img src="https://www.imo.universite-paris-saclay.fr/~macri/emolo.jpg"
style="height:200px"></a>
<br>
<h4> Mentors </h4>
<a href="http://pbelmans.ncag.info">
<img src="https://pbelmans.ncag.info/assets/photo.jpg" style="height:200px"></a>
<a href="<http://www-personal.umich.edu/~arper/">
<img src="http://www-personal.umich.edu/~arper/arp.jpg" style="height:200px"></a>
<a href="http://www.mat.unimi.it/users/pertusi/">
<img src="assets/images/pertusi.jpg" style="height:200px"></a>
<a href="https://pcwww.liv.ac.uk/~arizzard/">
<img src="https://www.liverpool.ac.uk/media/livacuk/maths/images/staffimages/rizzardo.jpg" style="height:200px"></a>
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</section>
<section data-background-image="assets/images/Dock_with_Trees_0.jpg">
<p> Applications open soon!!</a>
</section>
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