16
$\begingroup$

I'm looking for an explanation or a reference to why there is this equivelence of triangulated categories: $${D}^b(\mathrm {Coh}(\Bbb P^1))\simeq {D}^b(\mathrm {Rep}(\bullet\rightrightarrows \bullet))$$ It is my understanding that the only reason why $\Bbb P^1$ appears at all is because it is used to index the regular irreducible representations of the Kronecker quiver. I have also heard that this equivalence can be used to understand the geometry of $\Bbb P^1$.

I suppose more generally, adding arrows to the Kronecker quiver gives a similar result for $\Bbb P^n$.

$\endgroup$
1
20
$\begingroup$

Let $\mathcal O$ be the structure sheaf of $\mathbb P^1$. Then $\mathcal O \oplus \mathcal O(1)$ is rigid and generates the derived category of coherent sheaves on $\mathbb P^1$. Thus, it is a tilting object, and so the derived category is equivalent to the category of modules over its endomorphism ring, which is the path algebra of the Kronecker quiver.

For $\mathbb P^n$, you need $n+1$ objects to generate the derived category. You can take $\mathcal O(i)$ for $0\leq i \leq n$; this will be a tilting object again, but its endomorphism ring will not be hereditary; you will get a quiver with relations. This shouldn't be surprising, though: path algebras of quivers have global dimension one, so you shouldn't expect their derived categories to agree with derived categories of sheaves on higher-dimensional varieties.

$\endgroup$
10
$\begingroup$

The original reference for the general result is

A.A. Beilinson, Coherent sheaves on $P^n$ and problems of linear algebra, Func. Anal. Appl. 12 (1978), pp. 214-216.

Google search brings many related papers and lecture notes, e.g. a very elementary exposition http://math.harvard.edu/~hirolee/pdfs/280x-13-14-dbcoh.pdf , and also the paper https://arxiv.org/pdf/math/0702861.pdf where in the very beginning you see the quiver with relations needed for $P^n$ (your guess, as you will see in that paper, is not quite correct).

$\endgroup$
8
$\begingroup$

One version of the question you might be asking is where such equivalences come from in general, as opposed to a proof of this one in particular. The general setup is "derived Morita theory": here is an underived version of the basic result.

As setup, let $C$ be a cocomplete abelian category and let $S$ be an essentially small full subcategory of $C$. This gives a restricted Yoneda embedding

$$Y : C \ni c \mapsto (s \mapsto \text{Hom}(s, c)) \in [S^{op}, \text{Ab}]$$

and we would like to know when this is an equivalence of categories. Roughly speaking this means that $S$ "freely generates" $C$.

Theorem (Freyd): The restricted Yoneda embedding is an equivalence of categories if and only if the following conditions hold:

  • each object $s \in S$ is compact projective, and
  • if $Y(c) = 0$, then $c = 0$.

For a long meandering proof see this blog post. These conditions are unfortunately quite restrictive: for example, for modules compactness is equivalent to being finitely presented, so the only compact projectives are the finitely presented projectives. Note that

  • if $S$ has one object $s$ the conclusion of the theorem is that $C$ is equivalent to the category of right $\text{End}(s)$-modules,
  • if $S$ has finitely many objects $s_1, \dots s_n$ a small argument shows that the conclusion of the theorem is that $C$ is equivalent to the category of right $\text{End}(s_1 \oplus \dots \oplus s_n)$-modules, and
  • if $S$ is the free $\text{Ab}$-enriched category on a quiver the conclusion of the theorem is that $C$ is equivalent to the category of (right) representations of this quiver (over $\mathbb{Z}$, but you can replace $\text{Ab}$ with vector spaces and everything goes through the same).

Applications of this theorem include Serre's affineness criterion and, I believe, the Dold-Kan theorem, although I haven't seen the details written down.

Now, there is an analogous theorem in the derived (stable $\infty$) setting which has the pleasant additional property that the projectivity hypothesis can be dropped. This makes it much easier to find full subcategories $S$ satisfying the hypotheses of the theorem; see, for example, Schwede-Shipley for a discussion in the setting of model categories. In particular, many schemes (stacks, etc.) now have the property that their derived categories of quasicoherent sheaves satisfy the hypotheses of the theorem; these are said to be "compactly generated."

$\endgroup$
6
$\begingroup$

I'd just like to say thank you for the answers I have gotten for this question. After doing some more reading I have come across a few results which I shall summarise here.

  1. Beilinson [1] gives the result that for projective space $\Bbb P^n$ the line bundles $\mathcal O_{\Bbb P^n}, \dots,\mathcal O_{\Bbb P^n}(n)$ freely generate $D^b(\mathrm{Coh}(\Bbb P^n))$.

  2. Bondal [2] generalises this by observing that if line bundles $L_0,\dots,L_r$ freely generate $D^b(\mathrm {Coh} (X))$, then there is an equivilence of triangulated categories given by $${\operatorname{{\bf R}Hom}}_{\mathcal O_X}(\oplus_i L_i, -):D^b(\mathrm {Coh} (X))\longrightarrow D^b(\mathrm {mod}_A)$$ where $\mathrm {mod}_A$ is the category of finitely generated right modules over the algebra $A = \mathrm{End}(\oplus_i L_i)$.

  3. King [3,4] shows that Hirzebruch surfaces $\Bbb F_n $ and the smooth Fano toric surfaces have a collection of line bundles that generate $D^b(\mathrm{ Coh}(X))$ as above. To do this, King showed that each of these toric surfaces is a fine moduli space of $\theta$-stable representations of a quiver with relations $Q$ whose path algebra is the endomorphism algebra $A$. The tautological bundles of the moduli space are the line bundles that generate $D^b(\mathrm{Coh}(X))$.

  4. Craw-Smith [5] generalise King's construction to projective toric varieties of arbitrary dimension. They show that $X$ is isomorphic to a component of the fine moduli space of $\theta$-stable representations of a quiver with relations. (I think this quiver is the Bondal quiver?)

  5. Craw-Winn [6] further generalise this result to Mori-dream spaces.

References:

[1] A. Beilinson. Coherent sheaves on $\Bbb P^n$ and problems in linear algebra. Funktsional. Anal. i Prilozhen., 12 (1978) 68-69.

[2] A. Bondal. Representations of associative algebras and coherent sheaves. Izv. Akad. Nauk SSSR Ser. Mat. 45 (1989) 25-44.

[3] A. King. Moduli of representations of finite-dimensional algebras. Quarterly J. Math. Oxford 45 (1994) 515-530.

[4] A. King. Tilting bundles on some rational surfaces. Unpublished article available from Alastair King's homepage http://people.bath.ac.uk/masadk/papers/ (near the bottom), (1997).

[5] A. Craw and G.G. Smith. Projective toric varieties as fine moduli spaces of quiver representations American Journal of Mathematics, 130 (2008) 1509-1534., arXiv:0608183

[6] A. Craw and D. Winn. Journal of Pure and Applied Algebra 217 (2013) 172-189 arXiv:1104.2490

Interestingly most of the authors names sound like mine.

$\endgroup$
4
$\begingroup$

Your question about what happens when you have two vertices and $n$ arrows between them was answered independently by Minamoto (https://arxiv.org/abs/math/0702861) and Piontkovski (https://arxiv.org/abs/math/0606279). They showed that one still gets a derived equivalence as above when one replaces $\mathbb{P}^{1}$ by Piontkovski's $n$th noncommutative projective line $\mathbb{P}^{1}_{n}$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.