Why are coherent sheaves on $\Bbb P^1$ derived equivalent to representations of the Kronecker quiver?

Question

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 $\Bbb P^n$.

Answer 1

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.

Answer 2

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).

Answer 3

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."

Answer 4

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.

Answer 5

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 $\mathbb{P}^{1}_{n}$.