What is \(\mathbb{P}^1\)
Throuought this page we will assume that \(k=\bar{k}\), unless that's unneccecary in which case I didnt assume it you did
There are many faces to \(\mathbb{P}^1\) most of which I'm sure that I will never see in my lifetime but here's some, I'm writing it all down because sometimes a man doesn't want to drag himself through Hartshorne to find the definition he's looking for
We start with the basics, classically speaking the \(k\) points of \(\mathbb{P}^1\), denoted \(\mathbb{P}^1(k)\) are the nonzero points \([(x_1,x_2)]\) modulo scaling by \(k\). We can cover this by affines to give a structure sheaf. Explicitly algebraically \(\mathbb{P}^1(k)\) is covered by the affines \(k[T]\) and \(k[T^{-1}]\). Or in a more general case, ie allowing for analytic functions, we glue together \(\mathbb{A}^1_{k,t}\) and \(\mathbb{A}^1_{k,t^{-1}}\)
It's sometimes more convenient to think of \(\mathbb{P}^1\) as one would an affine scheme, ie we want some projective generalisation of \(\operatorname{spec}\). To do so we define for a graded ring \(S\), $$\operatorname{Proj}S = \{\mathfrak{p} \mid \mathfrak{p}\text{ is a homogeneous prime ideal not containing the irrelevant ideal}\}$$ with a topology defined by the basic opens \(D(f) = \operatorname{spec}((S_f)_0) = \{\mathfrak{p} \in \operatorname{Proj} S \mid f \in S\}\) for \(f\) a homogenous element of \(S\). This lets us define a natural structure sheaf on \(X=\operatorname{Proj}S\) by \(\Gamma(D(f),\mathcal{O}_X) = (S_f)_0\) And more generally for any graded \(S\) module \(M\) a sheaf of \(\mathcal{O}_X\) modules given by \(\Gamma(D(f),\tilde{M}) = (M_f)_0\). One of the critical things about this scheme is that since the sheaves are (basically) graded modules we can shift the grading to get a different sheaf. This is called twisting, for a module \(M\) we can define the twisted modules as the graded module so that \(M(n)_i = M_{n+i}\), note for generalisation in a second that this is just \(M \otimes S(n)\). This then defines a sheaf by \(\Gamma(D(f), \tilde{M}(n)) = (M_f)_n\) or more generally for a sheaf \(\mathcal{F}\) we have \(\mathcal{F}(n) = \mathcal{F}\otimes (\mathcal{O}_X(n))\) where the tensor is understood to be the sheafification of the tensor product of presheaves. Note that twisting gives a homomorphism \(\mathbb{Z} \to \operatorname{Pic}(X)\) as all \(\mathcal{O}(n)\) are invertible (this is in fact an isomorphism). One can compute that in the case of projective space $$\mathbb{P}^n_k = \operatorname{Proj}k[x_0,\dots,x_n]$$we have \(\Gamma(\mathbb{P}^n_k , \mathcal{O}(n)) = k[x_0,...x_n]_k\). In particular there are no global sections for \(n\le -1\). Using this explicit description we can also find exact sequences $$ 0 \to \Omega^1 \to \mathcal{O}(-1)^{\oplus (n+1)} \to \mathcal{O} \to 0 $$ $$ 0 \to \mathcal{O} \to \mathcal{O}(1)^{\oplus (n+1)} \to \mathcal{T} \to 0 $$ Where \(\mathcal{T}\) is the tangent sheaf and \(\Omega^1\) is the sheaf of differentials. Taking the top exterior powers we see that $$\mathcal{O}(-n-1) \cong \mathcal{O}(-1)^{\otimes(n+1)} \cong [\Lambda \mathcal{O}(-1)]^{\otimes(n+1)} \cong \Lambda [\mathcal{O}(-1)^{\oplus (n+1)}] \cong \Lambda \Omega \otimes \Lambda \mathcal{O} \cong \Lambda \Omega = \omega$$ the canonical sheaf. So in particular on \(\mathbb{P}^1, \ \omega = \Omega\) we have \(\Omega^1 \cong \mathcal{O}(-2)\)
\(\mathcal{D}^b(\mathbb{P}^1)\)
Given a morphism of schemes \(f:X \to Y\) we get a six functor formalism on the derived categories, importantly for now we have four functors, well only two that matter for this. I will be supressing derived stuff in notation as to not get overwhelmed, additionally note that the derived category for a scheme will be derived category of coherent sheaf and I will refer to an object in this category as a sheaf (that is unless it's competing for attention with an actual sheaf in which case the distinction will be made). We have the pullback \(f^*\) and pushforward \(f_*\) with \(f^*\) the left adjoint of \(f_*\). As we can go both ways we can play this nice trick, we have projection maps \(X\ \leftarrow X \times Y \rightarrow Y\) and hense we can take a sheaf on \(X\), pull it back to the product, mess with it there and then push it down to \(Y\) (integrate along the fiber). This is the structure of a Fourier-Mukai transform
Written explicitly given a kernel sheaf \(K \in \mathcal{D}^b(X \times Y)\) we can construct the transform \(X \to Y\) via the formula \(\pi_{Y*}(K \otimes \pi_X^*(-))\). The strategy of the proof is as follows;
We can restrict to the special case of \(X=Y\) as this gives us a nice general set up. If we take the pushforward of the structure sheaf, aka the diagonal sheaf \(\mathcal{O}_\Delta\) over kernel we see that this is the skyscraper sheaf over the graph \((x,x)\) and hense we change nothing and get the identity functor. This \(\mathcal{O}_\Delta\) sheaf however is not projective so we can take a projection resolution and recover an equivalent functor that looks different. Restricting now to \(X = \mathbb{P}^1\) we get a resolution $$ 0 \to \mathcal{O}_{\mathbb{P}^1 \times \mathbb{P}^1}(-1,-1) \to \mathcal{O}_{\mathbb{P}^1 \times \mathbb{P}^1} \to \mathcal{O}_\Delta \to 0 $$ these two sheaves, give us two points which we can use to construct resolutions of the Kronecker quiver. Now to write that properly:
Seeing that the diagonal sheaf does nothing is clear as this is an honest sheaf the tensor product is easy to find so for a complex \(\mathcal{F}\) the tensor gives that complex with components supported at \((x,x)\) and as on each of these the fiber of the projection is a point there is no failure of exactness so \(\mathcal{F}\) just maps to itself
If we now replace \(\mathcal{O}_\Delta\) with its resolution, call it \(\mathcal{E}\) it's now useful to mention when something is derived, for a derived functor \(f\), write \(\bar{f}\) for the non derived version. We note that \(\mathcal{O}(-1,-1) = \bar{\pi^*} \mathcal{O}(-1) \otimes \bar{\pi^*} \mathcal{O}(-1)\) and \(\mathcal{O} = \bar{\pi^*} \mathcal{O} \otimes \bar{\pi^*} \mathcal{O}\) we find
$$
\begin{aligned}
\Phi_{\mathcal{E}}(F) &= \pi_* (\mathcal{E}\otimes \pi^* \mathcal{F}) \\
&= \pi_* ([\mathcal{O}(-1,-1) \to \mathcal{O}]\otimes \pi^* \mathcal{F}) \\
&= \pi_* ([\bar{\pi^*} \mathcal{O}(-1) \otimes \bar{\pi^*}\mathcal{O}(-1) \to \bar{\pi^*} \mathcal{O} \otimes \bar{\pi^*} \mathcal{O}]\otimes \pi^* \mathcal{F}) \\
&= \pi_* [\bar{\pi^*} \mathcal{O}(-1) \otimes \pi^*\mathcal{F}(-1) \to \bar{\pi^*} \mathcal{O} \otimes \pi^* \mathcal{F}]\\
&= [\mathcal{O}(-1) \otimes \pi_* \pi^*\mathcal{F}(-1) \to \mathcal{O} \otimes \pi_* \pi^* \mathcal{F}]\\
\end{aligned}
$$
Whats important here is that we can realise any complex as a complex of the form \([\mathcal{O}(-1) \otimes V \to \mathcal{O} \otimes U]\) For some homological data \(V,U\). That is the derived category is generated by these two sheaves where \(\operatorname{Ext}(\mathcal{O}(-1), \mathcal{O})=\{i=0;k^2, i\neq 0;0\}, \ \operatorname{Ext}(\mathcal{O}, \mathcal{O}(-1))=0\). Similarly \(\mathcal{D}^b(\bullet \rightrightarrows \bullet)\) [This is the derived category of representations of this quiver] is generated by the representations \([0 \rightrightarrows k], [k \rightrightarrows 0]\) which have that same Ext, hense we get an equivalence of the derived categories
This proof might inspire something in the reader, we started out with a derived category, found inside it two objects with two dimensions of maps in one direction and none in the other and thereby found that the category is the same as the derived category of the representations of a quiver with two vertices and two maps going in one direction and none in the other. This is not a coincidence, to explain this properly we need to introduce the theory of semiorthogonal decompositons
Take an arbitrary \(k\)-linear(triangulated/stable infinity) category We say that an object is exceptional if it has no ext and a one dimensional hom space, an ordered collection of exceptional objects is called exceptional if theres no \(R\hom\) in the increasing direction. We say it's full if it generates the category.
In the case of \(\mathbb{P}^1\) we found that \(\mathcal{O}\) and \(\mathcal{O}(-1)\) are a full exceptional collection. This is not the only pair, in fact if we define
$$
L_E F = \operatorname{cone}(R\hom(E,F) \otimes E \to F)
$$
$$
R_E F = \operatorname{cocone}(E \to R\hom(E,F)^{\vee} \otimes F)
$$
Note that \(R\hom\) is a bifunctor to the derived category of \(k\) vector spaces so the dual is well defined, note that the dual of a compex reverses the grading. Then this allows us to mutate our exceptional collection that preserves exceptionality and fullness. Given \(E_0 ,..., E_n\) exceptional we can reassign
$$
E_i,E_{i+1} \iff E_{i+1}, R_{E_{i+1}}E_i
$$
$$
E_i,E_{i+1} \iff L_{E_{i}}E_{i+1},E_{i}
$$
Ie these allow us to push objects left and right. In our example this lets us change our \(\langle\mathcal{O}(-1), \mathcal{O}\rangle\) into a slightly more aestetic looking \(\langle\mathcal{O},\mathcal{O}(1)\rangle\) and in fact by doing a similar thing as before, taking a resolution of the diagonal, we get that the standard full exceptional collection on \(\mathbb{P}^n\) is \(\langle \mathcal{O},...,\mathcal{O}(n) \rangle\) we could use negative twists but to be inkeeping with what you'll find in the wild we can use this. We say a collection is strong if there is no ext in any direction. This mimics what we have for \(\mathbb{P}^n\). For a strong exceptional collection \(E_0...E_n\) we have that the (triangulated/stable) category they generate is equivalent to the bounded derived category of modules over \(E=\operatorname{End}(\bigoplus E_i)\). This should remind you of the result of Freyd making equivalences out of compact projective generators, however note that we did not need projectivity, we mostly care about generating (this is an instantiaton of the more general trend that for a stable (infinity/model) category you only need compact generators to get to get an equivalence to some category of modules, for more info see Schwede-Shipley). In proving this theorem we will show that this ring is in fact the path algebra for a quiver with relations and so we should look at the structure of such an object
For a quick quiver crash course; a quiver is a directed graph, a representation is when you let the vertices be vector spaces and edges be linear maps, the path algebra is the free algebra on the edges where multiplication is concatenation of paths or zero if not possible, modules for this algebra are the same as representatoins of the quiver, relations are just an ideal on the path algebra generated by stuff of not too large degree. Ok, given a quiver with relations we want to look at the derived category of representations. As it turns out this isn't too tricky as inside of our path algebra \(A\) we have the length zero paths, starting and ending at \(\alpha\), written \(e_\alpha\) (note these are basically neccecary to include as we need an identity and \(1 = \sum_\alpha e_\alpha\)) These elements are orthogonal idempotents and therefor allow us to decompose \(A\) into \(\bigoplus e_\alpha A\) and in fact any representation \(V\)decomposes into \(\bigoplus e_\alpha V\), this is where the equivalence comes from as the vector space attatched to vertex \(\alpha\) is \(e_\alpha V\) and attatched to an arrow \(f: \alpha \to \beta\) we get the linear map \(V_\alpha \to V_\beta\) attached to the action of multiplication by \(f \in A\). In our exploration of exceptional collections note that the order is very important, this is mimicked in the world of quivers by disalowing quivers with cycles, algebraically this means everything is finite over \(k\) and in fact allows us to give an order to the vertices where the arrows only go up, im sure you can see where this is going. Projective modules over this quiver are just the summands of free modules but since we have a decomposition by orthogonal idempotents we this will be generated by the summands of \(A\) so we have a collection of compact projective generators \(P_i = e_{\alpha_i} A\) these give our vertices, and
$$
A = \bigoplus_{ij}\hom(P_i,P_j)
$$
So the arrows are just maps between these projective generators. In the case of our derived category the equivalence is induced by pretending that these \(E_i\) are the projectives \(P_i\), as we have isomorphisms \(R\hom(E_i, E_j) \to R\hom(P_i, P_j)\) using the theory of semiorthogonal decompositions we can construct the equivalence. This is to say that given a derived category \(\mathcal{D}^b\) if we can find a strong full exceptional collection of objects, we can draw a quiver where vertices are these objects and arrows are a \(k\) basis of morphisms \(E_i \to E_j\), then the relations will be whatever relations we get on the maps in \(\mathcal{D}^b\) giving us an equivalence to the derived category of quiver representations. Doing this for \(\mathbb{P}^n\) with our standard exceptional collection gives vertices \(\alpha_0 ,..., \alpha_n\) with \(n(n-1)\) arrows going \(\alpha_i \to \alpha_{i+1}\), written \(\phi_i^j\) with relations \(\{\phi_{i+1}^{j} \phi_i^k - \phi_{i+1}^k \phi_{i}^j\}\)
I realise I started this out talking saying semiorthogonal decompositions but never actually said what that was. Put simply its a generalisation of this setup where we decompose a category into an ordered series of subcategories that generate the category and have no maps going up, an exceptional collection gives a semiorthogonal decomposition \(\langle \langle E_n,...,E_1 \rangle^{\bot},E_n,...,E_1 \rangle\) where \((X)^\bot\) is those objects who have no homs with \(X\)
The flag variety of \(\operatorname{Sl}_2\)
Given a vector space \(k^n\) we can construct the \((n-1)\)-dimensional projective space as the space of lines. Similarly we can construct the Grassmanian \(\mathbf{Gr}(\ell,n)\) as the space of \(\ell\)-dimensional subspaces in \(k^n\). There's a natural way to glue these together as an \(\ell\)-dimensional subspace is contained in a bunch of \((\ell+1)\)-dimensional subspaces, which are then contained in some ammount of \(\ell+2\)-dimensional subspaces. So a natural way to look at all of these Grassmanians at once is to consider
\[
F(k^n)=\{(x_0,x_2,...,x_n) \vert x_\ell \subset x_{\ell+1}\} \subset \mathbf{Gr}(0,n) \times \mathbf{Gr}(1,n) \times ... \times \mathbf{Gr}(n-1,n) \times \mathbf{Gr}(n,n)
\]
This is called the flag variety and each of the sequences \((x_0,...,x_n)\) is called a flag. To get an explicit description we recall an explicit description of \(\mathbf{Gr}(\ell,n)\). An \(\ell\)-dimensional subspace is described by a list of \(\ell\) vectors in \(k^n\) that are linearly independant, by multiplying this by some matrix \(A \in sl_n\) we can transform this subspace into the space spanned by the first \(\ell\) basis vectors. That is to say any subspace is some matrix times some basic subspace and so is just isomorphic to \(sl_n / B_\ell\) for some subgroup that fixes any subspace. We can compute this explicitly by taking our standard subset as the first \(\ell\) basis vectors
\[
\begin{pmatrix}
a_{11} & a_{12} & \dots & a_{1n} \\
a_{21} & a_{22} & \dots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n1} & a_{n2} & \dots & a_{nn} \\
\end{pmatrix} \cdot \begin{pmatrix}
1 & 0 & \dots & 0 \\
0 & 1 & \dots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \dots & 1 \\
0 & 0 & \dots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \dots & 0 \\
\end{pmatrix} = \begin{pmatrix}
a_{11} & a_{12} & \dots & a_{1\ell} \\
a_{21} & a_{22} & \dots & a_{2\ell} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n1} & a_{n2} & \dots & a_{n\ell} \\
\end{pmatrix}
\] So we get the subgroup \(B_\ell\) of block matrices
\[
\left\{\left(\begin{array}{@{}c|c@{}}
A_{\ell \times \ell}
& * \\
\hline
0_{(n-\ell) \times \ell} &
B_{(n-\ell) \times (n-\ell)}
\end{array}\right) \vert \operatorname{det}(A)\operatorname{det}(B)=1\right\}
\]
The reason we go through this work is to see what it means to take a subspace, ie if we have some matrix \(A\) representing an \(\ell\) dimensional subspace then it can also represent an \((\ell-1)\)-dimensional subspace and so on, in fact this will give a sequence of subspaces contained in eachother, that's a flag. So in fact any matrix \(A\) represents a flag as a transformation of the canonical flag \(0\subset\langle e_1 \rangle\subset\langle e_1, e_2 \rangle\subset \dots \subset \langle e_1, e_2, ..., e_n\rangle\). This means we can write it as \(sl_n/B\) for some group \(B\) that stabilisies the canonical flag. That we can just rip from our explicit computation, we're looking for matrices that stabilise subspaces at every level, hense we take the intersection of all \(B_\ell\) which is just the upper triangular matrices. As it turns out this is a very important kind of subgroup called a Borel subgroup and (considering \(sl_n\) with Zariski topology as algebraic group) the (up to conjugacy) maximal subgroup that is closed, connected and soluable. This motivates for a general algebraic group \(G\) the definition \(F(G)=G/B\) for \(B\) the Borel subgroup. This object has many nice properties, for exaple if we have \(G\) is say, a semisimple algebraic group, then the variety \(\operatorname{Nilp(\mathfrak{g})} \subset \mathfrak{g}\) of ad-nilpotent objects is singular but using the flag variety we have resolution \(\Omega (G/B) \to \operatorname{Nilp}(\mathfrak{g})\) where \(\Omega(G/B)\) is the cotangent bundle so smooth giving a resolution. Morally this holds because for an algebraic/lie group we can mostly work over the identity so as \(\Omega_B(G/B)=(\mathfrak{g}/\mathfrak{b})^\vee = \mathfrak{b}^\bot\), that is those linear maps that kill \(\mathfrak{b}\), using the non degenerate killing form on \(\mathfrak{g}\) this is those things orthogonal to \(\mathfrak{b}\) which is well known to be the nilradial of \(\mathfrak{b}\). This means that, at least at the identity, the cotangents are precicely the nilpotents contained in our chosen Borel. This lets us see that when we take an arbitrary Borel coset and pull it back to the identity the cotangents are those nilpotents contained in the adjoint image of \(\mathfrak{b}\), this gives an identification \(\Omega(G/B) \leftrightarrow \{(x,\mathfrak{b}) \vert x \text{ nilpotent }, x \in \mathfrak{b}\}\) and hense a projection map \(\Omega(G/B) \to \operatorname{Nilp}(\mathfrak{g})\). This is interesting to geometers already and interesting to representation theorists as the cohomology of \(\Omega(G/B) \times_{\operatorname{Nilp}(\mathfrak{g})} \Omega(G/B)\) acts on the cohomology of the fibres by convolution (at the level of sheaves this looks like matrix multiplication) and in fact the cohomology is just the group algebra of the Weyl group and even better the cohomology of the fiber over \(x\) decomposes into \(\operatorname{Stab}_G(x)/\operatorname{Stab}_G(x)_o\) isotypic subspaces which are a complete set of irreducible representations of the Weyl group!
Here I must confess I am sad, as flag varieties \(\mathbb{P}^1\) corresponds to \(sl_2\) which is the semisimple lie algebra corresponding to \(A_1\). This is sad because dynkin diagrams are quivers and the Kronecker quiver looks like the \(B_2\) or maybe \(A_2\) diagram. The main issue here is that \(A_1\) is so simple that there's no real way to redraw it that the representations can be anything but \(k-\mathbf{Vect}\) which will clearly be distinct from anything associated to \(\bullet \rightrightarrows \bullet\). An interesting question is let \(F\) be the flag variety of a seimisimple lie/algebraic group \(G\) with corresponding dynkin diagram \(D\), is there a combinatorial way to find a quiver with relations \(\mathbf{Q}\) so that \(\mathcal{D}^b(\mathbf{Q})\) is equivalent to \(\mathcal{D}(F)\). Ie this map should send \([\bullet] \mapsto [\bullet \rightrightarrows \bullet]\). My guess is there's no such thing since we only obtain the quiver up to derived morita equivalence and in general there is no canonical choice, for example on \(\mathbb{P}^2\) if we choose \(\langle \mathcal{O}, \Omega(2), \mathcal{O}(1) \rangle\) we get quiver
With relations \(\{f_i g_j + f_j g_i\}\), wheras if we take \(\langle \mathcal{O}(-2), \mathcal{O}(-1), \mathcal{O} \rangle\) we get the same diagram but with relations \(\{f_i g_j - f_j g_i\}\) so we cannot hope to obtain a canonical quiver with relations given a projective variety. Additionally this derived morita equivalence class is a property of the group wheras the dynkin diagram is a property of the algebra which only gives back a canonical group if you restrict to semisimple, connected, simply connected algebraic groups. For example \(\mathfrak{su}_2 = \mathfrak{so}_3\) as \(SU(2)\) is the universal cover of \(SO(3)\) and it's clear that in general a non simply connected group will have the same lie algebra as its universal cover.
John Baez describes a fundamental property of the dynkin diagram, where you can use it to read off maximal parbolics of your algebraic group as well as using it to read off the irreducible representations. This seems arguably nicer than what we get with the Kronecker quiver since that lets us read off what? Homologically projective complexes?
