Splitting on the line

Posted on June 7, 2023 by Parth

Grothendieck proved any algebraic vector bundle on \(\mathbb{P}^1\) splits into a direct sum of line bundles, and consequently any coherent sheaf on the variety is a direct sum of line bundles and torsion sheaves. This behaviour, and its ramifications on the derived category, exhibit how \(\mathbb{P}^1\) is very close to a principal ideal domain. The results above can be proven with as much or as little machinery as one desires– indeed one can take the point of view that this classification of coherent sheaves is a mere consequence of existence of Jordan normal forms of matrices. In this article however we look at a more cohomological proof, showing how Serre duality shines in these situations.

Serre’s twisting sheaves

We start with a quick recap of divisors on the projective line– recall that Serre characterises quasicoherent sheaves on \(\mathbb{P}^1\) as coming from graded modules over the homogeneous coordinate ring \(S=k[x,y]\) via a construction akin to the \(\text{Proj}\) construction. Explicitly, given a graded \(S\)-module \(M\), the corresponding sheaf is given on the distinguished open \(U_f\) corresponding to a homogeneous element \(f\in S\) by the degree \(0\) part of \(M\otimes_S S[\frac{1}{f}]\). Thus for instance writing \(S(n)\) for the graded module given in degree \(k\) by \(S_{n+k}\), we see that the corresponding sheaf \(\mathscr{O}(n)\) is given on \(U_y=\mathbb{P}^1\setminus \{\infty\}\) by \(y^nk[\frac{x}{y}]\) so that it has global sections given by degree \(n\) homogeneous polynomials in \(S\). These are evidently locally free of rank \(1\), and we have a sequence \(...\mathscr{O}(-1),\mathscr{O},\mathscr{O}(1),...\) of line bundles called Serre’s twisting sheaves. These are so named because any sheaf \(\mathscr{F}\in \text{QCoh}\mathbb{P}^1\) can be twisted to obtain a new sheaf \(\mathscr{F}(n)=\mathscr{F}\otimes\mathscr{O}(n)\). The notation is consistent in that the structure sheaf \(\mathscr{O}\) is \(\mathscr{O}(0)\), and \(\mathscr{O}(m)\otimes \mathscr{O}(n)=\mathscr{O}(m+n)\) by noting that the functor sending a graded \(S\)-module to the corresponding sheaf of \(\mathscr{O}\)-modules is monoidal.

\(\mathbb{P}^1\) is a smooth integral variety, in particular the total quotient ring \(\mathscr{K}\) is the constant sheaf given by the function field (i.e. the stalk of \(\mathscr{O}\) at the generic point) \(k(x,y)\). The sheaf \(\mathscr{K}\) is quasicoherent, coming from the graded \(S\)-module \(K=k[x,y]\cdot k(\frac{x}{y})\) which can be seen as the subring of \(k(x,y)\) containing rational functions with homogeneous denominators. Now any line bundle on \(\mathbb{P}^1\) embeds into \(\mathscr{K}\) (and such embeddings are in bijection with rational sections of the bundle); and by definition a Cartier divisor is a rank \(1\) locally free subsheaf of \(\mathscr{K}\). For instance any homogeneous element \(f\in K_n\) defines an injection \(S(-n)\hookrightarrow K\) given by \(1\mapsto f\) and hence an injection \(\mathscr{O}(-n)\hookrightarrow \mathscr{K}\) corresponding to the rational section \(\frac{1}{f}\). We write \(\mathscr{O}(-D)\) for the image of this injection where \(D\) is the divisor of zeros and poles of \(f\); this is consistent with the standard correspondence between Weil and Cartier divisors on a smooth variety where an effective Weil divisor \(D\) has ideal sheaf \(\mathscr{O}(-D)\subset \mathscr{O}\) which happens to be locally free (hence a Cartier divisor.) Given a Weil divisor \(D\) it is rather straightforward to describe a homogeneous element \(f\in K\) with zeros and poles given by \(-D\), so that \(\mathscr{O}(D)\) must as a line bundle be isomorphic to the twisting sheaf \(\mathscr{O}(\text{deg}D)\). It follows that \(\{\mathscr{O}(n)\;|\; n\in \mathbb{Z}\}\) are all the line bundles (up to isomorphism) on \(\mathbb{P}^1\), and the Picard group is isomorphic to \(\mathbb{Z}\).

Cohomology and Serre duality

The functor \(H^0(\mathbb{P}^1,-)\) on \(\text{Coh}\mathbb{P}^1\) computes global sections of a coherent sheaf, equivalently \(\mathscr{O}\)-module homomorphisms from \(\mathscr{O}\) into the sheaf. Examining the description of the line bundles above, we see that the dimension \(h^0(\mathbb{P}^1,\mathscr{O}(n))\) is then the number of degree \(n\) monomials in \(k[x,y]\).

What about other cohomology groups? While it is easy to explicitly calculate them using a Cech cover, this is a good place to state the Serre duality theorem which we extensively use in what follows. Recall the canonical bundle \(\omega\) of \(\mathbb{P}^1\) is in this case the contangent bundle (it is the determinant of the cotangent bundle general projective varieties), hence it sits in the Euler exact sequence \[0\to \omega \to \mathscr{O}(-1)^{\oplus 2}\to \mathscr{O}\to 0.\] Taking determinants, we must have \(\omega \otimes \mathscr{O} = \text{det}(\mathscr{O}(-1)^{\oplus 2}) = (\mathscr{O}(-1))^{\otimes 2}\), i.e. \(\omega = \mathscr{O}(-2)\) (there are easier ways to see this!)

The Serre duality theorem for smooth projective varieties of dimension \(n\) states that for any locally free coherent sheaves \(F,G\) there are isomorphisms \(\text{Ext}^i(F,G) \cong \text{Ext}^{n-i}(G,F\otimes \omega )^{\check{}}\). In the case of \(\mathbb{P}^1\), this firstly shows that \(H^i(\mathbb{P}^1,F)=\text{Ext}^i(\mathscr{O},F)=\text{Ext}^{1-i}(F, \omega)^{\check{}}\) vanishes if \(i<0\) or \(i>1\), and moreover we have \(H^1(\mathbb{P}^1, F)=\text{Ext}^0(\mathscr{O},F\otimes\omega)^{\check{}}=H^0(F^{\check{}}(-2))^{\check{}}\) whenever \(F\) is locally free. Thus for the line bundles \(\mathscr{O}(n)\), the cohomology groups look like \[h^i(\mathbb{P}^1, \mathscr{O}(n))= \begin{cases} n+1, & i=0, \quad n\geq 0\\ -n+1, & i=1, \quad n\leq -1 \\ 0, &\text{otherwise} \end{cases}.\]

A series of decompositions

Equipped with an understanding of line bundles, we can now set off to prove our main theorem on the structure of \(\mathbf{D}^b(\text{Coh}\mathbb{P}^1)\), showing that the derived category of \(\mathbb{P}^1\) behaves very similarly to that of a point.

Theorem The only indecomposable objects in \(\mathbf{D}^b(\text{Coh}\mathbb{P}^1)\) are shifts of \(\mathscr{O}(n)\) and skyscrapers, and every object in \(\mathbf{D}^b(\text{Coh}\mathbb{P}^1)\) is isomorphic to a finite direct sum of indecomposables.

Note the theorem does not say all skyscrapers are indecomposable, rather it says any indecomposable that is not (up to shift) a line bundle must be a skyscraper.

Thus everything splits into line bundles and skyscrapers. The proof will go in a sequence of splitting results– we will first show that any complex in the derived category is a direct sum of its cohomology sheaves (which are simply coherent sheaves), and any coherent sheaf is the direct sum of a vector bundle and some skyscrapers (these results hold more generally for all smooth curves and will be proven in that generality). Finally, we prove Grothendieck’s theorem that all vector bundles on \(\mathbb{P}^1\) split into line bundles thus completing the proof.

A reader not interested in derived categories can skip the next section and still walk away with an understanding of \(\text{Coh}\mathbb{P}^1\); indeed the reduction from \(\mathbf{D}^b(\text{Coh}\mathbb{P}^1)\) to \(\text{Coh}\mathbb{P}^1\) is formal homological algebra and the interesting geometric arguments happen after this reduction.

Small dimension miracles

Recall that the homological dimension of an Abelian category \(\mathscr{A}\) is the smallest integer \(d\) such that \(\text{Ext}^i(F,G)=0\) whenever \(F,G\in \mathscr{A}\) and \(i>d\). By our discussion above it is clear that the category of coherent sheaves on a smooth curve has homological dimension \(1\), more generally the same Serre duality argument shows \(\text{Coh}X\) has homological dimension \(n\) whenever \(X\) is a smooth projective variety of dimension \(n\). In small homological dimensions, everything in \(\mathbf{D}^b(\mathscr{A})\) splits into a direct sum of its cohomologies.

Theorem [Huy, cor 3.15] If \(\mathscr{A}\) has homological dimension \(\leq 1\), then any object of \(\mathbf{D}^b(\mathscr{A})\) is isomorphic to a direct sum \(\bigoplus A_i[i]\) where the \(A_i\) are objects in \(\mathscr{A}\).

Proof We induct on the length of the complex \(A_\bullet\), clear if it is acyclic (i.e. has length \(0\) up to quasi-isomorphism). If the complex is acyclic, we can up to shift assume \(A_0=H^0(A_\bullet)\neq 0\), \(H^i(A_\bullet)=0\) for \(i<0\). Then there is an exact triangle of the form \[A_0 \to A_\bullet \to A'_\bullet \to A_0[1]\] where \(A'_\bullet\) is supported in degrees \(>0\). It suffices to show this triangle splits, which we do by showing \(\text{Hom}(A'_\bullet, A_0[1])=0\). By induction hypothesis \(A'_\bullet = \bigoplus_{i<0}A_i[i]\) for \(A_i\in \mathscr{A}\) (remember \(A[-1]\) is concentrated in degree \(1\)), so that \(\text{Hom}(A'_\bullet, A_0[1])=\bigoplus_{i<0}\text{Hom}(A_i[i], A_0[1])\) which is equal to \(\bigoplus_{i<0}\text{Ext}^{1-i}(A_i,A_0)\) and hence vanishes by the assumption on homological dimension. \(\square\)

Thus we have our first splitting result on the level of derived categories, and the problem has been reduced to understanding the underlying Abelian category.

The standard torsion pair

We say a quasicoherent sheaf \(T\) on an integral scheme \(X\) is torsion if its stalk at the generic point vanishes. Given a quasicoherent sheaf \(E\) we can define a subsheaf \(T\subset E\) as the union of all torsion subsheaves; this is torsion since taking stalk commutes with taking unions (one way to see this– the stalk is defined as a pullback, and the pullback functor is a left adjoint so preserves colimits) and we call \(T\) the torsion part of \(E\).

Lemma The construction of the torsion part is stalk-local, i.e. at any point \(p\in X\) the \(\mathscr{O}_p\) module \(T_p\) is the torsion part of \(E_p\).

Proof Suppose \(f_p\in \mathscr{O}_p\setminus 0\) and \(t_p\in \mathscr{E}_p\) are such that \(f_p\cdot t_p =0\), we want to show \(t_p\in T_p\). Extend \(f_p,t_p\) to sections \(f\in A\setminus 0\), \(t\in E(U)\) respectively over some affine open neighbourhood \(\text{Spec }A=U\ni p\); now take the subsheaf \(\mathscr{O}_U t\subset E|_U\) (coming from the submodule \(A t\subset E(U)\)) and push it forward along the inclusion \(i:U\hookrightarrow X\) to a subsheaf \(i_\ast\, \mathscr{O}_Ut \subset i_\ast E|_U \subset E\). The sheaf \(i_\ast\, \mathscr{O}_Ut\) is torsion since its stalk at the generic point is the localisation \((A t)_{(0)}=(A_pt_p)_{(0)}=0\), hence it follows that \(i_\ast \mathscr{O}_U t\subset T\). Take stalks at \(p\) to conclude. \(\square\)

Consequently, all stalks of the quotient \(F=E/T\) are torsion-free and we say \(F\) is a torsion-free sheaf. Both the full subcategories \(\mathfrak{T}=\{T\in \text{Coh}X\;|\; T\text{ is torsion}\}\) and \(\mathfrak{F}=\{F\in \text{Coh}X\;|\; F\text{ is torsion-free}\}\) are additive, and any \(E\in \text{Coh}X\) can be uniquely expressed as the extension of a torsion-free sheaf by a torsion sheaf. Moreover, if \(T\in \mathfrak{T}\) and \(F\in \mathfrak{F}\) then \(\text{Hom}(T,F)=0\) (since such a map must be zero on stalks) and thus \((\mathfrak{T}, \mathfrak{F})\) form a torsion pair on \(\text{Coh}X\).

If \(X\) is a smooth algebraic curve we can say more– our second splitting result!

Theorem Every coherent sheaf on a smooth algebraic curve can be expressed as the direct sum of a vector bundle and a torsion sheaf.

Proof We examine the two components of the standard torsion pair on \(\text{Coh}X\). Note each stalk \(\mathscr{O}_p\) is a discrete valuation ring (in particular a PID) so that torsion-free \(\mathscr{O}_p\)-modules are in fact free. Thus a coherent sheaf on \(X\) is torsion-free if and only if it is locally free i.e. an algebraic vector bundle.

If \(T\) is a torsion sheaf, by definition support is a closed subset not equal to \(X\) so \(T\) is supported on a finite set of closed points. Hence we can write \(T=\bigoplus_{p\in X}T_p\) where \(T_p\) is the skyscraper at \(p\) taking value equal to the stalk \(T_p\). In particular, \(T\) is a finite direct sum of flasque sheaves. Whenever \(F\) is locally free of rank \(n\), this allows us to compute \[\begin{align*} \text{Ext}^1(F,T)&=\text{Ext}^1(\mathscr{O}, F^{\check{}}\otimes T) \\ &=\text{H}^1(X,F^{\check{}} \otimes T) \\ &= \bigoplus_{p\in X}\text{H}^1(X,T_p^{\oplus n}) \\ &= 0. \end{align*}\] Thus any sequence \(0\to T\to E \to F \to 0\) must split (possibly non-canonically) and the result follows. \(\square\)

The splitting of vector bundles, or Grothendieck’s theorem

On a general curve this is the furthest we can go without invoking more sophisticated technology, indeed (stable, in particular indecomposable) vector bundles of a fixed rank on curves of positive genus form moduli. However the genus zero case is particularly nice– we will show that the only indecomposable vector bundles have rank \(1\) and every vector bundle splits as a direct sum of line bundles. This was first proved by Grothendieck in 1957 [Gro], and can be proven using very elementary techniques by considering Jordan normal forms of transition matrices as in [Haz]. We take the approach suggested in exercise V.2.6 of [Har], by considering invertible subsheaves and trying to split the inclusion. Section 18.5.5 of [Vak] walks the reader through this proof by breaking it down into a sequence of exercises, but still uses transition matrices to show the splitting– we avoid this by simply using more homological algebra.

The following lemma is a useful characterisation of when invertible subsheaves of a certain degree exist.

Lemma For a torsion-free coherent sheaf \(F\) on \(\mathbb{P}^1\) and \(k\in \mathbb{Z}\), we have \(h^0(\mathbb{P}^1, F(k))>0\) if and only if \(\mathscr{O}(-k)\) injects into \(F\).

An injection \(\mathscr{O}(-k)\hookrightarrow F\) can be twisted to get an injection \(\mathscr{O}\hookrightarrow F(k)\), which corresponds to a non-zero element of \(H^0(\mathbb{P}^1, F(k))\). Conversely if \(h^0(\mathbb{P}^1, F(k))>0\) then choosing a non-zero global section of \(F(k)\) gives a non-zero map \(\mathscr{O}\to F(k)\). Claim such a map must be injective– the kernel of the map is a proper submodule of \(\mathscr{O}\), i.e. an ideal sheaf for some non-empty closed subscheme \(Z\subset \mathbb{P}^1\). Thus the image of the map is the structure sheaf of \(Z\) (pushed forward to \(\mathbb{P}^1\) along the inclusion) but \(F(k)\) is torsion-free so all its subsheaves (in particular the image of the map) is torsion-free. This is possible only if \(Z=\mathbb{P}^1\), i.e. the kernel was trivial. Then twisting the injection \(\mathscr{O}\hookrightarrow F(k)\) by \(\mathscr{O}(-k)\) proves the lemma. \(\square\)

We can now prove the main result– the key idea is that quotienting a locally free sheaf by an invertible subsheaf of maximal degree necessarily leaves you with something locally free.

Theorem (Grothendieck) For any rank \(r\) vector bundle \(F\) on \(\mathbb{P}^1\), there is a uniquely determined non-decreasing sequence of integers \(n_1\leq n_2\leq ... \leq n_r\) such that \(F\cong \bigoplus_{i=1}^r\mathscr{O}(n_i)\).

Proof The uniqueness is straightforward– suppose \(F=\bigoplus_{i=1}^r\mathscr{O}(n_i)=\bigoplus_{i=1}^r\mathscr{O}(m_i)\) with \(n_1\leq n_2\leq ... \leq n_r\) and \(m_1\leq m_2 \leq ... \leq m_r\). If \(n_i\neq m_i\) for some \(i\), then we can assume without loss of generality \(n_r>m_r\) and twisting if necessary, \(n_r=0\). Thus \(m_i<0\) for all \(i\), while \(n_i<0\) for all \(i<r\). Computing global sections then gives a contradiction.

Thus it suffices to show every vector bundle splits as a direct sum of line bundles since the factors can always be reordered to satisfy the theorem. We induct on the rank \(r\) and note that the case \(r=1\) is simply the computation of the Picard group. Suppose \(F\) has rank \(r\geq 2\), and consider \[\{k\;|\; \mathscr{O}(k) \text{ injects into }F\} \subset \mathbb{Z}.\] By ampleness of \(\mathscr{O}(1)\), the set is non-empty. On the other hand an injection \(\mathscr{O}(k)\to F\) induces an injection of global sections, so the set must be bounded above (since \(h^0(\mathbb{P}^1, \mathscr{O}(k))\to \infty\) as \(k\to \infty\)). Thus the set has a maximum element \(n\). Choose an injection \(\mathscr{O}(n)\to F\) with image \(L\), claim the cokernel \(F'\) is torsion-free; to see this note that it surjects onto its torsion-free part \(F''\) and we have a composite surjection \(F\twoheadrightarrow F''\) of locally free sheaves. The kernel \(L'\) of this is a subsheaf of \(F\) so is necessarily torsion-free, moreover \(F"\) has rank \(r-1\) (since \(F'\) is locally free of rank \(r-1\) on the complement of finitely many points) so the \(L'\) is a line bundle \(\mathscr{O}(N)\) for some \(N\). Thus \(F/L\) surjects onto \(F/L'\), so we must have \(L\subset L' \subset F\) i.e. there is a sequence of injections \(\mathscr{O}(n)\hookrightarrow \mathscr{O}(N)\hookrightarrow F\). But the first injection necessarily implies \(N\geq n\) (by considering global sections after twisting if necessary), and equality holds by choice of \(n\). Thus \(F'=F''\) is torsion-free of rank \(r-1\).

By induction hypothesis \(F'=\bigoplus_{i=1}^{r-1}\mathscr{O}(n_i)\) is a sum of line bundles, and moreover \(n\geq n_i\) for all \(i\): to see this, note that applying \(\text{Hom}(\mathscr{O}(n+1),-)\) to the exact sequence \(0\to \mathscr{O}(n)\to F\to F'\to 0\) gives a long exact sequence \[\cdots \to \text{Hom}(\mathscr{O}(n+1), F) \to \text{Hom}(\mathscr{O}(n+1), F') \to \text{Ext}^1(\mathscr{O}(n+1),\mathscr{O}(n)) \to \cdots.\] Now the first term vanishes because any homomorphism \(\mathscr{O}(k)\to F\) is necessarily injective and \(n\) was maximal to admit such an injection. The last term vanishes because \(\text{Ext}^1(\mathscr{O}(n+1), \mathscr{O}(n))= H^1(\mathscr{O}(-1))=0\). Thus \(\mathscr{O}(n+1)\) does not inject into \(F'\), which is possible only if each \(n_i\) is strictly smaller than \(n+1\).

It remains to show \(0\to \mathscr{O}(n)\to F\to F' \to 0\) splits, we do this by computing \(\text{Ext}^1(F', \mathscr{O}(n))\) and showing it vanishes. By Serre duality it suffices to show \(\text{Hom}(\mathscr{O}, F'^{\check{}}\otimes \mathscr{O}(n)\otimes \omega)=H^0(F'^\check{}(n-2))\) vanishes. But this is clear, since \(F'^{\check{}}(n-2) = \bigoplus_{i=1}^{r-1}\mathscr{O}(n-n_i-2)\) and \(n-n_i-2<0\) for all \(i\). \(\square\)

This finishes the proof of the main result.

Full exceptional collections

We have seen how the entire derived category of \(\mathbb{P}^1\) can be built using skyscrapers and line bundles, if one is allowed to take shifts and direct sums. The definition of a triangulated category comes with more operations– if one also allows taking cones of morphisms, then the entire category can be generated by just two line bundles! Indeed, any skyscraper at a point \(i:P\hookrightarrow \mathbb{P}^1\) can be built as an iterated self-extension of the structure sheaf \(i\ast\mathscr{O}|_P\), which itself sits in an exact sequence \[0\to \mathscr{O}(-1)\to \mathscr{O}\to i_\ast \mathscr{O}|_P\to 0.\] Likewise, the Euler sequence shows \(\omega=\mathscr{O}(-2)\) is an extension of \(\mathscr{O}\) and \(\mathscr{O}(-1)^{\oplus 2}\), and taking twists of the Euler sequence we can build any \(\mathscr{O}(n)\) as an extension of shifts of summands of \(\mathscr{O}\) and \(\mathscr{O}(-1)\); in other words the collection \(\{\mathscr{O},\mathscr{O}(-1)\}\) is full.

Moreover, writing \(E_0=\mathscr{O}(-1), E_1=\mathscr{O}\) our cohomology calculation shows \[\text{ext}^n(E_i, E_j)=\begin{cases} 1, &n=0, \quad i=j \\ 0, &i>j \text{ or } n\neq 0, i=j \end{cases}.\] Such sequences in a triangulated category are exceptional, and full exceptional collections in a triangulated category are akin to a basis for a vector spaces.

Both the statements above remains true up to twists, giving us the following.

Theorem \(\{\mathscr{O}(n),\mathscr{O}(n+1)\}\) is a full exceptional collection in \(\mathbf{D}^b\text{Coh}\mathbb{P}^1\).

An analogous statement holds for \(\mathbb{P}^n\), and is a consequence of Beilinson’s theorem (see my part III essay.)


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