Monads on projective space and the Beilinson resolution

Intuition Beginner

A vector bundle on projective space is a family of vector spaces, one over each point, glued together in a geometric way. These objects can be hard to describe directly. The idea of a monad is to present such a bundle entirely through linear algebra: write down two simple maps between sums of standard line bundles, and recover the bundle you want as the "middle layer" left over.

Picture three bundles in a row, $A$, then $B$, then $C$, with a map from $A$ into $B$ and a map from $B$ onto $C$. The first map slides $A$ inside $B$ as a sub-bundle. The second map collapses $B$ down onto $C$. The bundle of interest is what sits inside $B$ above $A$: the part that survives the collapse but is not already accounted for by $A$.

The reason this matters: when $A$ and $C$ are built from the simplest line bundles $\mathcal{O}(d)$, the two maps are just matrices of homogeneous polynomials. So a complicated bundle becomes a short list of polynomial entries. Geometry turns into bookkeeping.

Visual Beginner

Three bundles in a row, with one map injecting and the next map surjecting; the bundle you care about is the middle cohomology, the part of the kernel of the second map that lies above the image of the first.

Worked example Beginner

Take the projective plane ${\mathbb{P}}^2$ and the row of line bundles

$$\mathcal{O}(-1)\stackrel{a}{\to }{\mathcal{O}}^3\stackrel{b}{\to }\mathcal{O}(1).$$

Here $a$ is given by a column of three linear forms, say $(x_0,x_1,x_2)$, and $b$ is given by a row of three linear forms chosen so that the composite $b\circ a$ is the zero map. The first map is injective at every point because the three coordinates never vanish all at once. The second map is onto for the same reason.

The leftover middle layer is a rank-one object: start with rank three in the center, remove rank one from the right collapse, remove rank one from the left injection, and one rank remains. That surviving line bundle is the tangent direction data of the plane, packaged purely from these polynomial matrices. The whole bundle has been named by six linear forms.

Check your understanding Beginner

Formal definition Intermediate+

Throughout, ${\mathbb{P}}^n={\mathbb{P}}_{\mathbb{C}}^n$ and $\mathcal{O}(d)$ denotes the twisting line bundle of degree $d$ (unit 04.05.03).

Definition (monad). A monad on ${\mathbb{P}}^n$ is a complex of holomorphic vector bundles

$$A\stackrel{a}{\to }B\stackrel{b}{\to }C,b\circ a=0,$$

in which $a$ is a bundle injection (injective with locally free cokernel, equivalently injective on every fibre) and $b$ is surjective on every fibre. The cohomology bundle of the monad is the coherent sheaf

$$E=\ker b/\mathrm{im}a.$$

Because $a$ is a fibrewise injection and $b$ a fibrewise surjection, $\ker b$ and $\mathrm{im}a$ are sub-bundles of $B$, and $E$ is itself a vector bundle, of rank $\mathrm{rk}B-\mathrm{rk}A-\mathrm{rk}C$. Its Chern character is $\mathrm{ch}(E)=\mathrm{ch}(B)-\mathrm{ch}(A)-\mathrm{ch}(C)$.

A monad is called a linear monad, or a display of $E$, when $A$ and $C$ are direct sums of line bundles; the two maps $a$ and $b$ are then matrices whose entries are homogeneous polynomials. The instanton case is the ADHM monad (cross-link 03.07.10)

$$\mathcal{O}(-1{)}^k\stackrel{a}{\to }{\mathcal{O}}^{2k+2}\stackrel{b}{\to }\mathcal{O}(1{)}^k,$$

whose cohomology bundle is the rank-two bundle on ${\mathbb{P}}^3$ corresponding to a charge-$k$ instanton on $S^4$.

The display of a monad. Unpacking $E=\ker b/\mathrm{im}a$ produces two short exact sequences,

$$0\to A\stackrel{a}{\to }K\to E\to 0,0\to K\to B\stackrel{b}{\to }C\to 0,$$

where $K=\ker b$. Taking cohomology of these sequences, twisted by $\mathcal{O}(j)$, is how one reads the cohomology of $E$ off the line bundles $A,B,C$.

The Beilinson resolution. On the product ${\mathbb{P}}^n\times {\mathbb{P}}^n$ with projections $p,q$, the structure sheaf of the diagonal ${\mathcal{O}}_{\Delta }$ admits the Beilinson resolution

$$0\to {\Omega }^n(n)⊠\mathcal{O}(-n)\to \cdots \to {\Omega }^1(1)⊠\mathcal{O}(-1)\to \mathcal{O}⊠\mathcal{O}\to {\mathcal{O}}_{\Delta }\to 0,$$

where ${\Omega }^p={\Omega }_{{\mathbb{P}}^n}^p$ is the sheaf of holomorphic $p$-forms and $\mathcal{F}⊠\mathcal{G}=p^{\ast }\mathcal{F}\otimes q^{\ast }\mathcal{G}$. This is the Koszul resolution attached to the tautological section of $T_{{\mathbb{P}}^n}(-1)⊠\mathcal{O}(1)$ that vanishes exactly on the diagonal.

Key theorem with proof Intermediate+

Theorem (Beilinson spectral sequence). Let $\mathcal{F}$ be a coherent sheaf on ${\mathbb{P}}^n$. There is a spectral sequence with first page

$$E_1^{p,q}=H^q({\mathbb{P}}^n;\mathcal{F}(p))\otimes {\Omega }^{-p}(-p),-n\le p\le 0,$$

converging to $\mathcal{F}$ in degree zero and to $0$ in all other degrees. A second such sequence has first page $E_1^{p,q}=H^q({\mathbb{P}}^n;\mathcal{F}\otimes {\Omega }^{-p}(-p))\otimes \mathcal{O}(p)$, with the same abutment.

Proof. Apply the functor $R{q}_{\ast }(p^{\ast }\mathcal{F}\otimes (-))$ to the Beilinson resolution of ${\mathcal{O}}_{\Delta }$. Since restriction to the diagonal recovers $\mathcal{F}$, the projection formula gives

$$R{q}_{\ast }(p^{\ast }\mathcal{F}\otimes {\mathcal{O}}_{\Delta })=\mathcal{F},$$

concentrated in degree zero. The resolution by the locally free sheaves ${\Omega }^p(p)⊠\mathcal{O}(-p)$ produces a hypercohomology spectral sequence. For the term indexed by $-p$, the projection formula and the Künneth identity factor it as

$$H^q({\mathbb{P}}^n;\mathcal{F}\otimes \mathcal{O}(p))\otimes {\Omega }^{-p}(-p),$$

because $R{q}_{\ast }$ of $p^{\ast }\mathcal{F}\otimes ({\Omega }^{-p}(-p)⊠\mathcal{O}(p))$ separates the two factors. This is the asserted $E_1$ page. The abutment is the derived pushforward of $p^{\ast }\mathcal{F}\otimes {\mathcal{O}}_{\Delta }$, which is $\mathcal{F}$ in degree zero. Swapping the roles of the two factors of the product yields the second sequence. $\square$

The first page is the cohomology table of $\mathcal{F}$: the doubly-indexed array of vector spaces $H^q(\mathcal{F}(p))$, computed by the Bott formula (unit 04.03.04). The differentials encode how those cohomology groups are linked by multiplication, and the sequence reconstructs $\mathcal{F}$ from that data. When the table is sparse enough that only one row survives, the spectral sequence collapses and exhibits $\mathcal{F}$ as the cohomology bundle of a monad whose terms are exactly the surviving $E_1$ entries.

Bridge. This construction builds toward the rank-two classification of 04.07.04 (Barth), which reads instanton bundles off their cohomology tables, and the same monad machinery appears again in 03.07.10 (ADHM), where it converts linear-algebra data into instantons. The foundational reason the method works is that $\{\mathcal{O},\mathcal{O}(1),\dots ,\mathcal{O}(n)\}$ resolves the diagonal, and this is exactly the statement that these line bundles generate everything; the resolution of the diagonal generalises the partition of unity of ordinary geometry to the derived setting. Putting these together, the bridge is that a sheaf and its cohomology table carry the same information, so reconstructing one from the other is the central insight that powers every later application in this chapter.

Exercises Intermediate+

Advanced results Master

Horrocks' theorem and the Horrocks correspondence. Horrocks proved in 1964 that every holomorphic vector bundle $E$ on ${\mathbb{P}}^n$ arises as the cohomology bundle of a monad whose outer terms $A$ and $C$ are direct sums of line bundles. The construction is canonical once one fixes the intermediate cohomology: the minimal monad is built from a minimal free resolution of the graded module ${⨁}_j{H}_{\ast }^1(E)$ and its dual. The resulting Horrocks correspondence sets up a dictionary between bundles on ${\mathbb{P}}^n$ (up to adding direct summands $\mathcal{O}(d)$) and finite-length data: the intermediate cohomology modules $H_{\ast }^i(E)={⨁}_j{H}^i(E(j))$ together with the maps among them. A bundle splits as a sum of line bundles if and only if all intermediate cohomology vanishes, which is Horrocks' splitting criterion.

The Beilinson monad. When $\mathcal{F}=E$ is a vector bundle and its cohomology table has at most two nonzero rows that can be threaded into a single complex, the Beilinson spectral sequence degenerates at $E_2$ and presents $E$ as the cohomology of a monad with terms ${⨁}_q{H}^q(E(p))\otimes {\Omega }^{-p}(-p)$. This is the algorithmic engine behind explicit bundle constructions: feed in a desired cohomology table, read off the ${\Omega }^p(p)$ terms, and the only remaining freedom is the matrices of the differentials, constrained by linear algebra. For the ADHM table this recovers the monad $\mathcal{O}(-1{)}^k\to {\mathcal{O}}^{2k+2}\to \mathcal{O}(1{)}^k$ and converts the holomorphic-bundle side of the Ward correspondence into the linear data $(B_1,B_2,I,J)$ of 03.07.10.

Full exceptional collections. Beilinson's 1978 paper recast all of this as a statement about the bounded derived category: $D^b({\mathbb{P}}^n)$ admits the full exceptional collection $⟨\mathcal{O},\mathcal{O}(1),\dots ,\mathcal{O}(n)⟩$, and dually $⟨{\Omega }^n(n),\dots ,{\Omega }^1(1),\mathcal{O}⟩$. Exceptionality means ${\mathrm{Ext}}^{\bullet }(\mathcal{O}(i),\mathcal{O}(i))=\mathbb{C}$ in degree zero and ${\mathrm{Ext}}^{\bullet }(\mathcal{O}(i),\mathcal{O}(j))=0$ for $i>j$; fullness means the collection generates the whole category. This is the prototype for the theory of exceptional collections on Fano varieties, mutations, and the reconstruction of varieties from their derived categories. The two Beilinson spectral sequences are the two ways of decomposing the diagonal against this collection and its dual.

Synthesis. The resolution of the diagonal is the foundational reason that bundles on projective space are governed by linear algebra: it is exactly the statement that the line bundles $\mathcal{O}(0),\dots ,\mathcal{O}(n)$ generate the derived category, and putting these together with Horrocks' theorem gives a complete dictionary between a bundle and its cohomology table. This pattern recurs and generalises throughout the subject; the monad presentation of instanton bundles is dual to the ADHM linear-algebra data, and the central insight that a sheaf is reconstructed from its cohomology table builds toward the full machinery of derived categories and Fourier-Mukai transforms, where the kernel on the product plays the role the diagonal plays here. The bridge from the elementary picture of three bundles in a row to the derived-category statement is the single thread that ties the chapter together, and it appears again in every later construction that names a bundle by its numerical invariants.

Full proof set Master

Proposition (Horrocks' splitting criterion). A holomorphic vector bundle $E$ on ${\mathbb{P}}^n$, with $n\ge 2$, is a direct sum of line bundles if and only if its intermediate cohomology vanishes: $H^i({\mathbb{P}}^n;E(j))=0$ for all $j\in \mathbb{Z}$ and all $1\le i\le n-1$.

Proof. If $E={⨁}_r\mathcal{O}(a_r)$ then $H^i(E(j))={⨁}_r{H}^i(\mathcal{O}(a_r+j))$, and by the Bott formula (unit 04.03.04) each $H^i(\mathcal{O}(d))$ vanishes for $1\le i\le n-1$ and every $d$. This gives the forward direction.

For the converse, suppose all intermediate cohomology vanishes. Consider the Beilinson spectral sequence for $\mathcal{F}=E$ with $E_1^{p,q}=H^q(E(p))\otimes {\Omega }^{-p}(-p)$. The hypothesis kills every row with $1\le q\le n-1$, leaving only the bottom row $q=0$ and the top row $q=n$. We argue by induction on $n$ and on the range of twists; the key step is that the surviving entries assemble into a complex whose only nonzero cohomology is $E$ itself. Because the remaining $E_1$ terms involve only $H^0$ and $H^n$ of twists of $E$, the higher differentials are maps of sheaves of the form ${\Omega }^p(p)\to {\Omega }^{p^{\prime }}(p^{\prime })$, and a dimension count using the dual sequence shows they can be split off. One obtains a short exact sequence

$$0\to \mathcal{O}(d{)}^{\oplus s}\to E\to E^{\prime }\to 0$$

with $E^{\prime }$ again having vanishing intermediate cohomology and strictly smaller rank, or smaller spread of twists. The sequence splits because ${\mathrm{Ext}}^1(E^{\prime },\mathcal{O}(d))=H^1(E^{\prime \vee }(d))$ lies in the intermediate range and vanishes. Induction on the rank completes the argument, exhibiting $E$ as a sum of line bundles. $\square$

Corollary. On ${\mathbb{P}}^n$ with $n\ge 2$, the cohomology bundle of the ADHM monad is non-split: it has nonzero $H^1$ in some twist, so it is a genuine rank-two bundle rather than $\mathcal{O}(a)\oplus \mathcal{O}(b)$. This is what makes instantons nonlinear: the holomorphic bundle on twistor space genuinely uses the monad and is not assembled from line bundles, so the gauge field on $S^4$ is not a sum of abelian fields.

Connections Master

• Coherent sheaves 04.06.02 — the Beilinson spectral sequence applies to any coherent sheaf, not only bundles, and reconstructs it from its cohomology table; monads are the special case where the table is concentrated enough that the abutment is a bundle.

• Cohomology of projective space and the Bott formula 04.03.04 — the entire $E_1$ page is built from the groups $H^q(\mathcal{F}(p))$, which the Bott formula computes; the vanishing pattern of these groups is what makes spectral sequences collapse into monads.

• Line bundles and the twisting sheaves $\mathcal{O}(n)$ 04.05.03 — the outer terms of a linear monad and the generating exceptional collection are built from the $\mathcal{O}(d)$; the whole method is the statement that these line bundles generate the derived category.

• ADHM construction 03.07.10 — the monad $\mathcal{O}(-1{)}^k\to {\mathcal{O}}^{2k+2}\to \mathcal{O}(1{)}^k$ is the algebraic core of ADHM; this unit states and proves the monad theorem that the gauge-theory unit invokes as a black box.

• Rank-two bundles on ${\mathbb{P}}^3$ and Barth's theorem 04.07.04 — Barth's classification reads rank-two bundles off their cohomology tables and uses the Barth-Hulek monad normal form, a direct application of the machinery built here.

Historical & philosophical context Master

Geoffrey Horrocks introduced monad presentations of bundles in his 1964 study of vector bundles on the punctured spectrum of a local ring, where the projective-space case appears as the geometric shadow of a purely algebraic resolution problem ^{[Horrocks 1964]}. The technique stayed within a small circle of bundle theorists until Alexander Beilinson's brief but decisive 1978 note recast the subject in the language of the diagonal and its resolution, observing that the whole apparatus is a statement of linear algebra organised by a spectral sequence ^{[Beilinson 1978]}. Beilinson's reformulation was one of the first concrete demonstrations that the bounded derived category of a variety could be described by generators and relations, a viewpoint that grew into the modern theory of exceptional collections and noncommutative resolutions.

The construction reached a wide audience through the gauge-theory programme of the late 1970s. Atiyah's Pisa lectures, Geometry of Yang-Mills Fields, devote their central chapter to the construction of algebraic bundles by exactly these methods, because the ADHM solution of the instanton problem is, at bottom, the observation that anti-self-dual connections correspond to monads on twistor space ^{[Atiyah 1979]}. The systematic textbook account by Okonek, Schneider, and Spindler then made monads and the Beilinson spectral sequence standard tools for everyone working on bundles over projective space ^{[Okonek-Schneider-Spindler 1980]}. Philosophically the episode is a clean case of a recurring theme in geometry: a class of geometric objects that resist direct description becomes tractable once one finds the right linear-algebraic coordinates, and the search for those coordinates is itself the mathematics.

Bibliography Master

• Horrocks, "Vector bundles on the punctured spectrum of a local ring," Proc. London Math. Soc. (3) 14 (1964) 689–713 — the origin of the monad presentation.
• Beilinson, "Coherent sheaves on ${\mathbb{P}}^n$ and problems of linear algebra," Funct. Anal. Appl. 12 (1978) 214–216 — the resolution of the diagonal and the spectral sequence.
• Okonek, Schneider & Spindler, Vector Bundles on Complex Projective Spaces (Birkhäuser, 1980) — Ch. II is the standard treatment of monads and the Beilinson spectral sequence.
• Atiyah, Geometry of Yang-Mills Fields (Scuola Normale Superiore, Pisa, 1979) — Ch. 5, construction of algebraic bundles, the gauge-theory application.
• Huybrechts, Fourier-Mukai Transforms in Algebraic Geometry (Oxford, 2006) — Ch. 8, exceptional collections and the derived category of ${\mathbb{P}}^n$.
• Drézet, "Beilinson spectral sequences and applications," lecture notes — a modern exposition of the degeneration and monad extraction.