Simplicity and exceptionality of syzygy bundles over ${\mathbb P}^n$

Abstract

In this work, we will prove results that ensure the simplicity and the exceptionality of vector bundles, which are defined by the splitting of pure resolutions. We will call such objects syzygy bundles.

1 Introduction

The study of particular families of vector bundles over projective varieties has always taken a great part in algebraic geometry. In particular, many authors focused on the family of syzygy bundles, defined as the kernel of an epimorphism of the form

$$\begin{aligned} \phi : \oplus _{i=1}^t {\mathcal O}_{{\mathbb P}^n}(-d_i) \longrightarrow {\mathcal O}_{{\mathbb P}^n}, \end{aligned}$$

that have been studied in the last decades. Brenner in [4] gives combinatorial conditions for (semi)stability of the syzygy bundles on ${\mathbb P}^n$ when they are given by monomial ideals. Coanda in [5] studies stability for syzygies on ${\mathbb P}^n$ defined by polynomials of the same degree, of any possible rank for $n \ge 3$. Costa, Marques, and Miró-Roig, see [6], also study stability of syzygies on ${\mathbb P}^n$ given by polynomials of same degree and studied moduli spaces.

Ein, Lazarsfeld, and Mustopa in [8, 9] extend the problem for smooth projective varieties X, studying the stability of the syzygy bundles that are given by the kernel of the evaluation map $\text{ eval }_L: H^0(L)\otimes _{{\mathbb K}} \mathcal {O}_X \longrightarrow L$ where $L$ is a very ample line bundle over $X$.

We define the syzygy bundles on ${\mathbb P}^n$ as the vector bundles coming from the splitting of pure resolutions of the form

(1)

into short exact sequences. Observe that the first syzygy bundle $F$ in (1) obtained as

$$\begin{aligned} 0\longrightarrow F \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _{1}}(-d_{1}) \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _{0}} \longrightarrow 0 \end{aligned}$$

is also a syzygy bundle in the sense of [4, 5] and [6].

In the second section, we recall some notions on pure resolutions, and we introduce in detail what we will mean by syzygy bundle on the projective space.

In the third section, we will prove two results, see Theorems 3.1 and 3.6, which ensure the simplicity of the syzygy bundles previously defined. Recall that a vector bundle $E$ on $\mathbb {P}^n$ is simple if $\dim \mathrm{Hom }(E,E) = 1$. The results here generalize the ones proved in Section 4 and Section 5 of [13].

In particular, we provide an answer to a question proposed by Herzog and Kühl in [11], where they wonder whether the modules coming from linear pure resolution of monomial ideals are indecomposable or not. We will be able to ensure such property under specific hypotheses, see Remark 3.9.

In the fourth section, we will show necessary and sufficient conditions to prove exceptionality of the bundles, see Theorem 4.1, and we will state a conjecture, which relates syzygy bundles with Steiner bundles. There is a long-standing conjecture saying that every exceptional bundle on $\mathbb {P}^n$ is stable. For the case $n=2$, it was proved by Drèzet and Le Potier in [7], and for $n=3$, it was proved by Zube in [17]. There are other results in this sense for some families of vector bundles; for example, Brambilla [1] proved that exceptional Steiner bundles $S$ on $\mathbb {P}^n$, for $n \ge 2$, given by

are stable.

In the fifth section, we will consider some classical pure resolutions, studying when the bundles defined in their splitting are simple and when exceptional.

2 Preliminaries

In this section, we fix the notation that will be used in this work, and we recall some basic definitions and results. Let ${\mathbb K}$ be an algebraically closed field of characteristic 0 and let $R = {\mathbb K}[x_0, \ldots ,x_n]$ be the ring of polynomials in $n+1$ variables. Let $M$ be a graded $R$-module.

An $R$-module $N \ne 0$ is said to be a $k$-syzygy of $M$ if there is an exact sequence of graded $R$-modules

where the modules $F_i$ are free $R$-modules.

We say that $M$ has a finite projective dimension if there exist a free resolution over $R$

(2)

The least length $s$ of such resolutions is called the projective dimension of $M$ and denoted by $\mathrm{pd}(M)$. The resolution (2) is minimal if $\mathrm{im}\,\varphi _i \subset m F_{i-1}$, $\forall \, i$, where $m = (x_0, \ldots , x_n)$ is the irrelevant ideal of $R$. From the Hilbert syzygy Theorem, see for example [15, Theorem 1.1.8], we have that $\mathrm{pd}(M) \le n+1$. If $M$ has a graded minimal free resolution

then the integers $\beta _{i,j}(M) = \dim \mathrm{Tor}^R_i (M, {\mathbb K})_j$ are called the $(i,j)$-th graded Betti number of $M$, and $\beta _i := \sum _j \beta _{i,j}(M)$ is the $i$th total Betti number of $M$.

We say $M$ has a pure resolution of type $d = (d_0, \ldots , d_p)$ if it is given by

with $d_0 < d_1 < \cdots <d_p, d_i \in \mathbb {Z}$.

We say that $M$ has a linear resolution if it has a pure resolution of type $(0, 1, \ldots , p)$.

Eisenbud and Schreyer [10] proved the following result conjectured by Boij and Söderberg [3].

Theorem 2.1

For any degree sequence $d = (d_0,\ldots , d_p)$, there is a Cohen–Macaulay module $M$ with a pure resolution of type $d$.

Consider $S = {\mathbb K}[x_0, \ldots ,x_m]$ the ring of polynomials in $m+1$ variables. Let $d = (d_0, \ldots , d_p)$ be a degree sequence. Then, by Theorem 2.1, there is a Cohen–Macaulay module $M$ with pure resolution

Let $\overline{M}$ be the Artinian reduction of $M$. Then, $\overline{M}$ is an Artinian module with pure resolution

(3)

where $R= {\mathbb K}[x_0,\ldots , x_n]$ with $n = p-1$.

We now want to pass from modules to vector bundles and study pure resolutions involving them. Assume we have an Artinian module $\overline{M}$ with pure resolution (3); sheafifying the complex, we obtain

(4)

During this paper, we will be interested in such resolution, in particular, we will be interested in studying properties of the bundles coming by splitting the resolution in short exact sequences.

Definition 2.2

We will call syzygy bundles the vector bundles, which arise by splitting of resolutions of the type (4).

We conclude this section recalling the following notions and results on vector bundles.

Let $E$ be a vector bundle on $\mathbb {P}^{n}$. A resolution of $E$ is an exact sequence

(5)

where every $F_i$ splits as a direct sum of line bundles.

One can show that every vector bundle on ${\mathbb {P}^n}$ admits resolution of the form (5), see [12, Proposition 5.3]. The minimal number $d$ of such resolution is called homological dimension of $E$, and it is denoted by $\mathrm{hd}(E)$. For a coherent sheaf $F$ on ${\mathbb P}^n$, let us denote the graded $R$-module $\oplus _{j \in {\mathbb Z}}H^q(F(j)) = H^{q}_{*}(F) .$ Bohnhorst and Spindler proved the following two results, [2, Proposition 1.4] and [2, Corollary 1.7], respectively.

Proposition 2.3

Let $E$ be a vector bundle on ${\mathbb {P}^n}$. Then,

$$\begin{aligned} \mathrm{hd}(E) \le d \Longleftrightarrow H^{q}_{*}(E) = 0, \forall \; 1 \le q \le n-d-1. \end{aligned}$$

Proposition 2.4

Let $E$ be a nonsplitting vector bundle on ${\mathbb {P}^n}$. Then,

$$\begin{aligned} \mathrm{rk}(E) \ge n+1 - \mathrm{hd}(E). \end{aligned}$$

Recall that a vector bundle $E$ on ${\mathbb P}^n$ is simple if $\dim \mathrm{{Hom}}(E,E)=1$, and it is exceptional if it is simple and $\mathrm{Ext}^i(E,E)=0,$ for $i \ge 1.$ We also recall the notion of cokernel bundles and Steiner bundles, as defined, respectively, in [1] and [16].

Definition 2.5

Let $E_0$ and $E_1$ be two vector bundles on ${\mathbb P}^n$, with $n\ge 2$. A cokernel bundle of type $(E_0,E_1)$ on ${\mathbb P}^n$ is a vector bundle $C$ defined by the following short exact sequence

$$\begin{aligned} 0 \longrightarrow E_0^a \longrightarrow E_1^b \longrightarrow C \longrightarrow 0 \end{aligned}$$

where $b \mathrm{rk }E_1 - a \mathrm{rk }E_0 \ge n$, with $a,b \in \mathbb {N}$, and $E_0$, $E_1$ satisfy the following conditions:

• $E_0$ and $E_1$ are simple;

• $\mathrm{Hom }(E_1,E_0) = 0$;

• $\mathrm{Ext }^1(E_1,E_0) = 0$;

• the bundle $E_0^\vee \otimes E_1$ is globally generated;

• $W = \mathrm{Hom }(E_0,E_1)$ has dimension $w \ge 3$.

If, moreover,

$$\begin{aligned} \mathrm{Ext}^i(E_1,E_0) = 0, \,\,\text{ for } \text{ each }\,\, i \ge 2 \end{aligned}$$

and

$$\begin{aligned} \mathrm{Ext}^i(E_0,E_1) = 0, \,\,\text{ for } \text{ each }\,\, i \ge 1, \end{aligned}$$

the pair $(E_0,E_1)$ is called strongly exceptional, and the bundle $C$ is called a Steiner bundle of type $(E_0,E_1)$ on ${\mathbb P}^n$.

3 Simplicity of syzygy bundles

In this section, we will give some results that ensure the simplicity of the syzygy bundles of the following pure resolution

(6)

with $d_1<d_2<\ldots <d_{n+1}$ and $d_i>0$ for each $i$, which splits in short exact sequences

$$\begin{aligned} \begin{array}{c} 0 \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _{n+1}}(-d_{n+1}) \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _{n}}(-d_{n}) \longrightarrow G_1 \longrightarrow 0\\ \vdots \\ 0 \longrightarrow G_i \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _{n-i}}(-d_{n-i}) \longrightarrow G_{i+1} \longrightarrow 0\\ \vdots \\ 0 \longrightarrow G_{n-1} \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _{1}}(-d_{1}) \longrightarrow {\mathcal O}_{{\mathbb P}_n}^{\beta _0} \longrightarrow 0 \end{array} \end{aligned}$$

(7)

We will also consider the dual resolution of (6) and tensor it by ${\mathcal O}_{{\mathbb P}^n}(-d_{n+1})$, obtaining

(8)

which splits as

$$\begin{aligned} \begin{array}{c} 0 \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _0}(-d_{n+1}) \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _{1}}(d_1-d_{n+1}) \longrightarrow F_1 \longrightarrow 0\\ \vdots \\ 0 \longrightarrow F_j \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _{j+1}}(d_{j+1}-d_{n+1}) \longrightarrow F_{j+1} \longrightarrow 0\\ \vdots \\ 0 \longrightarrow F_{n-1} \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _{n}}(d_{n}-d_{n+1}) \longrightarrow {\mathcal O}_{{\mathbb P}_n}^{\beta _{n+1}} \longrightarrow 0 \end{array} \end{aligned}$$

(9)

Let us notice that we have supposed, without loss of generality, that $d_0=0$; else we can tensor the resolution (4) by ${\mathcal O}_{{\mathbb P}^n}(-d_0)$ in order to obtain a new resolution as in (6). Let us prove now some results which ensure the simplicity of the bundles $F_i$, for $i$ from 1 to $n-1$; in particular, the next theorem tell us when the syzigies are simple only looking at the first or the last Betti number.

Theorem 3.1

Consider a pure resolution as in (6). If $\beta _0 = 1$ or $\beta _{n+1}=1$, then all bundles $F_i$, for $i$ from 1 to $n-1$, are simple.

Proof

Let us consider first the case $\beta _0=1$, whose importance will be explained by Corollary 3.2.

Let us prove first that the bundle $F_1$ is simple.

Consider the exact sequence, obtained by (9),

$$\begin{aligned} 0 \longrightarrow (F^\vee _1)^{\beta _0}(-d_{n+1}) \longrightarrow (F^\vee _1)^{\beta _{1}}(d_1-d_{n+1}) \longrightarrow F^\vee _1 \otimes F_1 \longrightarrow 0 \end{aligned}$$

which induces the long exact sequence in cohomology

$$\begin{aligned}&0 \longrightarrow H^0((F_1^\vee )^{\beta _0}(-d_{n+1})) \longrightarrow H^0((F_1^\vee )^{\beta _1}(d_1-d_{n+1}))\\&\longrightarrow H^0(F_1^\vee \otimes F_1) \longrightarrow H^1((F^\vee _1)^{\beta _0}(-d_{n+1})) \longrightarrow \cdots \end{aligned}$$

Taking again the short exact sequences obtained by dualizing (9) and tensoring by ${\mathcal O}_{{\mathbb P}^n}(d_1 - d_{n+1})$

$$\begin{aligned} 0 \longrightarrow F^\vee _{i+1}(d_1-d_{n+1}) \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _{i+1}}(d_1-d_{i+1}) \longrightarrow F^\vee _{i}(d_1-d_{n+1}) \longrightarrow 0, \end{aligned}$$

for $i$ from 1 to $n-2$, we obtain the following chain of isomorphisms

$$\begin{aligned} H^0(F_1^\vee (d_1-d_{n+1})) \simeq H^1(F_2^\vee (d_1-d_{n+1})) \simeq \cdots \simeq H^{n-2}(F_{n-1}^\vee (d_1-d_{n+1}))= 0, \end{aligned}$$

and the isomorphisms are true (especially the first one) because $d_1 < d_{n+1}$ and the vanishing comes from the short exact sequences of (9). Combining these two results, we get an injective map

$$\begin{aligned} H^0(F_1^\vee \otimes F_1) \hookrightarrow H^1((F_1^\vee )^{\beta _0}(-d_{n+1})). \end{aligned}$$

Consider the short exact sequence

$$\begin{aligned} 0 \longrightarrow F_1^\vee (-d_{n+1}) \longrightarrow {\mathcal O}^{\beta _1}_{{\mathbb P}^n}(-d_1) \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _0} \longrightarrow 0 \end{aligned}$$

(10)

from which it is straightforward to obtain $\beta _0 = h^1\left( F_1^\vee (-d_{n-1})\right)$; this implies, in the case $\beta _0 = 1$, that the $F_1$ is a simple bundle.

Let us prove now that each bundle $F_i$ is simple, for $i$ from $2$ to $n-1$.

Consider the following exact sequence obtained from (9)

$$\begin{aligned} 0 \longrightarrow F_{i-1} \otimes F_i^\vee \longrightarrow (F_i^\vee )^{\beta _i}(d_i-d_{n+1}) \longrightarrow F_i \otimes F_i^\vee \longrightarrow 0 \end{aligned}$$

(11)

Taking the exact sequences of type

$$\begin{aligned} 0 \longrightarrow F^\vee _{i+1}(d_i-d_{n+1}) \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _{i+1}}(d_i-d_{i+1}) \longrightarrow F^\vee _{i}(d_i-d_{n+1}) \longrightarrow 0 \end{aligned}$$

for $i$ from 1 to $n-2$, and their induced long exact sequence in cohomology, we get a chain of isomorphisms of type

$$\begin{aligned} H^0(F_i^\vee (d_i-d_{n+1})) \simeq H^1(F_{i+1}^\vee (d_i-d_{n+1})) \simeq \cdots \simeq H^{n-1-i}(F_{n-1}^\vee (d_i-d_{n+1}) = 0, \end{aligned}$$

(12)

again because $d_i < d_{n+1}$ and the vanishing comes from short exact sequences of (9). Therefore, inducing the long exact sequence in cohomology of (11), we have an inclusion of type

$$\begin{aligned} H^0(F_i\otimes F_i^\vee ) \hookrightarrow H^1(F_{i-1}\otimes F_i^\vee ). \end{aligned}$$

Proceeding step by step, lowering by one the value of $i$, and using similar isomorphisms as in (12) that are consequence of the short exact sequences in (9), we manage to obtain the following inclusions

$$\begin{aligned} H^1(F_{i-1}\otimes F_i^\vee ) \hookrightarrow H^2(F_{i-2}\otimes F_i^\vee )\hookrightarrow \cdots \hookrightarrow H^{i-1}(F_{1}\otimes F_i^\vee ) \hookrightarrow H^i(F_i^\vee (-d_{n+1})). \end{aligned}$$

In order to compute the last cohomology group, we consider, as before, the exact sequences of the following form

$$\begin{aligned} 0 \longrightarrow F^\vee _{i+1}(-d_{n+1}) \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _{i+1}}(-d_{i+1}) \longrightarrow F^\vee _{i}(-d_{n+1}) \longrightarrow 0 \end{aligned}$$

for $i$ from 1 to $n-2$, obtaining, by our hypothesis $\beta _0 = 1$,

$$\begin{aligned} H^i(F_i^\vee (-d_{n+1})) \simeq H^{i-1}(F_{i-1}^\vee (-d_{n+1})) \simeq \cdots \simeq H^1(F_1^\vee (-d_{n+1})) \simeq \mathbb {C}. \end{aligned}$$

This proves that the bundle $F_i$ is simple.

The case $\beta _{n+1}=1$ can be proved, by duality, applying the same technique. Indeed, we can define $\tilde{d}_i = d_{n+1} - d_{n+1-i}$, and dualizing the resolution (8) and tensoring by ${\mathcal O}_{{\mathbb P}_n}(-\tilde{d}_{n+1})$, we obtain a new resolution of the form

(13)

where, as before, the integers $\tilde{d}_i$ satisfy $\tilde{d}_{n+1} > \tilde{d}_n > \ldots > \tilde{d}_1 >0$, and we apply the previous technique. $\square$

As a corollary, we have the following.

Corollary 3.2

Consider a quotient ring $A = R/I$ where $I$ is Artinian module and its pure resolution. Then, each vector bundle $F_i$, arising from the splitting of the resolution in short exact sequences, is simple.

Proof

Since $A$ is a quotient, we have that $\beta _0 =1$, then we can apply the previous theorem and obtain that all the bundles $F_i$ are simple. $\square$

With the next lemmas, we give an explicit description and boundaries for the Betti number arising in the resolution we are considering, which will be useful to prove a different theorem about simplicity of the syzygies.

Lemma 3.3

The syzygies in the short exact sequences (9) satisfies

$$\begin{aligned} h^{0}(F_{i-1}^\vee (d_i-d_{n+1})) = \beta _i, \; for \; i =2, \ldots ,n. \end{aligned}$$

Proof

Twisting the short exact sequences, we have

One can check that

$$\begin{aligned} h^{0}(F_i^\vee (d_i -d_{n+1})) = h^1(F_i^\vee (d_i-d_{n+1})) \end{aligned}$$

and the result follows. $\square$

Lemma 3.4

Consider the syzygies $G_i$ and $F_i$ from the short exact sequences (7) and (9). Then, hd$(G_i) =$hd$(F_i) =i$, for $i=1,\ldots ,n-1$.

Proof

Let us prove for $F_i$. The case of $G_i$ is analogous. We prove it by induction on $i$. From the short exact sequences (9), it is clear that hd$(F_1) = 1$. Let us suppose that hd$(F_{i-1}) = i-1$. We know that hd$(F_i) \le i$. Suppose hd$(F_i) \le i-1$. By Proposition 2.3,

$$\begin{aligned} H^{q}_{*}(F_i) = 0,\quad \forall \; 1 \le q \le n-i. \end{aligned}$$

Since $H^{n-i}_{*}(F_i) \simeq H^{n-i+1}_{*}(F_{i-1})$ from the sequences (9), and hd$(F_{i-1}) = i-1$, by induction, there exists $t \in \mathbb {Z}$ such that $H^{n-i+1}(F_{i-1}(t)) \ne 0$. Therefore, hd$(F_i) = i$. $\square$

Lemma 3.5

The Betti numbers $\beta _i$ from the sequence (6) satisfy the inequalities

$$\begin{aligned} \left. \begin{array}{rcll} &{}&{}\beta _1 - \beta _0 \ge n &{} \\ &{}&{}\beta _i \ge 2n-2i+3, &{} \text{ for } \;\; 2 \le i \le \frac{n+1}{2} \\ &{}&{}\beta _i \ge 2i+1, &{} \text{ for } \;\; \frac{n+1}{2} \le i \le n-1\\ &{}&{}\beta _n - \beta _{n+1} \ge n &{} \end{array}\right. \end{aligned}$$

In particular, $\beta _i \ge 3$, for $2 \le i \le n-1$.

Proof

We have by Lemma 3.4 that hd$(G_i) =$hd$(F_i) = i$, $1 \le i \le n-1$. With Proposition 2.4

$$\begin{aligned} \text{ rk }(E) \ge n+1 - \text{ hd }(E) \end{aligned}$$

and using the short exact sequences of $F_i$ for $1 \le i \le \frac{n+1}{2}$ and the sequences of $G_i$ for $\frac{n+1}{2} \le i \le n-1$, we prove the inequalities. $\square$

We are now ready to state the second theorem on the simplicity of the syzygy bundles.

Theorem 3.6

Consider the pure resolution (8) and the syzygies given by the short exact sequences. Then, if $F_1$ or $F_{n-1}$ are simple, all the syzygies are simple.

Proof

Suppose $F_1$ is simple. Let us prove by induction hypothesis that $F_i$ is simple for $i=1, \ldots , n-1$. Consider an injective generic map

$$\begin{aligned} \alpha _i : F_{i-1} \rightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _i}(d_i -d_{n+1}). \end{aligned}$$

We have the following properties:

$(i)$ :

$F_{i-1}, {\mathcal O}_{{\mathbb P}^n}(d_i-d_{n+1})$ are simple, the first bundle by induction hypothesis;

$(ii)$ :

$\mathrm{Hom }({\mathcal O}_{{\mathbb P}^n}(d_i -d_{n+1}),F_{i-1}) = 0.$ It follows from

$$\begin{aligned} H^{0}(F_{i-1}(d_{n+1}-d_i)) \simeq H^j(F_{i-j-1}(d_{n+1}-d_i)), \; 0 \le j \le i-2 \end{aligned}$$

and $H^{i-2}(F_1(d_{n+1}-d_i)) = H^{i-1}({\mathcal O}_{{\mathbb P}^n}(-d_i)) = 0$, by the sequence (9);

$(iii)$ :

$\mathrm{Ext }^1({\mathcal O}_{{\mathbb P}^n}(d_i -d_{n+1}),F_{i-1})=0.$ In fact, by the above sequence (9)

$$\begin{aligned} H^{1}(F_{i-1}(d_{n+1}-d_i)) \simeq H^j(F_{i-j}(d_{n+1}-d_i)), \; 0 \le j \le i-1 \end{aligned}$$

and $H^{i-1}(F_1(d_{n+1}-d_i)) = H^{i}({\mathcal O}_{{\mathbb P}^n}(-d_i)) = 0$;

$(iv)$ :

$F^\vee _{i-1} \otimes {\mathcal O}_{{\mathbb P}^n}(d_i -d_{n+1})$ is globally generated. This is clear from the short exact sequences also in (9).

$(iv)$ :

$\dim \mathrm{Hom }(F_{i-1}, {\mathcal O}_{{\mathbb P}^n}(d_i-d_{n+1})) \ge 3$. Follows from Lemmas 3.3 and 3.5.

Hence, we have that $F_{i-1}$ and ${\mathcal O}_{{\mathbb P}^n}(d_i-d_{n+1})$ satisfies the conditions of cokernel bundles (see Definition 2.5); therefore, $\overline{F}_i = \mathrm{coker} \alpha _i$ is a cokernel bundle and since $1 + {\beta _i}^2 - h^0(F^\vee _{i-1}(d_i-d_{n+1})) \beta _i = 1$ by Lemma 3.3, we have that $\overline{F_i}$ is simple, see [1, Theorem 4.3].

Notice that we have two short exact sequences of the form

Having a 1:1 correspondence between $\alpha _i$ and $h_i$, we can always get $h_i$ an isomorphism and therefore, $\overline{F_i} \simeq F_i$ and $F_i$ is simple.

Regarding the other case, i.e., supposing that $F_{n-1}$ is simple, we can define new coefficients $\tilde{d}_i$, in the same way as in the last part of the proof of Theorem 3.1, and take the dual of (8), tensor it by ${\mathcal O}_{{\mathbb P}_n}(-\tilde{d}_{n+1})$ and apply the same technique. $\square$

We would like to find conditions to grant simplicity for every syzygy bundle in the resolution; therefore, in the next results, we ask for conditions which give us either $F_1$ or $F_{n-1}$ simple bundles.

Corollary 3.7

Consider the complex (6). If $\beta _n-\beta _{n+1} = n$ or $\beta _1- \beta _0 = n$, then all the syzygies are simple.

Proof

Under this hypothesis, it follows from [2, Theorem 2.7] that either $F_{n-1}$ is stable or $F_1$ is stable, then all the syzygies are simple. $\square$

Corollary 3.8

Consider the complex (6). If the injective map

$$\begin{aligned} \alpha _{n+1}: {\mathcal O}_{{\mathbb P}^n}^{\beta _{n+1}}(-d_{n+1}) \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _{n}}(-d_{n}) \end{aligned}$$

is generic and ${\beta _{n+1}}^2 + \beta _{n}^2 -h^0({\mathcal O}_{{\mathbb P}^n}(d_{n+1}-d_{n})) \beta _{n+1} \beta _{n} \le 1$, or the injective map

$$\begin{aligned} \alpha _1^\vee : {\mathcal O}_{{\mathbb P}^n}^{\beta _0} \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _1}(d_1) \end{aligned}$$

is generic and ${\beta _0}^2 + \beta _{1}^2 - h^0({\mathcal O}_{{\mathbb P}^n}(d_1)) \beta _0 \beta _{1} \le 1$, then all the syzygies are simple.

Proof

If we have the hypothesis above, $\mathrm{coker}\alpha _{n+1} = F^\vee _{n-1}(-d_{n+1})$ or $\mathrm{coker}\alpha ^\vee _1 = F_1$ are simple cokernel bundles, see [1, Theorem 4.3], and the previous theorem applies. $\square$

We conclude this part with the following observation.

Remark 3.9

Consider the syzygy modules $N_i$, for $i$ from 1 to $p-2$, which are obtained by the pure resolution

Recalling the equivalence of category between modules and their sheafifications, we get that, if the vector bundles $F_i$ are simple, the modules $N_i$ are indecomposable.

4 Exceptionality of syzygy bundles

In this section, we state and prove sufficient and necessary conditions to ensure the exceptionality of the syzygy bundles $F_i$. We obtain the following result.

Theorem 4.1

Consider the syzygy bundles $F_i$ as defined in (9), for $i$ from 1 to $n-1$. Suppose also that $F_i$ are simple for each $i$, then every $F_i$ is exceptional if and only if each one of the following conditions hold

• (i) $\beta _0^2 + \beta _1^2 - \left( {\begin{array}{c}d_1+n\\ n\end{array}}\right) \beta _0\beta _1 = 1$;

• (ii) $d_1 \le n;$

• (iii)
  $$\begin{aligned} \left\{ \begin{array}{ll} H^{n-i+1}(F_{i-1}(d_{n+1}-d_i)) = H^i(F_i^\vee (d_i - d_{n+1})) = 0 &{} \text{ if }\,\;n\,\;\text{ is } \text{ even; }\\ H^{n-i+1}(F_{i-1}(d_{n+1}-d_i)) = H^i(F_i^\vee (d_i - d_{n+1})) = 0 &{} \text{ if }\,\;n\,\;\text{ is } \text{ odd } \text{ and } \;\, i \ne \frac{n+1}{2};\\ H^{n-i+1}(F_{i-1}(d_{n+1}-d_i)) \mathop {\simeq }\limits ^{H^i(\varphi )} H^i(F_i^\vee (d_i - d_{n+1})) &{} \text{ if }\,\;n\,\;\text{ is } \text{ odd } \text{ and } \;\, i=\frac{n+1}{2}. \end{array} \right. \end{aligned}$$

  If $n$ is odd and $i = \frac{n+1}{2}$, then $H^{n-i+1}(F_{i-1}(d_{n+1}-d_i)) \simeq H^i(F_{i-1} \otimes F_i^\vee )$ and the morphism $H^i(\varphi ): H^i(F_i^\vee \otimes F_{i-1}) \rightarrow H^i((F_i^\vee )^{\beta _i} (d_i - d_{n+1}))$ is the one obtained by the short exact sequence

  $$\begin{aligned} 0 \longrightarrow F_i^\vee \otimes F_{i-1} \mathop {\longrightarrow }\limits ^{\varphi } (F_i^\vee )^{\beta _i} (d_i - d_{n+1}) \longrightarrow F_i^\vee \otimes F_i \longrightarrow 0 \end{aligned}$$

  considering the long exact induced in cohomology.

Proof

We will first look for conditions, which are equivalent to the exceptionality of the bundle $F_1$; therefore, we must compute the cohomology of the bundle $F_1^\vee \otimes F_1$. Consider the short exact sequence

$$\begin{aligned} 0 \longrightarrow (F_1^\vee )^{\beta _0}(-d_{n+1}) \longrightarrow (F_1^\vee )^{\beta _1}(d_1-d_{n+1}) \longrightarrow F_1^\vee \otimes F_1 \longrightarrow 0. \end{aligned}$$

(14)

The first step consists now in calculating the cohomology of $F_1^\vee (-d_{n+1})$. We now consider the sequence

$$\begin{aligned} 0 \longrightarrow F_1^\vee (-d_{n+1}) \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _1}(-d_1) \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _0} \longrightarrow 0 \end{aligned}$$

obtaining that

$$\begin{aligned} \begin{array}{ll} H^i(F_1^\vee (-d_{n+1})) = 0 &{} \text{ for }\,\, i=0,2,3,\ldots ,n-2,n-1\\ H^1(F_1^\vee (-d_{n+1}) \simeq H^0({\mathcal O}_{{\mathbb P}^n}^{\beta _0}) \simeq {\mathbb K}^{\beta _0} \\ H^n(F_1^\vee (-d_{n+1}) \simeq H^n({\mathcal O}_{{\mathbb P}^n}^{\beta _1}(-d_1)) \end{array} \end{aligned}$$

We must now compute the cohomology of the second bundle appearing in (14), and we will do it using the short exact sequence

$$\begin{aligned} 0 \longrightarrow F_1^\vee (d_1-d_{n+1}) \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _1} \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _0}(d_1) \longrightarrow 0 \end{aligned}$$

from which we obtain that

$$\begin{aligned} H^0(F_1^\vee (d_1-d_{n+1})) \simeq \cdots \simeq H^{n-2}(F_{n-1}^\vee (d_1-d_{n+1})) = 0, \end{aligned}$$

that we have already computed in the proof of Theorem 3.1, and moreover

$$\begin{aligned} h^1(F_1^\vee (d_1-d_{n+1})) = \mathrm{dim} H^1(F_1^\vee (d_1-d_{n+1})) =\beta _0 \left( {\begin{array}{c}d_1+n\\ n\end{array}}\right) - \beta _1,\\ H^2(F_1^\vee (d_1-d_{n+1})) \simeq \cdots \simeq H^n(F_1^\vee (d_1-d_{n+1})) = 0. \end{aligned}$$

From the cohomology we have already calculated, we get that

$$\begin{aligned} H^2(F_1^\vee \otimes F_1) \simeq \cdots \simeq H^{n-2}(F_1^\vee \otimes F_1) \simeq H^n(F_1^\vee \otimes F_1) = 0. \end{aligned}$$

Recall that we supposed $F_1$ to be simple, hence we have the following exact sequence in cohomology

$$\begin{aligned} 0 \!\longrightarrow \! {\mathbb K}\!\longrightarrow \! H^1((F_1^\vee )^{\beta _0}(-d_{n+1})) \!\longrightarrow \! H^1((F_1^\vee )^{\beta _1}(d_1-d_{n+1})) \!\longrightarrow \! H^1(F_1^\vee \otimes F_1) \longrightarrow 0. \end{aligned}$$

Therefore, $H^1(F_1^\vee \otimes F_1)$ vanishes if and only if

$$\begin{aligned} \beta _0^2 + \beta _1^2 - \left( {\begin{array}{c}d_1+n\\ n\end{array}}\right) \beta _0\beta _1 = 1. \end{aligned}$$

The other cohomology which we need to vanish is given by

$$\begin{aligned} H^{n-1}(F_1^\vee \otimes F_1) \simeq H^n(F_1^\vee (-d_{n+1}) \simeq H^n({\mathcal O}_{{\mathbb P}^n}^{\beta _1}(-d_1)), \end{aligned}$$

that is equal to zero if and only if $d_1\le n$.

Let us suppose the bundle $F_{i-1}$ to be exceptional, and let us find conditions ensuring the exceptionality of $F_i$. In order to do so, consider an $i$ fixed from $2$ to $n-1$ and consider also the following short exact sequences

$$\begin{aligned} 0 \longrightarrow F_i^\vee \otimes F_{i-1} \longrightarrow (F_i^\vee )^{\beta _i}(d_i-d_{n+1}) \longrightarrow F_i^\vee \otimes F_i \longrightarrow 0\end{aligned}$$

(15)

$$\begin{aligned} 0 \longrightarrow F_i^\vee \otimes F_{i-1} \longrightarrow F_{i-1}^{\beta _i}(d_{n+1}-d_i) \longrightarrow F_{i-1}^\vee \otimes F_{i-1} \longrightarrow 0 \end{aligned}$$

(16)

Recall that by induction hypothesis, we have that

$$\begin{aligned} H^0(F_{i-1}^\vee \otimes F_{i-1}) \simeq {\mathbb K}\,\,\quad \text{ and }\quad \,\, H^k(F_{i-1}^\vee \otimes F_{i-1}) = 0 \,\,\text{ for } \text{ each }\,\, k >0. \end{aligned}$$

Let us compute the cohomology of the bundle $F_i^\vee (d_i-d_{n+1})$; using the usual short exact sequences defined in (9), we have that:

• if $k<i$ then $H^k(F_i^\vee (d_i-d_{n+1})) \simeq H^{k+n-i-1}(F_{n-1}^\vee (d_i-d_{n+1})) = 0$, because $k+n-1-i \le n-2$;

• similarly, if $k>i$ then $H^k(F_i^\vee (d_i-d_{n+1})) \simeq H^{k+1-i}(F_1^\vee (d_i-d_{n+1})) = 0$, because $k+1-i \ge 2$;

hence, we notice that the only possible nonvanishing cohomology of the bundle is given exactly by $H^i(F_i^\vee (d_i-d_{n+1})).$

Using Serre duality and a similar argument, we obtain that

$$\begin{aligned} H^k(F_{i-1}(d_{n+1}-d_i)) = 0, \,\,\text{ if }\,\, k \ne n-i+1 \end{aligned}$$

the only possible nonvanishing cohomology of the bundle is given exactly by $H^{n-i+1}(F_{i-1}(d_{n+1}-d_i)).$ Considering this and the induction hypothesis, we get from the exact sequence (16), that

$$\begin{aligned} H^1(F_i^\vee \otimes F_{i-1}) \simeq H^0(F_{i-1}^\vee \otimes F_{i-1}) \simeq {\mathbb K}\end{aligned}$$

and

$$\begin{aligned} H^k(F_i^\vee \otimes F_{i-1}) \simeq H^k(F_{i-1}(d_{n+1}-d_i)) \,\,\forall \,i=2,\ldots ,n. \end{aligned}$$

Notice that the following cohomology groups are isomorphic

$$\begin{aligned} H^i(F_i^\vee (d_i - d_{n+1})) \simeq H^1(F_1^\vee (d_i-d_{n+1})) \simeq H^{n-1}(F_{n-1}^\vee (d_i-d_{n+1})). \end{aligned}$$

Let us first look for an explicit expression of the group $H^1(F_1^\vee (d_i-d_{n+1}))$ and in order to do so, take the sequence

$$\begin{aligned} 0 \longrightarrow F_1^\vee (d_i-d_{n+1}) \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _1}(d_i-d_1) \longrightarrow {\mathcal O}^{\beta _0}_{{\mathbb P}^n}(d_i) \longrightarrow 0 \end{aligned}$$

and we are interested in the following part of the induced sequence in cohomology

$$\begin{aligned}&0 \longrightarrow H^0(F_1^\vee (d_i-d_{n+1})) \longrightarrow H^0({\mathcal O}_{{\mathbb P}^n}^{\beta _1}(d_i-d_1)) \longrightarrow H^0({\mathcal O}^{\beta _0}_{{\mathbb P}^n}(d_i))\\&\longrightarrow H^1(F_1^\vee (d_i-d_{n+1})) \longrightarrow 0 \end{aligned}$$

The problem now moves to the computation of the dimension of the vector space $H^0(F_1^\vee (d_i-d_{n+1}))$.

Consider the exact sequences

$$\begin{aligned} \begin{array}{c} 0 \longrightarrow F_2^\vee (d_i - d_{n+1}) \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _2}(d_i - d_2) \longrightarrow F_1^\vee (d_i-d_{n+1}) \longrightarrow 0\\ \vdots \\ 0 \longrightarrow F_{i-1}^\vee (d_i - d_{n+1}) \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _{i-1}}(d_i - d_{i-1}) \longrightarrow F_{i-2}^\vee (d_i-d_{n+1}) \longrightarrow 0\\ 0 \longrightarrow F_i^\vee (d_i - d_{n+1}) \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _i} \longrightarrow F_{i-1}^\vee (d_i-d_{n+1}) \longrightarrow 0 \\ 0 \longrightarrow F_{i+1}^\vee (d_i - d_{n+1}) \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _{i+1}}(\underbrace{d_i - d_{i+1}}_{<0}) \longrightarrow F_i^\vee (d_i-d_{n+1}) \longrightarrow 0 \end{array} \end{aligned}$$

We have already proven that $H^0(F_i^\vee (d_i-d_{n+1})) = H^1(F_i^\vee (d_i-d_{n+1}))= 0$ and also $H^1(F_j^\vee (d_i-d_{n+1}))=0$ for each $j = 2, \ldots , i+1$; hence, we obtain that

$$\begin{aligned} h^i(F_i^\vee (d_i-d_{n+1})) = h^1(F_1^\vee (d_i-d_n+1)) = \sum _{k=0}^i (-1)^k \beta _k \left( {\begin{array}{c}d_i-d_k+n\\ n\end{array}}\right) . \end{aligned}$$

Let us now “go to the other side”, arriving to $H^{n-1}(F_{n-1}^\vee (d_i-d_{n+1}))$.

Consider the short exact sequence

$$\begin{aligned} 0 \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _{n+1}}(d_i-d_{n+1}) \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _n}(d_i-d_n) \longrightarrow F_{n-1}^\vee (d_i-d_{n+1}) \longrightarrow 0 \end{aligned}$$

from which we induce the following part induced in cohomology

$$\begin{aligned}&0 \longrightarrow H^{n-1}(F_{n-1}^\vee (d_i-d_{n+1})) \longrightarrow H^n({\mathcal O}_{{\mathbb P}^n}^{\beta _{n+1}}(d_i-d_{n+1})) \longrightarrow H^n({\mathcal O}_{{\mathbb P}^n}^{\beta _n}(d_i-d_n)) \\&\longrightarrow H^n(F_{n-1}^\vee (d_i-d_{n+1})) \longrightarrow 0. \end{aligned}$$

As before, take

$$\begin{aligned} \begin{array}{c} 0 \longrightarrow F_{n-1}^\vee (d_i-d_{n+1}) \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _{n-1}}(d_i-d_{n+1}) \longrightarrow F_{n-2}^\vee (d_i-d_{n+1}) \longrightarrow 0\\ \vdots \\ 0 \longrightarrow F_{i+2}^\vee (d_i-d_{n+1}) \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _{i+2}}(d_i-d_{i+2}) \longrightarrow F_{i+1}^\vee (d_i-d_{n+1}) \longrightarrow 0\\ 0 \longrightarrow F_{i+1}^\vee (d_i-d_{n+1}) \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _{i+1}}(d_i-d_{i+1}) \longrightarrow F_{i}^\vee (d_i-d_{n+1}) \longrightarrow 0\\ 0 \longrightarrow F_{i}^\vee (d_i-d_{n+1}) \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _i} \longrightarrow F_{i-1}^\vee (d_i-d_{n+1}) \longrightarrow 0 \end{array} \end{aligned}$$

Suppose that $i<n-1$ (or else the computation comes directly considering only the first exact sequence and we will obtain the same result), we have that

$$\begin{aligned} H^n(F_i^\vee (d_i-d_{n+1})) \simeq H^{n-1}(F_i^\vee (d_i-d_{n+1})) = 0 \end{aligned}$$

and also

$$\begin{aligned} H^{n-1}(F_j^\vee (d_i-d_{n+1}))=0, \,\,\text{ for } \text{ each }\,\, j=i+1,\ldots ,n-2. \end{aligned}$$

We can conclude that

$$\begin{aligned} h^i(F_i^\vee (d_i-d_{n+1})) \!=\! h^{n-1}(F_{n-1}^\vee (d_i-d_{n+1})) = \left\{ \begin{array}{ll} \sum _{k=i+1}^{n+1} (-1)^{k+1} \beta _k \left( {\begin{array}{c}d_k-d_i-1\\ n\end{array}}\right) &{} \text{ for } \,\,n\,\,\text{ even }\\ \sum _{k=i+1}^{n+1} (-1)^{k} \beta _k \left( {\begin{array}{c}d_k-d_i-1\\ n\end{array}}\right)&\text{ for } \,\,n\,\,\text{ odd }. \end{array} \right. \end{aligned}$$

Let us focus now on the cohomology of the bundle $F_{i-1}(d_{n+1}-d_i)$.

We obtain by Serre duality that

$$\begin{aligned} H^k(F_{i-1}(d_{n+1}-d_i)) \simeq H^{n-k}(F_{i-1}^\vee (d_i-d_{n+1}-n-1)); \end{aligned}$$

therefore, we already know that

• $H^k(F_{i-1}(d_{n+1}-d_i))=0$ if $k\ne n-i+1$,

• $H^{n-i+1}(F_{i-1}(d_{n+1}-d_i)) \simeq H^{i-1}(F_{i-1}^\vee (d_i-d_{n+1}-n-1)).$

As before, we have the isomorphisms

$$\begin{aligned} H^{i-1}(F_{i-1}^\vee (d_i-d_{n+1}-n-1))&\simeq H^1(F_1^{\vee }(d_i - d_{n+1} - n- 1))\\&\simeq H^{n-1}(F_{n-1}^{\vee }(d_i - d_{n+1} - n - 1)). \end{aligned}$$

Using the same techniques as before, if we focus on the first isomorphism, then we have to consider the exact sequences

$$\begin{aligned} \begin{array}{c} 0 \longrightarrow F_1^\vee (d_i-d_{n+1}-n-1) \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _1}(d_i-d_1-n-1) \longrightarrow {\mathcal O}^{\beta _0}_{{\mathbb P}^n}(d_i-n-1) \longrightarrow 0\\ 0 \longrightarrow F_2^\vee (d_i - d_{n+1}-n-1) \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _2}(d_i - d_2-n-1) \longrightarrow F_1^\vee (d_i-d_{n+1}-n-1) \longrightarrow 0\\ \vdots \\ 0 \longrightarrow F_{i-1}^\vee (d_i - d_{n+1}-n-1) \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _{i-1}}(d_i - d_{i-1}-n-1) \longrightarrow F_{i-2}^\vee (d_i-d_{n+1}\!-n\!-1) \longrightarrow 0\\ 0 \longrightarrow F_i^\vee (d_i - d_{n+1}-n-1) \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _i}(-n-1) \longrightarrow F_{i-1}^\vee (d_i-d_{n+1}-n-1) \longrightarrow 0 \end{array} \end{aligned}$$

and knowing that if $i\ge 3$ (or else, as before, we only consider the first short exact sequence and obtain the same result), we have $H^0(F_{i-1}^\vee (d_i-d_{n+1}-n-1)) = H^1(F_{i-1}^\vee (d_i-d_{n+1}-n-1)) = 0$, and therefore,

$$\begin{aligned} h^{n-i+1}(F_{i-1}(d_{n+1}-d_i)) = h^1(F_1^\vee (d_i-d_{n+1}-n-1)) = \sum _{k=0}^{i-1}(-1)^k \beta _k \left( {\begin{array}{c}d_i-d_k-1\\ n\end{array}}\right) . \end{aligned}$$

Let us focus now on the other isomorphism, computing $H^{n-1}(F_{n-1}^\vee (d_i-d_{n+1}-n-1))$. Take the sequences

$$\begin{aligned} \begin{array}{l} 0 \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _{n+1}}(d_i-d_{n+1}-n-1) \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _n}(d_i-d_n-n-1) \longrightarrow F_{n-1}^\vee (d_i-d_{n+1}-n-1) \longrightarrow 0 \\ 0 \longrightarrow F_{n-1}^\vee (d_i-d_{n+1}-n-1) \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _{n-1}}(d_i-d_{n+1}-n-1) \longrightarrow F_{n-2}^\vee (d_i-d_{n+1}-n-1) \longrightarrow 0 \\ \vdots \\ 0 \longrightarrow F_{i+2}^\vee (d_i-d_{n+1}-n-1) \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _{i+2}}(d_i-d_{i+2}-n-1) \longrightarrow F_{i+1}^\vee (d_i-d_{n+1}-n-1) \longrightarrow 0\\ 0 \longrightarrow F_{i+1}^\vee (d_i-d_{n+1}-n-1) \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _{i+1}}(d_i-d_{i+1}-n-1) \longrightarrow F_{i}^\vee (d_i-d_{n+1}-n-1) \longrightarrow 0\\ 0 \longrightarrow F_{i}^\vee (d_i-d_{n+1}-n-1) \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _i}(-n-1) \longrightarrow F_{i-1}^\vee (d_i-d_{n+1}-n-1) \longrightarrow 0 \end{array} \end{aligned}$$

and, being $i-1<n-1$ we can state that $H^{n-1}(F_{i-1}^\vee (d_i-d_{n+1}-n-1)) = H^n(F_{i-1}^\vee (d_i-d_{n+1}-n-1))=0$ and also that

$$\begin{aligned} H^{n-1}(F_j^\vee (d_i-d_{n+1}-n-1))=0 \,\,\text{ for } \text{ each }\,\,j=i,\ldots ,n-2. \end{aligned}$$

We obtain that

$$\begin{aligned} h^{n-i+1}(F_{i-1}(d_{n+1}-d_i))&= h^{n-1}(F_{n-1}^\vee (d_i-d_{n+1}-n-1)) \\&= \left\{ \begin{array}{ll} \sum _{k=i}^{n+1} (-1)^{k+1} \beta _k \left( {\begin{array}{c}d_k-d_i+n\\ n\end{array}}\right) &{} \text{ for }\,\,n\,\,\text{ even } \\ \sum _{k=i}^{n+1} (-1)^{k} \beta _k \left( {\begin{array}{c}d_k-d_i+n\\ n\end{array}}\right)&\text{ for }\,\,n\,\,\text{ odd } \end{array} \right. \end{aligned}$$

Let us fix some notation, for each $i$ fixed, we will call

$$\begin{aligned} \Sigma _{i,1}&= h^i(F_i^\vee (d_i-d_{n+1}))\\ \Sigma _{i,2}&= h^{n-i+1}(F_{i-1}(d_{n+1}-d_i)). \end{aligned}$$

We have learned that for each $i$ fixed from $2$ to $n-1$ the cohomology group of $F_i^\vee (d_i-d_{n+1})$ which may not vanish is the $i$-th group; hence, the important part of the exact sequence induced in cohomology by (15) is

$$\begin{aligned} \begin{array}{c} \longrightarrow \underbrace{H^{i-1}((F_i^\vee )^{\beta _i}(d_i-d_{n+1}))}_{=0} \longrightarrow H^{i-1}(F_i^\vee \otimes F_i) \longrightarrow H^i(F_{i-1}^{\beta _i}(d_{n+1}-d_i))\\ \longrightarrow \underbrace{H^i((F_i^\vee )^{\beta _i}(d_i-d_{n+1}))}_{\text{ dimension }\,\,\beta _i \Sigma _{i,1}} \longrightarrow \longrightarrow H^{i}(F_i^\vee \otimes F_i) \longrightarrow H^{i+1}(F_{i-1}^{\beta _i}(d_{n+1}-d_i))\\ \longrightarrow \underbrace{H^{i+1}((F_i^\vee )^{\beta _i}(d_i-d_{n+1}))}_{=0} \longrightarrow \end{array} \end{aligned}$$

(17)

We now need to check out how the nonvanishing group in cohomology, associated with the bundle $F_{i-1} (d_{n+1}-d_i)$, relates to the first group, we can have the following situations.

Case 1 If $i\ne \frac{n+1}{2}$ and $i\ne \frac{n}{2}$, which means that $n-i \ne i-1$ and $n-i-1 \ne i-1$, then the two groups belong to two different exact sequences of type (17), and we have

$$\begin{aligned} H^{n-i}(F_i^\vee \otimes F_i) \simeq H^{n-i+1}(F_{i-1}^{\beta _i}(d_{n+1}-d_i)) \end{aligned}$$

and

$$\begin{aligned} H^i(F_i^\vee \otimes F_i) = H^i((F_i^\vee )^{\beta _i}(d_i - d_{n+1})) \end{aligned}$$

hence $F_i$ is exceptional if and only if $H^{n-i+1}(F_{i-1}(d_{n+1}-d_i)) = H^i(F_i^\vee (d_i - d_{n+1})) = 0$.

Case 2 If $i=\frac{n}{2}$, so only in the even cases, we are in the following situation

$$\begin{aligned} 0 \longrightarrow \underbrace{H^i((F_i^\vee )^{\beta _i}(d_i-d_{n+1}))}_{\text{ dimension }\,\,\beta _i \Sigma _{i,1}} \longrightarrow H^{i}(F_i^\vee \otimes F_i) \longrightarrow \underbrace{H^{i+1}(F_{i-1}^{\beta _i}(d_{n+1}-d_i))}_{\beta _i \Sigma _{i,2}} \longrightarrow 0. \end{aligned}$$

Being $\Sigma _{i,p}\ge 0$ for $p=1,2$, we can state that $F_i$ is exceptional if and only if

$$\begin{aligned} H^i((F_i^\vee )^{\beta _i}(d_i-d_{n+1}))=H^{i+1}(F_{i-1}^{\beta _i}(d_{n+1}-d_i))=0. \end{aligned}$$

Case 3 If $i=\frac{n+1}{2}$, so only in the odd cases, we are in the following situation

$$\begin{aligned}&0 \longrightarrow H^{i-1}(F_i^\vee \otimes F_i) \longrightarrow \underbrace{H^i(F_{i-1}^{\beta _i}(d_{n+1}-d_i))}_{\text{ dimension }\,\,\beta _i \Sigma _{i,2}} \mathop {\longrightarrow }\limits ^{H^i(\varphi )} \underbrace{H^i((F_i^\vee )^{\beta _i}(d_i-d_{n+1}))}_{\text{ dimension }\,\,\beta _i \Sigma _{i,1}}\\&\longrightarrow H^{i}(F_i^\vee \otimes F_i) \longrightarrow 0, \end{aligned}$$

where $H^i(\varphi )$ is the morphism induced in cohomology by $\varphi : F_i^\vee \otimes F_{i-1} \rightarrow (F_i^\vee )^{\beta _i}(d_i-d_{n+1})$. Therefore $F_i$ is exceptional if and only if $H^{i-1}(F_i^\vee \otimes F_i) = H^i(F_i^\vee \otimes F_i)=0$ if and only if $H^i(\varphi )$ is an isomorphism.

This concludes the proof. $\square$

Corollary 4.2

If each bundle $F_i$ for $i = 2, \ldots , n-1,$ defined as

$$\begin{aligned} 0\longrightarrow F_{i-1} \longrightarrow {\mathcal O}_{{\mathbb P}^n}^{\beta _i}(d_i-d_{n+1}) \longrightarrow F_i \longrightarrow 0, \end{aligned}$$

is a Steiner bundle of type $(F_{i-1}, {\mathcal O}_{{\mathbb P}^n}(d_i-d_{n+1}))$ and $\beta _0^2 + \beta _1^2 - \left( {\begin{array}{c}d_1+n\\ n\end{array}}\right) \beta _0\beta _1 = 1$; then all bundles $F_i$ are exceptional, for $i = 1, \ldots ,n-1$.

Proof

The cohomological vanishings appearing in the definition of strongly exceptional pairs, used to define Steiner bundles (recall Definition 2.5), imply the hypothesis (iii) of Theorem 4.1. $\square$

We would like to know if the viceversa of the previous result holds, but at the moment, we are only able to state the following.

Conjecture 1

The syzygy bundles $F_i$ are Steiner if and only if they are also exceptional.

The conjecture would be true if, considering $n$ odd and $i=\frac{n+1}{2}$, we prove that the two cohomology groups

$$\begin{aligned} H^{n-i+1}(F_{i-1}(d_{n+1}-d_i))\,\,\, \text{ and } \,\,\,H^i(F_i^\vee (d_i - d_{n+1})) \end{aligned}$$

are isomorphic if and only if they are zero.

As for the results implying simplicity, also for the last theorem, we have a correspondent result obtained considering the dual resolution. Recall that the bundles $F_i$ are simple or exceptional if and only if the bundles $G_i$ are.

Theorem 4.3

Consider the syzygy bundles $G_i$ as defined in (7), for $i$ from 1 to $n-1$. Suppose also that $G_i$ are simple for each $i$; then $G_i$, for $i=1,\ldots ,n-1$, is exceptional if and only if each one of the following conditions hold

• (i) $\beta _{n+1}^2 + \beta _n^2 - \left( {\begin{array}{c}d_{n+1}-d_n+n\\ n\end{array}}\right) \beta _{n+1}\beta _n = 1$;

• (ii) $d_{n+1}-d_n \le n;$

• (iii)
  $$\begin{aligned} \left\{ \begin{array}{ll} H^{n-i+1}(G_{i-1}(d_{n+1-i})) = H^i(G_i^\vee (-d_{n+1-i})) = 0 &{} \text{ if }\,\;n\,\;\text{ is } \text{ even; }\\ H^{n-i+1}(G_{i-1}(d_{n+1-i})) = H^i(G_i^\vee (-d_{n+1-i})) = 0 &{} \text{ if }\,\;n\,\;\text{ is } \text{ odd } \text{ and } \;\, i \ne \frac{n+1}{2};\\ H^{n-i+1}(G_{i-1}(d_{n+1-i})) \mathop {\simeq }\limits ^{H^i(\varphi )} H^i(G_i^\vee (-d_{n+1-i})) &{} \text{ if }\,\;n\,\;\text{ is } \text{ odd } \text{ and } \;\, i = \frac{n+1}{2}. \end{array} \right. \end{aligned}$$

  where, if $n$ is odd and $i = \frac{n+1}{2}$, we get $H^{n-i+1}(G_{i-1}(d_{n+1-i})) \simeq H^i(G_{i-1} \otimes G_i^\vee )$ and the morphism $H^i(\varphi ): H^i(G_i^\vee \otimes G_{i-1}) \rightarrow H^i((G_i^\vee )^{\beta _i} (-d_{n+1-i}))$ is the one obtained by the short exact sequence

  $$\begin{aligned} 0 \longrightarrow G_i^\vee \otimes G_{i-1} \mathop {\longrightarrow }\limits ^{\varphi } (G_i^\vee )^{\beta _{n+1-i}} (-d_{n+1-i}) \longrightarrow G_i^\vee \otimes G_i \longrightarrow 0 \end{aligned}$$

  considering the long exact induced in cohomology.

5 Examples

In this section, we present some famous pure resolutions, and we will apply the results obtained to determine whenever the syzygies are simple or exceptional. Some of these resolutions were studied by [13].

5.1 Pure linear resolution

Let $R = {\mathbb K}[x_0, \ldots ,x_n]$ be the ring of polynomials and $I = (x_0, \ldots , x_n)$ be the ideal generated by the coordinate variables . The Koszul complex $K(x_0, \ldots , x_n)$ is given by

Sheafifying we get the exact sequence

(18)

Proposition 5.1

The syzygy bundles arising from the complex (18) are all simple and exceptional.

Proof

It is a simple computation that the complex satisfies the hypothesis of Theorem 3.1 and of Theorem 3.6 for simplicity, and the hypothesis of Theorem 4.1 for the exceptionality. $\square$

5.2 Compressed Gorenstein Artinian graded algebras

Let $I = (f_1, \ldots , f_{\alpha _1})$ be an ideal generated by $\alpha _1$ forms of degree $t+1$, such that the algebra $A = R/I$ is a compressed Gorenstein Artinian graded algebra of embedding dimension $n+1$ and socle degree $2t$. Thus, by Proposition $3.2$ of [14], the minimal free resolution of $A$ is

where

$$\begin{aligned} \alpha _i = \left( {\begin{array}{c}t+i - 1\\ i-1\end{array}}\right) \left( {\begin{array}{c}t+n+1\\ n+1-i\end{array}}\right) - \left( {\begin{array}{c}t+n-i\\ n+1-i\end{array}}\right) \left( {\begin{array}{c}t+n\\ i-1\end{array}}\right) , \quad \hbox {for} \, i=1, \ldots , n. \end{aligned}$$

Sheafifying the complex above, we have

(19)

where $\beta$ is the map given by the $\alpha _1$ forms of degree $t+1$.

Remark 5.2

By applying Theorem 3.1 , we have that the syzygies $F_i$ of the complex (19) are simple vector bundles. Moreover, $\mathrm{hd}(F_i) = n-i$, $h^0(F_i^*(-t-i)) =\alpha _i$ for $1 \le i \le n-1$. If we take $t=1,$ then we get the linear resolution and we already know that all syzygies are exceptional. Nevertheless, it is easy to loose the exceptionality. For instance, if we take $t$ such that $t>n-1$, the second condition of Theorem 4.1 is not satisfied. Moreover, being $\beta _0 = 1$, the first condition of Theorem 4.1 is equivalent to prove that

$$\begin{aligned} \left( {\begin{array}{c}t+n+1\\ n\end{array}}\right) - \left( {\begin{array}{c}t+n-1\\ n\end{array}}\right) = \beta _1 = \left( {\begin{array}{c}d_1+n\\ n\end{array}}\right) = \left( {\begin{array}{c}t+n+1\\ n\end{array}}\right) , \end{aligned}$$

which are not equal if $t \ge 1$. Hence, for this example, the only exceptional bundles come from the linear resolution.

5.3 Generalized Koszul complex

The reference for this section is [15].

Definition 5.3

Let $\mathcal {A}$ be a $p \times q$ matrix with entries in $R$. We say that $\mathcal {A}$ is a $t$-homogeneous matrix if the minors of size $j \times j$ are homogeneous polinomials for all $j \le t$. The matrix $\mathcal {A}$ is an homogeneous matrix if their minors of any size are homogeneous.

Let $\mathcal {A}$ be an homogeneous matrix. We denote by $I(\mathcal {A})$ the ideal of $R$ generated by the maximal minors of $\mathcal {A}$. Le $\mathcal {A}$ be a $t$-homogeneous matrix. For all $j \le t$, we denote by $I_j(\mathcal {A})$ the ideal of $R$ generated by the minors of size $j$ of $\mathcal {A}$.

Note that to any homogeneous $p \times q$ matrix $\mathcal {A}$, we have a morphism $\varphi : F \rightarrow G$ of free graded $R$-modulos of ranks $p$ and $q$, respectively. We write $I(\varphi ) = I(\mathcal {A})$.

An homogeneous ideal $I\subset R$ is called determinantal ideal if

$(1)$ :

there exists a $r$-homogeneous matrix $\mathcal {A}$ of size $p \times q$ with entries in $R$ such that $I = I_r(\mathcal {A})$ and

$(2)$ :

$ht(I) = (p - r + 1)(q - r + 1),$ where $ht(I)$ is the height of $I.$

An homogeneous determinantal ideal $I \subset R$ is called standard determinantal ideal if $r = \max \{p,q\}$. That is, an homogeneous ideal $I \subset R$ of codimension $c$ is called standard determinantal ideal if $I = I_r(\mathcal {A})$ for some homogeneous matrix $\mathcal {A}$ of size $r \times (r+ c-1)$.

Let $X \subset \mathbb {P}^{n+c}$, and $\mathcal {A}$ homogeneous matrix associated with $X$. Let $\varphi : F \rightarrow G$ be a morphism of free graded $R$-modules of ranks $t$ and $t+c-1$, respectively, defined by $\mathcal {A}.$ The generalized Koszul complex $C_i(\varphi ^*)$ is given by

From this complex, we have the complex $D_i(\varphi ^*)$

where $D_0(\varphi ^*)$ is called Eagon–Northcott complex and $D_1(\varphi ^*)$ is called Buchsbaum–Rim complex.

Let $\varphi : R(-d)^{a} \rightarrow R^{a+n}$ be a map, let $M$ be the matrix associated with the map, and $I = I_a(M)$ be the ideal generated by the maximal minors of $M$. The Eagon–Northcott complex $D_0(\varphi ^{*})$ gives us a minimal free resolution of $R/I$

Sheafifying, we get the complex

(20)

Remark 5.4

Applying Theorem 3.1, all syzygies $F_i$ of the complex (20) are simple. If we take $d=a=1,$ then we get the linear resolution, and we already know that all syzygies are exceptional. We obtain exceptionality, for example, also for $n=3, d=1$, and $a=2$. Nevertheless, it is easy to loose the exceptionality. For instance, if we take $d,a$ such that $da>n$, the second condition of Theorem 4.1 is not satisfied. Moreover, if we consider $n=3$, $d=2$, and $a=1$, the syzygy bundles are not exceptional because the first condition of Theorem 4.1 is not satisfied.