Boij-Söderberg theory: introduction and survey.

*(English)*Zbl 1260.13020
Francisco, Christopher (ed.) et al., Progress in commutative algebra 1. Combinatorics and homology. Berlin: Walter de Gruyter (ISBN 978-3-11-025034-3/hbk; 978-3-11-025040-4/ebook). De Gruyter Proceedings in Mathematics, 1-54 (2012).

The paper under review is a report on the Conjecture of M. Boij and J. Söderberg [J. Lond. Math. Soc., II. Ser. 78, No. 1, 85–106 (2008; Zbl 1189.13008)], on its proof, and on some further developments related to it. Although the author follows closely the presentation of the original papers, his work should be useful to those willing to understand the significance and implications of this inspired conjecture and of its equally inspired proof by D. Eisenbud and F.-O. Schreyer [J. Am. Math. Soc. 22, No. 3, 859–888 (2009; Zbl 1213.13032)].

The Conjecture of Boij and Söderberg, originally formulated in an e-print in November 2006, describes the diagrams of graded Betti numbers of finitely generated, Cohen-Macaulay, graded modules \(M\) over a polynomial ring \(S = k[X_1, \dots , X_n]\) over a field \(k\), up to multiplication by a positive rational number. Its formulation and proof can be easily reduced to the case where \(M\) is Artinian, which we shall assume from now on. The first part of the conjecture asserts that for every increasing sequence of integers \(\mathbf{d} = (d_0 < d_1 < \cdots < d_n)\) there exists a module \(M\) whose minimal free resolution is pure of type \(\mathbf{d}\), i.e., of the form: \[ 0 \rightarrow S(-d_n)^{b_n} \rightarrow \cdots \rightarrow S(-d_0)^{b_0} \rightarrow M \rightarrow 0\, . \] In this case, the Betti diagram of \(M\) turns out to be an integer multiple of the diagram \(\pi(\mathbf{d})\) defined by: \[ \pi(\mathbf{d})_{ij} = \frac{(-1)^im(\mathbf{d})}{ \prod_{l\neq i}(d_l-d_i)}\;\text{for}\;j=d_i\, ,\;\pi(\mathbf{d})_{ij} = 0\, ,\;\text{otherwise}\, , \] where \(m(\mathbf{d})\) is the least common multiple of the denominators. The second part of the conjecture asserts that if \((\beta_{ij})\) is the Betti diagram of an Artinian graded \(S\)-module \(M\) then there exists a chain of degree sequences \(\mathbf{d}^1 < \mathbf{d}^2 < \cdots < \mathbf{d}^r\) (for the standard partial order on \({\mathbb Z}^{n+1}\)) and positive rational numbers \(c_1, \dots ,c_r\) such that: \[ (\beta_{ij}) = c_1\pi(\mathbf{d}^1) + \cdots + c_r\pi(\mathbf{d}^r)\, . \]

The first part of the conjecture was proved, in characteristic 0, by D. Eisenbud, G. Fløystad and J. Weyman [Ann. Inst. Fourier 61, No. 3, 905–926 (2011; Zbl 1239.13023)] who constructed \(\text{GL}(n)\)-equivariant pure free resolutions. Their paper appeared, originally, as an e-print in September 2007. Then Eisenbud and Schreyer provided, in an e-print from December 2007, later published in the above mentioned form, a characteristic free construction of pure free resolutions and also proved the second part of the conjecture. The construction of Eisenbud and Schreyer uses the following collapsing lemma: let \(f : X \rightarrow Y\) be a proper morphism of Noetherian schemes and let \({\mathcal F}^\bullet\) be a bounded complex of coherent sheaves on \(X\). Assume that: (1) \(\forall \, i\), \(\text{R}^jf_\ast{\mathcal F}^i \neq 0\) for at most one \(j\); (2) \(\forall \, i_1 < i_2\), if \(\text{R}^{j_1}f_\ast{\mathcal F}^{i_1} \neq 0\) and \(\text{R}^{j_2}f_\ast{\mathcal F}^{i_2} \neq 0\) then \(i_1 + j_1 < i_2 + j_2\). Then \(\text{R}f_\ast{\mathcal F}^\bullet\) is isomorphic, in the derived category, to a complex \({\mathcal G}^\bullet\) with \({\mathcal G}^p = \bigoplus_{i+j=p}\text{R}^jf_\ast{\mathcal F}^i\). Notice that, by assumption (2), the direct sum defining \({\mathcal G}^p\) contains at most one non-zero term. The lemma follows from a corresponding statement about double complexes.

Now, consider a degree sequence \(\mathbf{d} = (d_0 < \cdots < d_n)\) with \(d_0 = 0\), put \(m_i = d_{i+1} - d_i - 1\), \(i = 0, \dots , n-1\), and take \(X = {\mathbb P}^{n-1} \times {\mathbb P}^{m_0} \times \cdots \times {\mathbb P}^{m_{n-1}}\), \(Y = {\mathbb P}^{n-1}\) and \(f : X \rightarrow Y\) to be the projection. Since \(\dim X = d_n - 1\), one can find \(d_n\) global sections of \({\mathcal O}_X(1, \dots , 1)\) which generate this sheaf and define a Koszul complex: \[ {\mathcal K}^\bullet \;: \;0 \rightarrow {\mathcal K}^{-d_n} \rightarrow \cdots \rightarrow {\mathcal K}^0 = {\mathcal O}_X \rightarrow 0\, . \] Taking \({\mathcal F}^\bullet = {\mathcal K}^\bullet \otimes {\mathcal O}_X(0,d_0, \dots , d_{n-1})\) and applying the collapsing lemma one gets an exact complex \({\mathcal L}^\bullet\) on \({\mathbb P}^{n-1}\) which is pure of type \(\mathbf{d}\) in the sense that \({\mathcal L}^{-i} = \text{R}^{d_i-i}f_\ast{\mathcal F}^{-d_i} = {\mathcal O}_{\mathbb P}(-d_i)^{b_i}\), \(i = 0, \dots , n\). \({\mathcal L}^\bullet\) must be the sheafification of the minimal free resolution of an Artinian \(S\)-module \(M\).

This construction has been generalized by C. Berkesch, D. Erman, M. Kummini and S. V. Sam [“Tensor complexes: Multilinear free resolutions constructed from higher tensors”, (2011) arxiv:1101.4604]. It should be also mentioned that Eisenbud, Fløystad and Weyman conjectured that, for every degree sequence \(\mathbf{d} = (d_0 < \cdots < d_n)\), there exists an integer \(l(\mathbf{d})\) such that, \(\forall \, l \geq l(\mathbf{d})\), \(l\pi(\mathbf{d})\) is the Betti diagram of a module.

As for the Eisenbud’s and Schreyer’s proof of the second part of the conjecture, they discovered and used a duality between Betti diagrams of Artinian \(S\)-modules and cohomology tables of vector bundles on \({\mathbb P}^{n-1}\). As a byproduct of their method, they also got a complete classification (up to multiplication by a positive rational number) of this kind of cohomology tables, which is nowadays known under the name of Eisenbud-Schreyer theory. An important open question related to the second part of the conjecture is “when does the decomposition of the Betti diagram of a module into pure diagrams arise from some filtration of the module?” This question is analysed in [D. Eisenbud, D. Erman and F.-O. Schreyer, “Filtering free resolutions”, (Dec. 2011) arxiv:1001.0585].

Concerning the further developments, in March 2008, after Eisenbud and Schreyer proved their conjecture, M. Boij and J. Söderberg [Algebra Number Theory 6, No. 3, 437–454 (2012; Zbl 1259.13009)] extended the results to the case of Betti diagrams of arbitrary finitely generated graded \(S\)-modules (not necessarily Cohen-Macaulay). D. Erman [Algebra Number Theory 3, No. 3, 341–365 (2009; Zbl 1173.13013)] proved that (any conveniently bounded part of) the semigroup of actual Betti diagrams of Artinian modules and (of) the semigroup of integer diagrams in the Boij-Söderberg fan are both finitely generated. Finally, C. Berkesch et al. [Int. Math. Res. Not. 2012, No. 22, 5132–5160 (2012; Zbl 1258.13018)] showed that if \(\mathbf{d} = (d_0 < \cdots < d_n)\) and \(\mathbf{d^\prime} = (d_0^\prime < \cdots < d_n^\prime)\) are two degree sequences then \(\mathbf{d} \leq \mathbf{d^\prime}\) (i.e., \(d_i \leq d_i^\prime\), \(i = 0, \dots , n\)) if and only if there exist Artinian modules \(M\) and \(M^\prime\) with pure resolutions of types \(\mathbf{d}\) and \(\mathbf{d^\prime}\), respectively, and a non-zero morphism \(M^\prime \rightarrow M(a)\), for some \(a \leq 0\).

For the entire collection see [Zbl 1237.13005].

The Conjecture of Boij and Söderberg, originally formulated in an e-print in November 2006, describes the diagrams of graded Betti numbers of finitely generated, Cohen-Macaulay, graded modules \(M\) over a polynomial ring \(S = k[X_1, \dots , X_n]\) over a field \(k\), up to multiplication by a positive rational number. Its formulation and proof can be easily reduced to the case where \(M\) is Artinian, which we shall assume from now on. The first part of the conjecture asserts that for every increasing sequence of integers \(\mathbf{d} = (d_0 < d_1 < \cdots < d_n)\) there exists a module \(M\) whose minimal free resolution is pure of type \(\mathbf{d}\), i.e., of the form: \[ 0 \rightarrow S(-d_n)^{b_n} \rightarrow \cdots \rightarrow S(-d_0)^{b_0} \rightarrow M \rightarrow 0\, . \] In this case, the Betti diagram of \(M\) turns out to be an integer multiple of the diagram \(\pi(\mathbf{d})\) defined by: \[ \pi(\mathbf{d})_{ij} = \frac{(-1)^im(\mathbf{d})}{ \prod_{l\neq i}(d_l-d_i)}\;\text{for}\;j=d_i\, ,\;\pi(\mathbf{d})_{ij} = 0\, ,\;\text{otherwise}\, , \] where \(m(\mathbf{d})\) is the least common multiple of the denominators. The second part of the conjecture asserts that if \((\beta_{ij})\) is the Betti diagram of an Artinian graded \(S\)-module \(M\) then there exists a chain of degree sequences \(\mathbf{d}^1 < \mathbf{d}^2 < \cdots < \mathbf{d}^r\) (for the standard partial order on \({\mathbb Z}^{n+1}\)) and positive rational numbers \(c_1, \dots ,c_r\) such that: \[ (\beta_{ij}) = c_1\pi(\mathbf{d}^1) + \cdots + c_r\pi(\mathbf{d}^r)\, . \]

The first part of the conjecture was proved, in characteristic 0, by D. Eisenbud, G. Fløystad and J. Weyman [Ann. Inst. Fourier 61, No. 3, 905–926 (2011; Zbl 1239.13023)] who constructed \(\text{GL}(n)\)-equivariant pure free resolutions. Their paper appeared, originally, as an e-print in September 2007. Then Eisenbud and Schreyer provided, in an e-print from December 2007, later published in the above mentioned form, a characteristic free construction of pure free resolutions and also proved the second part of the conjecture. The construction of Eisenbud and Schreyer uses the following collapsing lemma: let \(f : X \rightarrow Y\) be a proper morphism of Noetherian schemes and let \({\mathcal F}^\bullet\) be a bounded complex of coherent sheaves on \(X\). Assume that: (1) \(\forall \, i\), \(\text{R}^jf_\ast{\mathcal F}^i \neq 0\) for at most one \(j\); (2) \(\forall \, i_1 < i_2\), if \(\text{R}^{j_1}f_\ast{\mathcal F}^{i_1} \neq 0\) and \(\text{R}^{j_2}f_\ast{\mathcal F}^{i_2} \neq 0\) then \(i_1 + j_1 < i_2 + j_2\). Then \(\text{R}f_\ast{\mathcal F}^\bullet\) is isomorphic, in the derived category, to a complex \({\mathcal G}^\bullet\) with \({\mathcal G}^p = \bigoplus_{i+j=p}\text{R}^jf_\ast{\mathcal F}^i\). Notice that, by assumption (2), the direct sum defining \({\mathcal G}^p\) contains at most one non-zero term. The lemma follows from a corresponding statement about double complexes.

Now, consider a degree sequence \(\mathbf{d} = (d_0 < \cdots < d_n)\) with \(d_0 = 0\), put \(m_i = d_{i+1} - d_i - 1\), \(i = 0, \dots , n-1\), and take \(X = {\mathbb P}^{n-1} \times {\mathbb P}^{m_0} \times \cdots \times {\mathbb P}^{m_{n-1}}\), \(Y = {\mathbb P}^{n-1}\) and \(f : X \rightarrow Y\) to be the projection. Since \(\dim X = d_n - 1\), one can find \(d_n\) global sections of \({\mathcal O}_X(1, \dots , 1)\) which generate this sheaf and define a Koszul complex: \[ {\mathcal K}^\bullet \;: \;0 \rightarrow {\mathcal K}^{-d_n} \rightarrow \cdots \rightarrow {\mathcal K}^0 = {\mathcal O}_X \rightarrow 0\, . \] Taking \({\mathcal F}^\bullet = {\mathcal K}^\bullet \otimes {\mathcal O}_X(0,d_0, \dots , d_{n-1})\) and applying the collapsing lemma one gets an exact complex \({\mathcal L}^\bullet\) on \({\mathbb P}^{n-1}\) which is pure of type \(\mathbf{d}\) in the sense that \({\mathcal L}^{-i} = \text{R}^{d_i-i}f_\ast{\mathcal F}^{-d_i} = {\mathcal O}_{\mathbb P}(-d_i)^{b_i}\), \(i = 0, \dots , n\). \({\mathcal L}^\bullet\) must be the sheafification of the minimal free resolution of an Artinian \(S\)-module \(M\).

This construction has been generalized by C. Berkesch, D. Erman, M. Kummini and S. V. Sam [“Tensor complexes: Multilinear free resolutions constructed from higher tensors”, (2011) arxiv:1101.4604]. It should be also mentioned that Eisenbud, Fløystad and Weyman conjectured that, for every degree sequence \(\mathbf{d} = (d_0 < \cdots < d_n)\), there exists an integer \(l(\mathbf{d})\) such that, \(\forall \, l \geq l(\mathbf{d})\), \(l\pi(\mathbf{d})\) is the Betti diagram of a module.

As for the Eisenbud’s and Schreyer’s proof of the second part of the conjecture, they discovered and used a duality between Betti diagrams of Artinian \(S\)-modules and cohomology tables of vector bundles on \({\mathbb P}^{n-1}\). As a byproduct of their method, they also got a complete classification (up to multiplication by a positive rational number) of this kind of cohomology tables, which is nowadays known under the name of Eisenbud-Schreyer theory. An important open question related to the second part of the conjecture is “when does the decomposition of the Betti diagram of a module into pure diagrams arise from some filtration of the module?” This question is analysed in [D. Eisenbud, D. Erman and F.-O. Schreyer, “Filtering free resolutions”, (Dec. 2011) arxiv:1001.0585].

Concerning the further developments, in March 2008, after Eisenbud and Schreyer proved their conjecture, M. Boij and J. Söderberg [Algebra Number Theory 6, No. 3, 437–454 (2012; Zbl 1259.13009)] extended the results to the case of Betti diagrams of arbitrary finitely generated graded \(S\)-modules (not necessarily Cohen-Macaulay). D. Erman [Algebra Number Theory 3, No. 3, 341–365 (2009; Zbl 1173.13013)] proved that (any conveniently bounded part of) the semigroup of actual Betti diagrams of Artinian modules and (of) the semigroup of integer diagrams in the Boij-Söderberg fan are both finitely generated. Finally, C. Berkesch et al. [Int. Math. Res. Not. 2012, No. 22, 5132–5160 (2012; Zbl 1258.13018)] showed that if \(\mathbf{d} = (d_0 < \cdots < d_n)\) and \(\mathbf{d^\prime} = (d_0^\prime < \cdots < d_n^\prime)\) are two degree sequences then \(\mathbf{d} \leq \mathbf{d^\prime}\) (i.e., \(d_i \leq d_i^\prime\), \(i = 0, \dots , n\)) if and only if there exist Artinian modules \(M\) and \(M^\prime\) with pure resolutions of types \(\mathbf{d}\) and \(\mathbf{d^\prime}\), respectively, and a non-zero morphism \(M^\prime \rightarrow M(a)\), for some \(a \leq 0\).

For the entire collection see [Zbl 1237.13005].

Reviewer: Iustin Coandă (Bucureşti)

##### MSC:

13D02 | Syzygies, resolutions, complexes and commutative rings |

13C14 | Cohen-Macaulay modules |

14N99 | Projective and enumerative algebraic geometry |

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |