# New Trends in Syzygies (18w5133)

Arriving in Banff, Alberta Sunday, June 24 and departing Friday June 29, 2018

## Objectives

Free resolutions play a central role in commutative algebra and have many applications in algebraic combinatorics, algebraic geometry, computational algebra, noncommutative algebra, mathematical physics, and subspace arrangements. Recently there has been tremendous progress in understanding their structure and connections to other areas of mathematics.

Given a graded module $M$ over a polynomial ring $S$ over a field, its minimal graded free resolution is an invariant which carries a lot of information about M. The length, or projective dimension, of the resolution is at most the number of variables by a classical result of Hilbert. The largest degree of a syzygy is measured by its Castelnuovo-Mumford regularity. When $M = I$ is an ideal of $S$, Stillman asked whether a bound on the projective dimension could be obtained in terms of the degrees of the generators but independent of the number of variables. The parallel question regarding a bound on regularity was shown to be equivalent by Caviglia. In a recent (2016) breakthrough, Ananyan-Hochster have proved an affirmative answer to Stillman's Question in characteristic 0 and in the case when the characteristic is larger than the degrees of the minimal generators of $I$. Their work shows that one can always find a polynomial subring of $S$ with a bounded number of variables which contains the generators of $I$, from which they derive their result. Whether tighter bounds than those given by Ananyan-Hochster are possible is a fundamental problem in computational algebra; better bounds would give insight into the complexity of Buchbergerâ€™s algorithm for the computation of Gr\"obner bases. Tight bounds given by Huneke et. al. are known in only a few specific cases; however examples by Beder et. al. show that the optimal answer will be quite large. Another interesting direction would be to consider whether better bounds are possible under assumptions on the ideal itself. For instance, de Stefani-Nunez-Betancourt recently showed that the defining ideal of an $F$-injective ring satisfies a nice projective dimension bound. It would be interesting to know if more general statements are possible. Even less is known about the corresponding problems for regularity.

There is a doubly exponential bound (in terms of degrees of generators and number of variables) on the regularity of homogeneous ideals in a polynomial ring, proved by Bayer-Mumford, Caviglia-Sbarra, Galligo, and Giusti. In the geometric case, when we consider a nondegenerate prime ideal $P$ over an algebraically closed field, much better bounds on the regularity were expected. This was expressed by the Eisenbud-Goto Conjecture, which predicted a linear bound on regularity in terms of the degree of $P$. In another recent breakthrough (2016), McCullough-Peeva gave counterexamples to the Eisenbud-Goto conjecture and showed that no polynomial bound on regularity in terms of degree was possible. Their construction closely resembles the well-studied Rees algebras, which are tied to the geometric blow-up but whose defining equations are difficult to find in general. However, they prove that their Rees-like algebras have well-structured generators and free resolutions. It is an intriguing prospect to discover what information about the well-understood Rees-like algebras will pass to the more intricate Rees algebras. Another important question is whether there is any bound at all on the regularity of nondegenerate primes in terms of degree. Such a bound is closely related to those in Stillman's Question. Finally one can study the conjecture in the case that $P$ defines a smooth projective variety. In this case the conjecture is still open and there has been recent progress by Kwak-Park and Noma.

Another breakthrough in the theory of graded free resolutions comes from the proof of the Boij-S\"oderberg Conjectures by Eisenbud-Schreyer and the previous results of Eisenbud-Floystad-Weyman. Boij-S\"oderberg's key insight was that the study of graded Betti tables of modules might be best understood up to constant multiple by studying the semigroup or positive rational cone generated by all Betti tables of Cohen-Macaulay modules of fixed codimension. Its extremal rays were shown to correspond to pure Cohen-Macaulay modules and are the basis of a decomposition algorithm capable of expressing any graded Betti table as a positive rational linear combination of those of pure resolutions. Since then there has been great interest in finding parallel theories in other settings. Boij-Floystad give a description of a Boij-S\"oderberg theory for bigraded modules of codimension two. Several authors are working on versions with combinatorial flavors including monomial ideals, toric ideals and equivariant versions. Recently Avramov-Gibbons-Wiegand describe the full semigroup of Betti tables over short Gorenstein rings. The original Eisenbud-Schreyer theorem, extended to arbitrary modules by Boij-S\"oderberg, has been applied to the study of the Horrocks conjecture by Erman. This powerful tool likely will have further exciting uses.

Another problem of major interest in commutative algebra and algebraic geometry is the Eisenbud-Green-Harris (EGH) Conjecture, which characterizes all possible Hilbert functions of homogeneous ideals in graded complete intersection rings. It is a simultaneous generalization of two results: the Cayley Bacharach theorem from algebraic geometry and the Kruskal-Katona theorem from algebraic combinatorics. Furthermore it suggests a sharp upper bound on the graded Betti numbers of such ideals. In the combinatorial setting (over Clements-Lindstr\"om rings) where the EGH Conjecture is known, these bounds are approved by Mermin-Murai for finite free resolutions. Both questions motivate the study of resolutions over complete intersections.

Complete intersection rings are ubiquitous in algebra and geometry. Contrary to the resolutions over a polynomial ring, resolutions over complete intersections are infinite. Obtaining a description, even asymptotically, of such resolutions is a difficult problem. The simplest case is that of a hypersurface ring. In 1980 Eisenbud showed that the minimal free resolutions over a hypersurface ring are eventually periodic of period two; moreover, the matrices representing the alternating maps in the periodic part form a matrix factorization. The concept of matrix factorization has many applications in the study of Cohen-Macaulay modules, singularity theory, cluster algebras and cluster tilting, Hodge theory, Khovanov-Rozansky homology, moduli of curves, singularity categories, quiver and group representations, and string theory. Recently Eisenbud-Peeva defined a notion of matrix factorizations that generalizes the hypersurface case to arbitrary complete intersections. In particular every (infinite) minimal free resolution over a complete intersection ring is eventually given by the finite amount of data in a matrix factorization of the defining regular sequence. They relate the resolutions of matrix factorization modules over complete intersections to those over regular rings. Their machinery is already yielding surprising structural results about high syzygies over complete intersections such as the recent work of Eisenbud-Peeva-Schreyer.

Recently there has been a surge of interest and real progress in the above areas. The workshop will be timely and will provide an opportunity to bring together researchers related to algebraic geometry and commutative algebra, in order to discuss exciting recent developments and explore new directions.

Given a graded module $M$ over a polynomial ring $S$ over a field, its minimal graded free resolution is an invariant which carries a lot of information about M. The length, or projective dimension, of the resolution is at most the number of variables by a classical result of Hilbert. The largest degree of a syzygy is measured by its Castelnuovo-Mumford regularity. When $M = I$ is an ideal of $S$, Stillman asked whether a bound on the projective dimension could be obtained in terms of the degrees of the generators but independent of the number of variables. The parallel question regarding a bound on regularity was shown to be equivalent by Caviglia. In a recent (2016) breakthrough, Ananyan-Hochster have proved an affirmative answer to Stillman's Question in characteristic 0 and in the case when the characteristic is larger than the degrees of the minimal generators of $I$. Their work shows that one can always find a polynomial subring of $S$ with a bounded number of variables which contains the generators of $I$, from which they derive their result. Whether tighter bounds than those given by Ananyan-Hochster are possible is a fundamental problem in computational algebra; better bounds would give insight into the complexity of Buchbergerâ€™s algorithm for the computation of Gr\"obner bases. Tight bounds given by Huneke et. al. are known in only a few specific cases; however examples by Beder et. al. show that the optimal answer will be quite large. Another interesting direction would be to consider whether better bounds are possible under assumptions on the ideal itself. For instance, de Stefani-Nunez-Betancourt recently showed that the defining ideal of an $F$-injective ring satisfies a nice projective dimension bound. It would be interesting to know if more general statements are possible. Even less is known about the corresponding problems for regularity.

There is a doubly exponential bound (in terms of degrees of generators and number of variables) on the regularity of homogeneous ideals in a polynomial ring, proved by Bayer-Mumford, Caviglia-Sbarra, Galligo, and Giusti. In the geometric case, when we consider a nondegenerate prime ideal $P$ over an algebraically closed field, much better bounds on the regularity were expected. This was expressed by the Eisenbud-Goto Conjecture, which predicted a linear bound on regularity in terms of the degree of $P$. In another recent breakthrough (2016), McCullough-Peeva gave counterexamples to the Eisenbud-Goto conjecture and showed that no polynomial bound on regularity in terms of degree was possible. Their construction closely resembles the well-studied Rees algebras, which are tied to the geometric blow-up but whose defining equations are difficult to find in general. However, they prove that their Rees-like algebras have well-structured generators and free resolutions. It is an intriguing prospect to discover what information about the well-understood Rees-like algebras will pass to the more intricate Rees algebras. Another important question is whether there is any bound at all on the regularity of nondegenerate primes in terms of degree. Such a bound is closely related to those in Stillman's Question. Finally one can study the conjecture in the case that $P$ defines a smooth projective variety. In this case the conjecture is still open and there has been recent progress by Kwak-Park and Noma.

Another breakthrough in the theory of graded free resolutions comes from the proof of the Boij-S\"oderberg Conjectures by Eisenbud-Schreyer and the previous results of Eisenbud-Floystad-Weyman. Boij-S\"oderberg's key insight was that the study of graded Betti tables of modules might be best understood up to constant multiple by studying the semigroup or positive rational cone generated by all Betti tables of Cohen-Macaulay modules of fixed codimension. Its extremal rays were shown to correspond to pure Cohen-Macaulay modules and are the basis of a decomposition algorithm capable of expressing any graded Betti table as a positive rational linear combination of those of pure resolutions. Since then there has been great interest in finding parallel theories in other settings. Boij-Floystad give a description of a Boij-S\"oderberg theory for bigraded modules of codimension two. Several authors are working on versions with combinatorial flavors including monomial ideals, toric ideals and equivariant versions. Recently Avramov-Gibbons-Wiegand describe the full semigroup of Betti tables over short Gorenstein rings. The original Eisenbud-Schreyer theorem, extended to arbitrary modules by Boij-S\"oderberg, has been applied to the study of the Horrocks conjecture by Erman. This powerful tool likely will have further exciting uses.

Another problem of major interest in commutative algebra and algebraic geometry is the Eisenbud-Green-Harris (EGH) Conjecture, which characterizes all possible Hilbert functions of homogeneous ideals in graded complete intersection rings. It is a simultaneous generalization of two results: the Cayley Bacharach theorem from algebraic geometry and the Kruskal-Katona theorem from algebraic combinatorics. Furthermore it suggests a sharp upper bound on the graded Betti numbers of such ideals. In the combinatorial setting (over Clements-Lindstr\"om rings) where the EGH Conjecture is known, these bounds are approved by Mermin-Murai for finite free resolutions. Both questions motivate the study of resolutions over complete intersections.

Complete intersection rings are ubiquitous in algebra and geometry. Contrary to the resolutions over a polynomial ring, resolutions over complete intersections are infinite. Obtaining a description, even asymptotically, of such resolutions is a difficult problem. The simplest case is that of a hypersurface ring. In 1980 Eisenbud showed that the minimal free resolutions over a hypersurface ring are eventually periodic of period two; moreover, the matrices representing the alternating maps in the periodic part form a matrix factorization. The concept of matrix factorization has many applications in the study of Cohen-Macaulay modules, singularity theory, cluster algebras and cluster tilting, Hodge theory, Khovanov-Rozansky homology, moduli of curves, singularity categories, quiver and group representations, and string theory. Recently Eisenbud-Peeva defined a notion of matrix factorizations that generalizes the hypersurface case to arbitrary complete intersections. In particular every (infinite) minimal free resolution over a complete intersection ring is eventually given by the finite amount of data in a matrix factorization of the defining regular sequence. They relate the resolutions of matrix factorization modules over complete intersections to those over regular rings. Their machinery is already yielding surprising structural results about high syzygies over complete intersections such as the recent work of Eisenbud-Peeva-Schreyer.

Recently there has been a surge of interest and real progress in the above areas. The workshop will be timely and will provide an opportunity to bring together researchers related to algebraic geometry and commutative algebra, in order to discuss exciting recent developments and explore new directions.