# Free Resolutions, Representations, and Asymptotic Algebra (16w5155)

Arriving in Banff, Alberta Sunday, April 3 and departing Friday April 8, 2016

## Objectives

In commutative algebra, as in many fields of mathematics, many modules of interest naturally occur in families $\{ M_n \}$ indexed by the nonnegative integers. Moreover, the numerical invariants or algebraic properties of the modules $M_n$ often stabilize for all sufficiently large $n$. Inspired by several parallel developments and striking new examples, this workshop aims to solidify, consolidate, and analyze this emerging asymptotic algebraic phenomenon. The novelty of the motivating results and the lack of any comparable conference create a unique opportunity.

The first source of examples comes from equivariant commutative algebra. For instance, Sam-Snowden consider equivariant modules $M_n$ over the polynomial ring in $n$ variables equipped with an action of the general linear group. For a fixed nonnegative integer $k$, the space $(n \times n)$-matrices of rank at most $k$ give an explicit family of interest. Sam-Snowden construct an algebraic framework for studying a limiting object $M_{\infty}$ over the polynomial ring in infinitely many variables with an action of the infinite general linear group. By proving Noetherian properties for the limiting object, they draw conclusions about numerical invariants of the individual modules $M_n$ for all sufficiently large $n$. Analogous structures appear in the work of Aschenbrennar-Hillar and Hillar-Sullivant on ideals fixed under an action of the infinite symmetric group, the paper by Draisma-Kuttler on degree of bounded rank tensors, and the articles by Sam-Snowden and Putman-Sam on representation stability. The work of Putman-Sam- Snowden will notably be the focus of a November 2014 Bourbaki seminar.

The theory of FI-modules as developed by Church-Ellenberg-Farb provides a second important source for these asymptotic ideas. Roughly speaking, an FI-module is a homological object that encodes a sequence of representations $M_n$ connected by families of linear maps. As a concrete example, the cohomology groups of configurations spaces, which parametrize unordered $n$-tuples of distinct points on a fixed manifold, form an FI-module. By characterizing finitely generated FI-modules, Church-Ellenberg-Farb obtain polynomial descriptions for characters and dimensions of $M_n$ for all sufficiently large $n$. These methods also apply to the Betti numbers of congruence subgroups, multi-graded Betti numbers of diagonal coinvariant algebras, and classical Weyl group characters.

A third source of examples arise from the various embeddings of an algebraic scheme. Specifically, for a smooth algebraic variety $X$ with a fixed very ample line bundle $L$, Ein-Lazarsfeld study the free resolution of the homogeneous coordinate for $X$ under the closed immersion given by $n$-fold tensor product of $L$. Although the numerical invariants of these free resolutions depend on the geometry of $X$ is remarkably subtle ways, Ein-Lazarsfeld prove a uniform non-vanishing theorem. As a consequence, these numerical invariants stabilize into a surprisingly simple picture for sufficiently large $n$. Similar results are established by Beck-Stapledon and McCabe-Smith for Hilbert functions, and by Erman for Boij-Soederberg decompositions.

Common features among these different sources point towards a unifying philosophy, but the details have yet to materialize. There are already nascent connections including direct links between twisted commutative algebras and FI-modules, Snowden's work on $\Delta$-modules and asymptotic syzygies of products of projective spaces, and Raicu's work on representation stability for asymptotic Veronese embeddings of projective space. However, there is no comprehensive theory or even a heuristic for when a family $\{M_n\}$ should exhibit this sort of asymptotic stability. The major goals of this workshop are to refine these asymptotic ideas, to articulate unifying themes, and to identify the most promising new directions for study. The workshop will also start dialogues and possibly collaborations between relatively distant communities by bringing interested researchers together for the first time. Because important families of examples appear in such a range of mathematical fields, these new asymptotic techniques in commutative algebra are poised to have a broad impact.

The first source of examples comes from equivariant commutative algebra. For instance, Sam-Snowden consider equivariant modules $M_n$ over the polynomial ring in $n$ variables equipped with an action of the general linear group. For a fixed nonnegative integer $k$, the space $(n \times n)$-matrices of rank at most $k$ give an explicit family of interest. Sam-Snowden construct an algebraic framework for studying a limiting object $M_{\infty}$ over the polynomial ring in infinitely many variables with an action of the infinite general linear group. By proving Noetherian properties for the limiting object, they draw conclusions about numerical invariants of the individual modules $M_n$ for all sufficiently large $n$. Analogous structures appear in the work of Aschenbrennar-Hillar and Hillar-Sullivant on ideals fixed under an action of the infinite symmetric group, the paper by Draisma-Kuttler on degree of bounded rank tensors, and the articles by Sam-Snowden and Putman-Sam on representation stability. The work of Putman-Sam- Snowden will notably be the focus of a November 2014 Bourbaki seminar.

The theory of FI-modules as developed by Church-Ellenberg-Farb provides a second important source for these asymptotic ideas. Roughly speaking, an FI-module is a homological object that encodes a sequence of representations $M_n$ connected by families of linear maps. As a concrete example, the cohomology groups of configurations spaces, which parametrize unordered $n$-tuples of distinct points on a fixed manifold, form an FI-module. By characterizing finitely generated FI-modules, Church-Ellenberg-Farb obtain polynomial descriptions for characters and dimensions of $M_n$ for all sufficiently large $n$. These methods also apply to the Betti numbers of congruence subgroups, multi-graded Betti numbers of diagonal coinvariant algebras, and classical Weyl group characters.

A third source of examples arise from the various embeddings of an algebraic scheme. Specifically, for a smooth algebraic variety $X$ with a fixed very ample line bundle $L$, Ein-Lazarsfeld study the free resolution of the homogeneous coordinate for $X$ under the closed immersion given by $n$-fold tensor product of $L$. Although the numerical invariants of these free resolutions depend on the geometry of $X$ is remarkably subtle ways, Ein-Lazarsfeld prove a uniform non-vanishing theorem. As a consequence, these numerical invariants stabilize into a surprisingly simple picture for sufficiently large $n$. Similar results are established by Beck-Stapledon and McCabe-Smith for Hilbert functions, and by Erman for Boij-Soederberg decompositions.

Common features among these different sources point towards a unifying philosophy, but the details have yet to materialize. There are already nascent connections including direct links between twisted commutative algebras and FI-modules, Snowden's work on $\Delta$-modules and asymptotic syzygies of products of projective spaces, and Raicu's work on representation stability for asymptotic Veronese embeddings of projective space. However, there is no comprehensive theory or even a heuristic for when a family $\{M_n\}$ should exhibit this sort of asymptotic stability. The major goals of this workshop are to refine these asymptotic ideas, to articulate unifying themes, and to identify the most promising new directions for study. The workshop will also start dialogues and possibly collaborations between relatively distant communities by bringing interested researchers together for the first time. Because important families of examples appear in such a range of mathematical fields, these new asymptotic techniques in commutative algebra are poised to have a broad impact.