# Bridges between Noncommutative Algebra and Algebraic Geometry (16w5088)

Arriving in Banff, Alberta Sunday, September 11 and departing Friday September 16, 2016

## Organizers

Jason Bell (University of Waterloo)

James Zhang (University of Washington)

Michael Artin (MIT)

Lance Small (University of California, San Diego)

Colin Ingalls (University of New Brunswick)

## Objectives

The theme of this workshop is the interplay between noncommutative algebra, algebraic geometry, and representation theory. Our objectives are to bring together researchers who apply geometric and homological methods in different areas of algebra; to identify new research directions; and to encourage interaction and collaborations. We now describe some of the topics that will be discussed at the workshop. The connection between noncommutative algebraic geometry and other disciplines has grown rapidly. This has led to advances in noncommutative algebraic geometry due to new ideas and techniques from other areas, conversely ideas from noncommutative algebraic geometry have played an important role in many recent results from representation theory, commutative algebra, and mathematical physics.

One of the more striking examples of this phenomenon is that noncommutative varieties have been shown to arise in mirror symmetry and in the study of the derived category of a commutative variety, which can be best understood through the framework of noncommutative algebra (or more generally, $A_infty$-algebras). In mirror symmetry, one does not generally have that mirrors are always commutative and so the larger picture supplied by noncommutative geometry becomes essential. A second example arises in Van den Bergh's approach to Orlov's conjecture by using noncommutative algebra and his construction of noncommutative crepant resolutions.

This result has spawned the area of noncommutative resolutions of singularities. The complex nature of singularities occurring in the minimal model program gives a serious obstruction to obtaining a concrete characterization of the local structure of a minimal model, and it is now known that some singularities can be easily resolved by allowing noncommutative coordinate rings. This has the potential to improve our grasp of the minimal model program and to deepen our understanding of the birational classification of varieties in all dimensions. The advantage of these new, noncommutative, structures is that they often exhibit better homological behaviour and they also allow one to study singularities via commonly known noncommutative structures and invariants. Obtaining deeper insight into singularities has long been a goal of traditional algebraic geometry and with these new ideas comes the promise of exciting new results in this area.

Recent work by Sierra and Walton used geometric methods to understand the representation theory of the enveloping algebra of the Virasoro algebra. In particular, they answered a twenty-year-old conjecture of Dean and Small by showing that this algebra is not noetherian. It is worth noting that, once again, this was a concrete ring theoretic question where traditional techniques had failed. By understanding the point modules of this enveloping algebra and showing that they are parametrized by a projective scheme, Sierra and Walton were able to construct an explicit ascending chain of left ideals that does not terminate. The recent work of Etingof and Walton used geometric methods and deep results about fusion categories to completely classify finite-dimensional semisimple Hopf algebras that act inner faithfully on a commutative domain. This has naturally led to other questions about possible connections between noncommutative invariant theory and, in particular, Hopf algebra actions and noncommutative algebra.

Classical invariant theory of commutative polynomial rings contains a great number of beautiful results. In particular, from the viewpoint of homological algebra, the following results are fundamental: Noether's theorem, the Shephard-Todd-Chevalley Theorem, the Watanabe Theorem, and the Kac-Watanabe-Gordeev Theorem, which give criteria for an invariant subring to be respectively integral, regular, Gorenstein, and a complete intersection. In the noncommutative setting, Hopf algebra actions are more natural than group actions. Some of the results from classical invariant theory have already been established in the Hopf setting. One immediate goal is to prove analogues of Shephard-Todd-Chevalley and Kac-Watanabe-Gordeev theorems in the Hopf setting, where only partial results exist thus far. These studies are closely connected to the representation theory of finite groups and Hopf algebras, homological aspects of noncommutative algebra and the study of noncommutative singularities. A recent development in this area comes from the study of Hopf actions on Artin-Schelter regular algebras, initiated by Kirkman, Kuzmanovich, and Zhang. Just as the noncommutative invariant theory that arises in the study of Hopf algebra actions gives a sweeping generalization of classical invariant theory, the work of Kirkman, Kuzmanovich , and Zhang gives an extension of earlier work on group actions on regular algebras. The classification of finite-dimensional Hopf actions on Artin-Shelter regular algebras of dimension two was completed by Chan, Kirkman, Walton and Wang last year, and this leads naturally to the study of Kleinian or DuVal singularities of Hopf actions on noncommutative surfaces. Hopf algebra actions on almost commutative domains and on central simple algebras were recently examined by Etingof, Walton and Cuadra.

In addition to the above recent advances, much new work has appeared in the study of central simple algebras, which was initiated by Wedderburn, Albert and Brauer in the beginning of the 20th century. These algebras, and the closely related notion of the Brauer group of a field, play an important role in algebraic geometry, the theory of algebraic groups, algebraic number theory and algebraic $K$-theory. A recurring theme in this area is the use of algebro-geometric techniques, usually via the notion of ramification. Recently, Saltman and Krashen have used intersection theory to settle previously unknown cases of Amitsur's conjecture about birational isomorphisms of central simple algebras. One of the most exciting developments in the theory of central simple algebras is related to bounding the index in terms of the period for function fields of surfaces of varieties. Lieblich, Starr, Saltman, Krashen and de Jong have period-index results for fields of low transcendence degree. A striking result of Matzri from this year gives the first known bound for the symbol length of a division algebra over a field in terms of the period and its transcendence degree. The implications of this result and its proof are still being explored.

While Brauer groups implicitly deal with finite-dimensional division algebras, there is an analogous study of infinite-dimensional division rings. The study of infinite-dimensional division algebras, however, is much less developed than the study of their finite-dimensional counterparts. These algebras arise naturally as the quotient division rings of Ore domains such as Weyl algebras, enveloping algebras of finite-dimensional Lie algebras, and many quantum groups. Resco computed transcendence degrees of the maximal subfields of the Weyl division rings and his approach has been extended in work of Yekutieli and Zhang. Their work lays the foundations for an appropriately noncommutative transcendence degree. Moreover, they raise interesting questions as to when tensor products of division rings with themselves remain noetherian and when extending the base field preserves noetherianity. Other questions of interest focus on the subfield structure of various division rings and the existence of free subalgebras on two generators; for example, are the maximal subfields of the division ring of quotients of the quantum plane purely transcendental? This is not the case for the Weyl division rings as a famous example of Dixmier shows. Indeed, Mironov has recently constructed subfields of the first Weyl algebra of arbitrarily high genus. Finally, there is the question of whether a finitely generated division algebra which is not algebraic over its centre must contain a free algebra on two generators. Such a result would say that noncommutative localization is much more pathological than its commutative counterpart. Recently, Bell and Rogalski have shown that this is the case for quotient division algebras of finitely generated complex noetherian domains.

Geometric ideas and techniques coming from algebraic geometry are also having a groundbreaking effect on the study of noncommutative graded domains of low Gelfand-Kirillov dimension. There are new classification results and structure theorems that mimic results from commutative algebraic geometry. In particular, we mention current work of Rogalski, Sierra, and Stafford on blowing down noncommutative surfaces (or connected graded domains of Gelfand-Kirillov dimension three) and earlier work by Van den Bergh on blowing up. It is hoped that these new advances will ultimately lead to a birational classification of noncommutative surfaces as conjectured by Artin. The completion of Artin's program will have a revolutionary impact on noncommutative algebraic geometry, just as its commutative counterpart has had in the study of algebraic varieties.

This workshop would provide a useful follow-up to several related conferences that have been held over the past few years. We mention, in particular, the five-day Workshop on ``Noncommutative Algebraic Geometry and Related Topics'' held in Manchester, UK, from August 6--10, 2012; the five-day BIRS workshop ``New Trends in Noncommutative Algebra and Algebraic Geometry (12w5049)'' at the Banff International Research Station, Canada, from October 28--November 2, 2012; the five-day MSRI workshop on ``Interactions between Noncommutative Algebra, Representation Theory, and Algebraic Geometry'' in Berkeley, USA, from April 8--12, 2013; the five-day RIMS workshop on ``Noncommutative Algebraic Geometry and Related Topics'', in Kyoto, Japan, from July 1--5, 2013; the five-day Oberwolfach workshop on ``Interactions between Algebraic Geometry and Noncommutative Algebra'' in Germany, from May 18--24, 2014; and the recent five-day Shanghai workshop on ``Noncommutative Algebraic Geometry'' in Shanghai, China, from August 25-29, 2014.

One of the more striking examples of this phenomenon is that noncommutative varieties have been shown to arise in mirror symmetry and in the study of the derived category of a commutative variety, which can be best understood through the framework of noncommutative algebra (or more generally, $A_infty$-algebras). In mirror symmetry, one does not generally have that mirrors are always commutative and so the larger picture supplied by noncommutative geometry becomes essential. A second example arises in Van den Bergh's approach to Orlov's conjecture by using noncommutative algebra and his construction of noncommutative crepant resolutions.

This result has spawned the area of noncommutative resolutions of singularities. The complex nature of singularities occurring in the minimal model program gives a serious obstruction to obtaining a concrete characterization of the local structure of a minimal model, and it is now known that some singularities can be easily resolved by allowing noncommutative coordinate rings. This has the potential to improve our grasp of the minimal model program and to deepen our understanding of the birational classification of varieties in all dimensions. The advantage of these new, noncommutative, structures is that they often exhibit better homological behaviour and they also allow one to study singularities via commonly known noncommutative structures and invariants. Obtaining deeper insight into singularities has long been a goal of traditional algebraic geometry and with these new ideas comes the promise of exciting new results in this area.

Recent work by Sierra and Walton used geometric methods to understand the representation theory of the enveloping algebra of the Virasoro algebra. In particular, they answered a twenty-year-old conjecture of Dean and Small by showing that this algebra is not noetherian. It is worth noting that, once again, this was a concrete ring theoretic question where traditional techniques had failed. By understanding the point modules of this enveloping algebra and showing that they are parametrized by a projective scheme, Sierra and Walton were able to construct an explicit ascending chain of left ideals that does not terminate. The recent work of Etingof and Walton used geometric methods and deep results about fusion categories to completely classify finite-dimensional semisimple Hopf algebras that act inner faithfully on a commutative domain. This has naturally led to other questions about possible connections between noncommutative invariant theory and, in particular, Hopf algebra actions and noncommutative algebra.

Classical invariant theory of commutative polynomial rings contains a great number of beautiful results. In particular, from the viewpoint of homological algebra, the following results are fundamental: Noether's theorem, the Shephard-Todd-Chevalley Theorem, the Watanabe Theorem, and the Kac-Watanabe-Gordeev Theorem, which give criteria for an invariant subring to be respectively integral, regular, Gorenstein, and a complete intersection. In the noncommutative setting, Hopf algebra actions are more natural than group actions. Some of the results from classical invariant theory have already been established in the Hopf setting. One immediate goal is to prove analogues of Shephard-Todd-Chevalley and Kac-Watanabe-Gordeev theorems in the Hopf setting, where only partial results exist thus far. These studies are closely connected to the representation theory of finite groups and Hopf algebras, homological aspects of noncommutative algebra and the study of noncommutative singularities. A recent development in this area comes from the study of Hopf actions on Artin-Schelter regular algebras, initiated by Kirkman, Kuzmanovich, and Zhang. Just as the noncommutative invariant theory that arises in the study of Hopf algebra actions gives a sweeping generalization of classical invariant theory, the work of Kirkman, Kuzmanovich , and Zhang gives an extension of earlier work on group actions on regular algebras. The classification of finite-dimensional Hopf actions on Artin-Shelter regular algebras of dimension two was completed by Chan, Kirkman, Walton and Wang last year, and this leads naturally to the study of Kleinian or DuVal singularities of Hopf actions on noncommutative surfaces. Hopf algebra actions on almost commutative domains and on central simple algebras were recently examined by Etingof, Walton and Cuadra.

In addition to the above recent advances, much new work has appeared in the study of central simple algebras, which was initiated by Wedderburn, Albert and Brauer in the beginning of the 20th century. These algebras, and the closely related notion of the Brauer group of a field, play an important role in algebraic geometry, the theory of algebraic groups, algebraic number theory and algebraic $K$-theory. A recurring theme in this area is the use of algebro-geometric techniques, usually via the notion of ramification. Recently, Saltman and Krashen have used intersection theory to settle previously unknown cases of Amitsur's conjecture about birational isomorphisms of central simple algebras. One of the most exciting developments in the theory of central simple algebras is related to bounding the index in terms of the period for function fields of surfaces of varieties. Lieblich, Starr, Saltman, Krashen and de Jong have period-index results for fields of low transcendence degree. A striking result of Matzri from this year gives the first known bound for the symbol length of a division algebra over a field in terms of the period and its transcendence degree. The implications of this result and its proof are still being explored.

While Brauer groups implicitly deal with finite-dimensional division algebras, there is an analogous study of infinite-dimensional division rings. The study of infinite-dimensional division algebras, however, is much less developed than the study of their finite-dimensional counterparts. These algebras arise naturally as the quotient division rings of Ore domains such as Weyl algebras, enveloping algebras of finite-dimensional Lie algebras, and many quantum groups. Resco computed transcendence degrees of the maximal subfields of the Weyl division rings and his approach has been extended in work of Yekutieli and Zhang. Their work lays the foundations for an appropriately noncommutative transcendence degree. Moreover, they raise interesting questions as to when tensor products of division rings with themselves remain noetherian and when extending the base field preserves noetherianity. Other questions of interest focus on the subfield structure of various division rings and the existence of free subalgebras on two generators; for example, are the maximal subfields of the division ring of quotients of the quantum plane purely transcendental? This is not the case for the Weyl division rings as a famous example of Dixmier shows. Indeed, Mironov has recently constructed subfields of the first Weyl algebra of arbitrarily high genus. Finally, there is the question of whether a finitely generated division algebra which is not algebraic over its centre must contain a free algebra on two generators. Such a result would say that noncommutative localization is much more pathological than its commutative counterpart. Recently, Bell and Rogalski have shown that this is the case for quotient division algebras of finitely generated complex noetherian domains.

Geometric ideas and techniques coming from algebraic geometry are also having a groundbreaking effect on the study of noncommutative graded domains of low Gelfand-Kirillov dimension. There are new classification results and structure theorems that mimic results from commutative algebraic geometry. In particular, we mention current work of Rogalski, Sierra, and Stafford on blowing down noncommutative surfaces (or connected graded domains of Gelfand-Kirillov dimension three) and earlier work by Van den Bergh on blowing up. It is hoped that these new advances will ultimately lead to a birational classification of noncommutative surfaces as conjectured by Artin. The completion of Artin's program will have a revolutionary impact on noncommutative algebraic geometry, just as its commutative counterpart has had in the study of algebraic varieties.

This workshop would provide a useful follow-up to several related conferences that have been held over the past few years. We mention, in particular, the five-day Workshop on ``Noncommutative Algebraic Geometry and Related Topics'' held in Manchester, UK, from August 6--10, 2012; the five-day BIRS workshop ``New Trends in Noncommutative Algebra and Algebraic Geometry (12w5049)'' at the Banff International Research Station, Canada, from October 28--November 2, 2012; the five-day MSRI workshop on ``Interactions between Noncommutative Algebra, Representation Theory, and Algebraic Geometry'' in Berkeley, USA, from April 8--12, 2013; the five-day RIMS workshop on ``Noncommutative Algebraic Geometry and Related Topics'', in Kyoto, Japan, from July 1--5, 2013; the five-day Oberwolfach workshop on ``Interactions between Algebraic Geometry and Noncommutative Algebra'' in Germany, from May 18--24, 2014; and the recent five-day Shanghai workshop on ``Noncommutative Algebraic Geometry'' in Shanghai, China, from August 25-29, 2014.