Periodic Reporting for period 1 - SINGREP (Linking singularity theory and representation theory with homological methods)
Periodo di rendicontazione: 2018-08-01 al 2020-07-31
This project lies at the crossroads of singularity theory, algebraic geometry, commutative algebra, and representation theory. The main goal was to develop homological methods to find novel links between singularity theory and representation theory. Further, the experienced researcher (in the following referred to as the PI) wanted to exploit these connections in both directions to increase the understanding of singular algebraic varieties as well as the representation theory of algebras.
During the fellowship, the PI focussed on three work packages:
(1) the construction and study of noncommutative (crepant) resolutions of singularities (NC(C)Rs),
(2) a McKay correspondence for reflection groups,
(3) the study of friezes and singularity categories.
(1) The aim was to develop new methods for finding Noncommutative (Crepant) Resolutions of singularities (NC(C)Rs) and to determine their properties in important cases (in particular for toric rings).
(2) The aim was to establish a McKay correspondence for reflection groups. The construction proposed by Buchweitz, the PI, and Ingalls, is both natural and surprising, since reflection groups have not been studied yet in context of the McKay correspondence. During the fellowship, the PI proposed to work on several algebraic aspects emerging from this correspondence.
(3) The first objective was to find a categorical interpretation of higher SL_k-friezes. Further, the PI proposed to find new links between friezes and singularities by studying categories of maximal Cohen-Macaulay modules over coordinate rings of singular varieties to obtain friezes and other combinatorial data.
During the fellowship, the PI focussed on three work packages:
(1) the construction and study of noncommutative (crepant) resolutions of singularities (NC(C)Rs),
(2) a McKay correspondence for reflection groups,
(3) the study of friezes and singularity categories.
(1) The aim was to develop new methods for finding Noncommutative (Crepant) Resolutions of singularities (NC(C)Rs) and to determine their properties in important cases (in particular for toric rings).
(2) The aim was to establish a McKay correspondence for reflection groups. The construction proposed by Buchweitz, the PI, and Ingalls, is both natural and surprising, since reflection groups have not been studied yet in context of the McKay correspondence. During the fellowship, the PI proposed to work on several algebraic aspects emerging from this correspondence.
(3) The first objective was to find a categorical interpretation of higher SL_k-friezes. Further, the PI proposed to find new links between friezes and singularities by studying categories of maximal Cohen-Macaulay modules over coordinate rings of singular varieties to obtain friezes and other combinatorial data.
There are currently 7 publicly available research papers linked to the project. Preliminary work for 3 publications on topics of the fellowship has been carried out and the PI will work on finishing these in the coming months.
(1) Regarding construction of NCRs, the PI considered 1-dimensional commutative local rings. The main results are a characterization of certain of these rings in terms of trace ideals, and a characterization of the curves for which NCRs of global dimension 3 exist.
In collaboration with G. Muller and K.E. Smith the PI studied NCRs for toric rings R. They prove with elementary methods that the endomorphism ring of the sum of all (isomorphism classes of) conic R-modules has finite global dimension equal to the dimension of R. It is shown that this is a NCCR if and only if the toric variety Spec(R) is simplicial. For toric varieties over a perfect field of prime characteristic, it is shown that the ring of differential operators has finite global dimension.
(2) The main result of the collaboration with Buchweitz and Ingalls is that if the finite group G is generated by reflections of order 2 (i.e. G is a true reflection group), then a quotient of the skew group ring is isomorphic to the endomorphism ring of the hyperplane arrangement over the discriminant of G. This yields a NCR of the discriminant of G. The indecomposable projective modules over the quotient of the skew group ring are in bijection with the non-trivial irreducible representations of G and also with maximal Cohen-Macaulay-modules over the discriminant. This gives a McKay correspondence for true reflection groups.
As a byproduct, Buchweitz, the PI, and Ingalls, studied the magic square of reflections and rotations in dimension 2 and they show how to interpret it in terms of Clifford algebras.
In order to understand the geometry for pseudo-reflection groups better, the PI studied McKay quivers, together with C. Ingalls and M. Lewis. The main results are: a complete combinatorial description of the McKay quivers of the infinite family of pseudo-reflection groups G(r,p,n), and the definition of Lusztig algebras, which are path algebras of the McKay quiver.
(3) In collaboration with K. Baur, S. Gratz, K. Serhiyenko, and G. Todorov, the PI constructed SL_k-friezes from Plücker coordinates,
using the cluster structure on the homogeneous coordinate ring of the Grassmannian of k-spaces in n-space A(k,n). One of the main results is that when the cluster algebra A(k,n) is of finite type, then SL_k-friezes are in bijection with mesh friezes of the corresponding Grassmannian cluster category.
Together with J. August, M.-W. Cheung, S. Gratz, and S. Schroll, the PI constructs Grassmannian categories of infinite rank, providing an infinite analogue of Grassmannian cluster categories. They show that generically free modules of rank 1 in a Grassmannian category of infinite rank are in bijection with the Plücker coordinates in an appropriate Grassmannian cluster algebra. This work combines the study of matrix factorizations and cluster categories.
Transfer-of-knowledge: During the fellowship, the PI started mentoring a Ph.D. student, who is working on problems related to topic (2). The fellowship supported the PI through travel money and time that she could spend on the organisation of 5 scientific events (3 workshops, 1 conference, and 1 special session).
Dissemination: the PI gave 25 talks at international conferences, workshops, and research seminars. We mention as highlights 3 talks at workshops in Oberwolfach and CMO Oaxaca, and talks at conference on singularity theory at TSIMF in Sanya and an online conference on the McKay correspondence hosted by Kavli-IPMU.
(1) Regarding construction of NCRs, the PI considered 1-dimensional commutative local rings. The main results are a characterization of certain of these rings in terms of trace ideals, and a characterization of the curves for which NCRs of global dimension 3 exist.
In collaboration with G. Muller and K.E. Smith the PI studied NCRs for toric rings R. They prove with elementary methods that the endomorphism ring of the sum of all (isomorphism classes of) conic R-modules has finite global dimension equal to the dimension of R. It is shown that this is a NCCR if and only if the toric variety Spec(R) is simplicial. For toric varieties over a perfect field of prime characteristic, it is shown that the ring of differential operators has finite global dimension.
(2) The main result of the collaboration with Buchweitz and Ingalls is that if the finite group G is generated by reflections of order 2 (i.e. G is a true reflection group), then a quotient of the skew group ring is isomorphic to the endomorphism ring of the hyperplane arrangement over the discriminant of G. This yields a NCR of the discriminant of G. The indecomposable projective modules over the quotient of the skew group ring are in bijection with the non-trivial irreducible representations of G and also with maximal Cohen-Macaulay-modules over the discriminant. This gives a McKay correspondence for true reflection groups.
As a byproduct, Buchweitz, the PI, and Ingalls, studied the magic square of reflections and rotations in dimension 2 and they show how to interpret it in terms of Clifford algebras.
In order to understand the geometry for pseudo-reflection groups better, the PI studied McKay quivers, together with C. Ingalls and M. Lewis. The main results are: a complete combinatorial description of the McKay quivers of the infinite family of pseudo-reflection groups G(r,p,n), and the definition of Lusztig algebras, which are path algebras of the McKay quiver.
(3) In collaboration with K. Baur, S. Gratz, K. Serhiyenko, and G. Todorov, the PI constructed SL_k-friezes from Plücker coordinates,
using the cluster structure on the homogeneous coordinate ring of the Grassmannian of k-spaces in n-space A(k,n). One of the main results is that when the cluster algebra A(k,n) is of finite type, then SL_k-friezes are in bijection with mesh friezes of the corresponding Grassmannian cluster category.
Together with J. August, M.-W. Cheung, S. Gratz, and S. Schroll, the PI constructs Grassmannian categories of infinite rank, providing an infinite analogue of Grassmannian cluster categories. They show that generically free modules of rank 1 in a Grassmannian category of infinite rank are in bijection with the Plücker coordinates in an appropriate Grassmannian cluster algebra. This work combines the study of matrix factorizations and cluster categories.
Transfer-of-knowledge: During the fellowship, the PI started mentoring a Ph.D. student, who is working on problems related to topic (2). The fellowship supported the PI through travel money and time that she could spend on the organisation of 5 scientific events (3 workshops, 1 conference, and 1 special session).
Dissemination: the PI gave 25 talks at international conferences, workshops, and research seminars. We mention as highlights 3 talks at workshops in Oberwolfach and CMO Oaxaca, and talks at conference on singularity theory at TSIMF in Sanya and an online conference on the McKay correspondence hosted by Kavli-IPMU.
During the fellowship, the PI has started 4 new collaborations - some have already produced outputs directly related to the fellowship and some are are going beyond to new frontiers.
Let us mention impacts of the results above:
The toric rings from the collaboration with Muller and Smith in (1) are the first surprising examples of non-regular commutative rings for which the global dimension of the ring differential operators is finite, with an explicit bound.
This and the combinatorial construction of the NCRs sparked interest from various mathematical communities, e.g. the PI presented this work at the Oberwolfach workshops on Commutative Algebra and on Toric Geometry.
The work in (2) is interesting for several mathematical communities, as highlights, the PI gave two 4 hour lecture courses for Ph.D. students on it, as well as numerous conference and seminar talks. The PI will continue to investigate the McKay correspondence for pseudo-reflection groups.
Moreover, from the collaboration with Ingalls and Lewis, the PI got interested in Koszul algebras. In order to understand these algebras better, the PI started a collaboration with M. Juhnke-Kubitzke, H. Lindo, C. Miller, R. R.G. and A. Seceleanu, which already lead to first results.
The work in (3) on friezes yields a new categorical perspective on higher friezes and explains all known non-unitary SL_3-friezes from this point of view.
The PI is further discussing with the supervisor about higher friezes: they are currently investigating how to categorically glue together friezes and have some promising first steps in this direction.
The PI is also continuing the collaboration to study the infinite Grassmannian category of type A-infinity, which is the Z-graded category of maximal CM-modules over a non-reduced commutative ring of dimension 1. This infinite Grassmannian category can be modeled by arcs in an infinity-gon. Thus one gets a new perspective on a seemingly well-known category. This will lead to new connections between representation theory, singularity theory, and combinatorics.
Let us mention impacts of the results above:
The toric rings from the collaboration with Muller and Smith in (1) are the first surprising examples of non-regular commutative rings for which the global dimension of the ring differential operators is finite, with an explicit bound.
This and the combinatorial construction of the NCRs sparked interest from various mathematical communities, e.g. the PI presented this work at the Oberwolfach workshops on Commutative Algebra and on Toric Geometry.
The work in (2) is interesting for several mathematical communities, as highlights, the PI gave two 4 hour lecture courses for Ph.D. students on it, as well as numerous conference and seminar talks. The PI will continue to investigate the McKay correspondence for pseudo-reflection groups.
Moreover, from the collaboration with Ingalls and Lewis, the PI got interested in Koszul algebras. In order to understand these algebras better, the PI started a collaboration with M. Juhnke-Kubitzke, H. Lindo, C. Miller, R. R.G. and A. Seceleanu, which already lead to first results.
The work in (3) on friezes yields a new categorical perspective on higher friezes and explains all known non-unitary SL_3-friezes from this point of view.
The PI is further discussing with the supervisor about higher friezes: they are currently investigating how to categorically glue together friezes and have some promising first steps in this direction.
The PI is also continuing the collaboration to study the infinite Grassmannian category of type A-infinity, which is the Z-graded category of maximal CM-modules over a non-reduced commutative ring of dimension 1. This infinite Grassmannian category can be modeled by arcs in an infinity-gon. Thus one gets a new perspective on a seemingly well-known category. This will lead to new connections between representation theory, singularity theory, and combinatorics.