# Preprojective Algebras Interacting with Singularities, Cohen-Macaulay Modules and Weighted projective Spaces (15w5116)

Arriving in Oaxaca, Mexico Sunday, October 4 and departing Friday October 9, 2015

## Organizers

Ragnar-Olaf Buchweitz (University of Toronto Scarborough)

José Antonio de la Peña (Centro de Investigación en Matemáticas)

Osamu Iyama (Nagoya University)

Helmut Lenzing (University of Paderborn)

## Objectives

1. Preprojective algebras

Classical preprojective algebras, introduced by Gelfand-Ponomarev cite{Gelfand:Ponomarev:1979}, are a well-studied topic of the representation theory of finite quivers or, in different terminology, of finite dimensional hereditary algebras. If $H$ is such an algebra, supposed to be of infinite representation type, the associated preprojective algebra $Pi(H)$ occurs in two differently looking, but isomorphic, incarnations, see cite{Baer:Geigle:Lenzing:1987}. The first one owes its existence to Auslander-Reiten theory: The orbit $(tau^{-n}H)$, $ngeq0$, of the inverse Auslander-Reiten translation, has the category of all preprojective $H$-modules as its additive closure, hence a well-understood category of combinatorial flavor. By setting $R=bigoplus_{ngeq0}R_n$ with $R_n=mathrm{Hom}(H,tau^{-n}H)$, we obtain the graded preprojective algebra as an orbit algebra. The second approach is homological in nature: If $D$ denotes the standard vector space duality, one first forms the $(H,H)$-bimodule $mathrm{Ext}^1(DH,H)$ describing the homological interaction between projective and injective $H$-modules, and then obtains the tensor algebra $mathbb{T}({mathrm{Ext}^1(DH,H)})$ as a natural algebraic object which encodes this kind of homological interplay.

A well-established approach to study the graded preprojective algebra, is by treating it as a cone singularity, that is, by studying its graded projective spectrum. This is done by applying Serre's construction cite{Serre:1955}, which associates to the setting an abelian category, to be interpreted as a category of coherent sheaves on a space $mathbb{X}$ incorporating the geometry of the problem. If $H$ is tame, that is, if $H$ is the path algebra of an extended Dynkin quiver, then this category $mathrm{coh},mathbb{X}$ is known to be equivalent to the category of coherent sheaves on a weighted projective line $mathbb{X}(a,b,c)$, whose weight type is given by a Dynkin triple $(a,b,c)$ cite{Geigle:Lenzing:1987},cite{Baer:Geigle:Lenzing:1987}. If $H$ is wild, the space $mathbb{X}$ is still mysterious. But even then, actually in both cases, the category $mathrm{coh},mathbb{X}$ is well understood, since it is derived equivalent to the category $mod{H}$ of finite dimensional $H$-modules cite{Lenzing:1986}, cite{Minamoto:2008}, cite{Minamoto:2012}.

One major attempt at this workshop will be to obtain a better understanding of the so-called emph{higher preprojective algebras}, see cite{Iyama:Oppermann:2013,Kel11}, by choosing a similar geometric approach. Simplifying, such a $d$-preprojective algebra $Pi^{(d)}(A)$ can be attached to any finite dimensional algebra $A$ of finite global dimension $d$, just by forming the tensor algebra of the $(A,A)$-bimodule $mathrm{Ext}^d(DA,A)$. Moreover, in good cases, $d$ also has a geometric interpretation of a (possibly non-commutative) smooth projective variety. This amounts to putting severe additional restrictions on $A$, because the ring-theoretic properties of $Pi^{(d)}(A)$ are usually not good enough to support a successful geometric analysis. In the framework of higher Auslander-Reiten theory, a group of researchers [HIMO] (Herschend-Iyama-Minamoto-Oppermann) is investigating the setting. By now, promising preliminary results have been obtained cite{Oppermann:2013}, cite{Iyama:2013}. Specializing to socalled $d$-emph{representation-infinite} algebras, one obtains noetherian algebras with the potential to undertake a geometric analysis via Serre construction. Up to now, however, only few cases have lead to a fairly complete analysis. As these partial results indicate, there is a good chance to identify more of the obtained spaces to be weighted projective spaces or, in a different setup, as orbifolds, a class of geometric objects typically arising as quotients of well-behaved geometrical group actions.

This explains current interest in this new kind of spaces, where currently two groups [HIMO] and [KLM] (Kussin-Lenzing-Meltzer), see cite{Kussin:Lenzing:Meltzer:2013adv}, cite{Lenzing:2013}, cite{Lenzing:2013a}, in particular two of the organizers, are involved.

2. Weighted projective spaces

Weighted projective varieties, containing the class of weighted projective spaces, apparently made their first appearance as Satake's V-spaces cite{Satake:1956} already in 1956.

Still at present, there is no generally accepted concept for the `right definition' of a weighted projective space. Correspondingly, the present knowledge on such spaces is very fragmentary. Only slight exaggerating, it is true that there is a chaos of competing and incompatible definitions, and a corresponding confusion concerns the results. In the setting of algebraic geometry, weighted projective spaces were introduced and studied by I. Dolgachev cite{Dolgachev:1982} and Beltrametti-Robbiano cite{Beltrametti:Robbiano:1986}. These spaces, let's call them having type (I), arise by allowing a grading for the projective coordinate algebra, an appropriate polynomial algebra, where the generators get arbitrary positive degrees. The resulting (classical) projective spectrum then, typically, is a singular space. More recently, it gets customary in this situation, alternatively, to form the attached category of coherent sheaves by means of the associated emph{graded} projective spectrum, equivalently by Serre construction cite{Auroux:Katzarkov:Orlov:2008}. The advantage of such an approach is to obtain a non-singular or smooth weighted projective space $mathbb{X}$ whose category $mathrm{coh},mathbb{X}$ of coherent sheaves thus has finite global dimension. We will say, that these spaces have type (II).

A similar approach also yielding smooth weighted projective spaces was already established in the late 1980's by Geigle-Lenzing cite{Geigle:Lenzing:1987} and D. Baer cite{Baer:1988} where the coordinate algebra of the `underlying' projective space $mathbb{P}^d$ is suitably graded by a finitely generated, rank one abelian group $mathbb{L}$ and then Serre's construction is applied to obtain the categories $mathrm{coh},mathbb{X}$. Let's call these spaces to have type (III). To complicate the situation, Geigle-Lenzing introduced the concept of a weighted projective line, resulting in a space $mathbb{X}$ with a finite number of `weighted' points. Weighted projective lines turn out to be an important tool in the representation theory of finite dimensional algebras, and by a theorem of Happel cite{Happel:2001} their categories of coherent sheaves $mathrm{coh},mathbb{X}$ exhaust together with the module categories $mod{H}$ over a finite quiver (without oriented cycles) --- up to derived equivalence --- the class of hereditary categories with a tilting object. Due to Happel's result, it is generally accepted, that the concept of weighted projective lines is the proper notion for dimension one. However, a weighted projective line is typically NOT a one-dimensional weighted projective space of type (I), (II) or (III). Let's therefore call the weighted projective lines to have type (iv).

Only very recently in ongoing research, weighted projective spaces of type (IV), where specialization to dimension one agrees with type (iv), were introduced independently by [HIMO] and [KLM]. Both groups are following complementary motivations. These spaces, called GL-spaces by [HIMO], will be said to have type (IV).

(i)The motivation of [KLM] is a study of the (suitably graded) singularity theory of Brieskorn singularities $f=x_1^{a_1}+x_2^{a_2}+cdots+x_t^{a_t}$. Following the well-established pattern of the study of triangle singularities (case $t=3$) cite{Kussin:Lenzing:Meltzer:2013adv}, one associates by Serre construction to a Brieskorn singularity, more generally, to a complete intersection of such ones, a non-commutative projective space $mathbb{X}$, that turns out to be a weighted projective space over $mathbb{P}^d$, $d=t-2$, where as in type (III) the weights (expressed by multiplicities of simple sheaves) are concentrated in $t$ hyperplanes of $mathbb{P}^d$.

(ii) The motivation for [HIMO] is to create a set-up to produce algebraic and geometric applications for a number of important concepts from the higher Auslander-theory, also developed by these researchers. Let us just mention the $d$-representation-finite and $d$-representation-infinite algebras and, very importantly, for such classes of finite dimensional algebras the $d$-preprojective algebras which are usually infinite dimensional. [HIMO] is, in particular, interested in the class of $d$-canonical algebras, defined to be the endomorphism algebra of a canonical tilting object $T$ on a GL-space. ($T$ is a direct sum of line bundles, naturally arising in the theory.) Of particular interest for this, are the $d$ canonical algebras $Lambda$, where the GL-space $mathbb{X}$ has emph{Fano type}, corresponding to positive Gorenstein parameter. Conjecturally, assuming Fano type, the algebra $Lambda$ is $d$-representation-infinite, implying a nice behavior of the associated $d$-preprojective algebra $Pi=Pi^{(d)}(Lambda)$. It is expected that the associated geometric object then is the weighted projective space $mathbb{X}$, again.

As a byproduct, the Workshop aims to clarify the relationship between weighted projective spaces of smooth type (II), (III) and (IV) by taking the respective weight loci, that is, the arrangements of subspaces (or subvarieties) endowed with weights as an ordering principle. For instance, it is easy to see that for types (III) and (IV) the weights are concentrated on hyperplanes; moreover type (III) is a subclass of type (IV) if we allow weights to be one. For type (II), a preliminary analysis shows that the weights may be concentrated in subspaces of many different dimensions (points, lines, planes, ldots). However, the present knowledge is fragmentary, and quite unsatisfactory, not yet allowing a global picture about the arising arrangements of subspaces.

-Singularity theory and Cohen-Macaulay representations

The theory of Cohen-Macaulay representations of isolated singularities is controlled by Auslander-Reiten theory cite{Aus78,Yos90}. Among others, by algebraic McKay correspondence due to Auslander cite{Aus86}, the stable category of Cohen-Macaulay modules over a simple surface singularity is equivalent to the 1-cluster category of the path algebra of a Dynkin quiver, that is the orbit category of the derived category by the action of the Auslander-Reiten translation. Recently, a similar type of equivalences was given in cite{AIR11,BH,TV10} between the stable category of Cohen-Macaulay

modules over a certain class of quotient singularity in dimension $d$ and the $(d-1)$-cluster category of a certain finite dimensional algebra by using the $d$-preprojective algebra.

As mentioned in the last section, a weighted projective space leads to a (graded) coordinate algebras $S$ that is determined by Brieskorn singularities or, more generally, to a complete intersections of Brieskorn singularities. Accordingly, $S$ is a graded hypersurface (resp. a Gorenstein) singularity. The proposed Workshop will thus focus on the singularity theory of hypersurface, respectively Gorenstein, singularities but not restrict to those coming from Brieskorn singularities. A general feature for them is that the natural habitat for their investigation is the study of associated Cohen-Macaulay representations, which in the situation, we focus upon, is built by the category of graded Cohen-Macaulay modules, carrying a natural exact structure, which is Frobenius and leads to the triangulated stable category of Cohen-Macaulay modules cite{Buchweitz:1986},cite{Orlov:2009}. This category, in turn, is in the center of many investigations, and related to the strange duality of Arnold, more generally to mirror symmetry and mathematical physics (matrix factorizations and Landau-Ginzburg models).

In particular, we are going to investigate situations where the stable categories are (fractionally) Calabi-Yau; an instance of this is given by the graded coordinate algebras of weighted projective spaces, given by a single Brieskorn singularity. Further, and then assuming the base field of complex numbers, we will look into recent work by Ebeling and Takahashi giving interpretations of the Milnor lattice for certain isolated singularity, equipped with the Seifert form, to be isomorphic to the Grothendieck group of the stable category of Cohen-Macaulay modules, equipped with the homological Euler form, giving a new grip to attack mirror symmetry from a new angle.

Finally, and complementing a fundamental theorem by Orlov cite{Orlov:2009}, relating the derived category $mathrm{coh},mathbb{X}$ and the singularity category of $S$ by perpendicular constructions, we want to investigate another relationship, sofar only established for triangle singularities cite{Kussin:Lenzing:Meltzer:2013adv}, to investigate conditions, allowing to interpret the category of (graded) Cohen-Macaulay modules directly as a full subcategory of the category of vector bundles on $mathbb{X}$. In particular, the search for tilting objects will then conjecturally lead to interesting classes of vector bundles, not identified sofar. In the spirit of cite{Lenzing:Pena:2008}, we will also investigate the K-theoretic implications.

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Cambridge University Press, Cambridge, 1990.}

Classical preprojective algebras, introduced by Gelfand-Ponomarev cite{Gelfand:Ponomarev:1979}, are a well-studied topic of the representation theory of finite quivers or, in different terminology, of finite dimensional hereditary algebras. If $H$ is such an algebra, supposed to be of infinite representation type, the associated preprojective algebra $Pi(H)$ occurs in two differently looking, but isomorphic, incarnations, see cite{Baer:Geigle:Lenzing:1987}. The first one owes its existence to Auslander-Reiten theory: The orbit $(tau^{-n}H)$, $ngeq0$, of the inverse Auslander-Reiten translation, has the category of all preprojective $H$-modules as its additive closure, hence a well-understood category of combinatorial flavor. By setting $R=bigoplus_{ngeq0}R_n$ with $R_n=mathrm{Hom}(H,tau^{-n}H)$, we obtain the graded preprojective algebra as an orbit algebra. The second approach is homological in nature: If $D$ denotes the standard vector space duality, one first forms the $(H,H)$-bimodule $mathrm{Ext}^1(DH,H)$ describing the homological interaction between projective and injective $H$-modules, and then obtains the tensor algebra $mathbb{T}({mathrm{Ext}^1(DH,H)})$ as a natural algebraic object which encodes this kind of homological interplay.

A well-established approach to study the graded preprojective algebra, is by treating it as a cone singularity, that is, by studying its graded projective spectrum. This is done by applying Serre's construction cite{Serre:1955}, which associates to the setting an abelian category, to be interpreted as a category of coherent sheaves on a space $mathbb{X}$ incorporating the geometry of the problem. If $H$ is tame, that is, if $H$ is the path algebra of an extended Dynkin quiver, then this category $mathrm{coh},mathbb{X}$ is known to be equivalent to the category of coherent sheaves on a weighted projective line $mathbb{X}(a,b,c)$, whose weight type is given by a Dynkin triple $(a,b,c)$ cite{Geigle:Lenzing:1987},cite{Baer:Geigle:Lenzing:1987}. If $H$ is wild, the space $mathbb{X}$ is still mysterious. But even then, actually in both cases, the category $mathrm{coh},mathbb{X}$ is well understood, since it is derived equivalent to the category $mod{H}$ of finite dimensional $H$-modules cite{Lenzing:1986}, cite{Minamoto:2008}, cite{Minamoto:2012}.

One major attempt at this workshop will be to obtain a better understanding of the so-called emph{higher preprojective algebras}, see cite{Iyama:Oppermann:2013,Kel11}, by choosing a similar geometric approach. Simplifying, such a $d$-preprojective algebra $Pi^{(d)}(A)$ can be attached to any finite dimensional algebra $A$ of finite global dimension $d$, just by forming the tensor algebra of the $(A,A)$-bimodule $mathrm{Ext}^d(DA,A)$. Moreover, in good cases, $d$ also has a geometric interpretation of a (possibly non-commutative) smooth projective variety. This amounts to putting severe additional restrictions on $A$, because the ring-theoretic properties of $Pi^{(d)}(A)$ are usually not good enough to support a successful geometric analysis. In the framework of higher Auslander-Reiten theory, a group of researchers [HIMO] (Herschend-Iyama-Minamoto-Oppermann) is investigating the setting. By now, promising preliminary results have been obtained cite{Oppermann:2013}, cite{Iyama:2013}. Specializing to socalled $d$-emph{representation-infinite} algebras, one obtains noetherian algebras with the potential to undertake a geometric analysis via Serre construction. Up to now, however, only few cases have lead to a fairly complete analysis. As these partial results indicate, there is a good chance to identify more of the obtained spaces to be weighted projective spaces or, in a different setup, as orbifolds, a class of geometric objects typically arising as quotients of well-behaved geometrical group actions.

This explains current interest in this new kind of spaces, where currently two groups [HIMO] and [KLM] (Kussin-Lenzing-Meltzer), see cite{Kussin:Lenzing:Meltzer:2013adv}, cite{Lenzing:2013}, cite{Lenzing:2013a}, in particular two of the organizers, are involved.

2. Weighted projective spaces

Weighted projective varieties, containing the class of weighted projective spaces, apparently made their first appearance as Satake's V-spaces cite{Satake:1956} already in 1956.

Still at present, there is no generally accepted concept for the `right definition' of a weighted projective space. Correspondingly, the present knowledge on such spaces is very fragmentary. Only slight exaggerating, it is true that there is a chaos of competing and incompatible definitions, and a corresponding confusion concerns the results. In the setting of algebraic geometry, weighted projective spaces were introduced and studied by I. Dolgachev cite{Dolgachev:1982} and Beltrametti-Robbiano cite{Beltrametti:Robbiano:1986}. These spaces, let's call them having type (I), arise by allowing a grading for the projective coordinate algebra, an appropriate polynomial algebra, where the generators get arbitrary positive degrees. The resulting (classical) projective spectrum then, typically, is a singular space. More recently, it gets customary in this situation, alternatively, to form the attached category of coherent sheaves by means of the associated emph{graded} projective spectrum, equivalently by Serre construction cite{Auroux:Katzarkov:Orlov:2008}. The advantage of such an approach is to obtain a non-singular or smooth weighted projective space $mathbb{X}$ whose category $mathrm{coh},mathbb{X}$ of coherent sheaves thus has finite global dimension. We will say, that these spaces have type (II).

A similar approach also yielding smooth weighted projective spaces was already established in the late 1980's by Geigle-Lenzing cite{Geigle:Lenzing:1987} and D. Baer cite{Baer:1988} where the coordinate algebra of the `underlying' projective space $mathbb{P}^d$ is suitably graded by a finitely generated, rank one abelian group $mathbb{L}$ and then Serre's construction is applied to obtain the categories $mathrm{coh},mathbb{X}$. Let's call these spaces to have type (III). To complicate the situation, Geigle-Lenzing introduced the concept of a weighted projective line, resulting in a space $mathbb{X}$ with a finite number of `weighted' points. Weighted projective lines turn out to be an important tool in the representation theory of finite dimensional algebras, and by a theorem of Happel cite{Happel:2001} their categories of coherent sheaves $mathrm{coh},mathbb{X}$ exhaust together with the module categories $mod{H}$ over a finite quiver (without oriented cycles) --- up to derived equivalence --- the class of hereditary categories with a tilting object. Due to Happel's result, it is generally accepted, that the concept of weighted projective lines is the proper notion for dimension one. However, a weighted projective line is typically NOT a one-dimensional weighted projective space of type (I), (II) or (III). Let's therefore call the weighted projective lines to have type (iv).

Only very recently in ongoing research, weighted projective spaces of type (IV), where specialization to dimension one agrees with type (iv), were introduced independently by [HIMO] and [KLM]. Both groups are following complementary motivations. These spaces, called GL-spaces by [HIMO], will be said to have type (IV).

(i)The motivation of [KLM] is a study of the (suitably graded) singularity theory of Brieskorn singularities $f=x_1^{a_1}+x_2^{a_2}+cdots+x_t^{a_t}$. Following the well-established pattern of the study of triangle singularities (case $t=3$) cite{Kussin:Lenzing:Meltzer:2013adv}, one associates by Serre construction to a Brieskorn singularity, more generally, to a complete intersection of such ones, a non-commutative projective space $mathbb{X}$, that turns out to be a weighted projective space over $mathbb{P}^d$, $d=t-2$, where as in type (III) the weights (expressed by multiplicities of simple sheaves) are concentrated in $t$ hyperplanes of $mathbb{P}^d$.

(ii) The motivation for [HIMO] is to create a set-up to produce algebraic and geometric applications for a number of important concepts from the higher Auslander-theory, also developed by these researchers. Let us just mention the $d$-representation-finite and $d$-representation-infinite algebras and, very importantly, for such classes of finite dimensional algebras the $d$-preprojective algebras which are usually infinite dimensional. [HIMO] is, in particular, interested in the class of $d$-canonical algebras, defined to be the endomorphism algebra of a canonical tilting object $T$ on a GL-space. ($T$ is a direct sum of line bundles, naturally arising in the theory.) Of particular interest for this, are the $d$ canonical algebras $Lambda$, where the GL-space $mathbb{X}$ has emph{Fano type}, corresponding to positive Gorenstein parameter. Conjecturally, assuming Fano type, the algebra $Lambda$ is $d$-representation-infinite, implying a nice behavior of the associated $d$-preprojective algebra $Pi=Pi^{(d)}(Lambda)$. It is expected that the associated geometric object then is the weighted projective space $mathbb{X}$, again.

As a byproduct, the Workshop aims to clarify the relationship between weighted projective spaces of smooth type (II), (III) and (IV) by taking the respective weight loci, that is, the arrangements of subspaces (or subvarieties) endowed with weights as an ordering principle. For instance, it is easy to see that for types (III) and (IV) the weights are concentrated on hyperplanes; moreover type (III) is a subclass of type (IV) if we allow weights to be one. For type (II), a preliminary analysis shows that the weights may be concentrated in subspaces of many different dimensions (points, lines, planes, ldots). However, the present knowledge is fragmentary, and quite unsatisfactory, not yet allowing a global picture about the arising arrangements of subspaces.

-Singularity theory and Cohen-Macaulay representations

The theory of Cohen-Macaulay representations of isolated singularities is controlled by Auslander-Reiten theory cite{Aus78,Yos90}. Among others, by algebraic McKay correspondence due to Auslander cite{Aus86}, the stable category of Cohen-Macaulay modules over a simple surface singularity is equivalent to the 1-cluster category of the path algebra of a Dynkin quiver, that is the orbit category of the derived category by the action of the Auslander-Reiten translation. Recently, a similar type of equivalences was given in cite{AIR11,BH,TV10} between the stable category of Cohen-Macaulay

modules over a certain class of quotient singularity in dimension $d$ and the $(d-1)$-cluster category of a certain finite dimensional algebra by using the $d$-preprojective algebra.

As mentioned in the last section, a weighted projective space leads to a (graded) coordinate algebras $S$ that is determined by Brieskorn singularities or, more generally, to a complete intersections of Brieskorn singularities. Accordingly, $S$ is a graded hypersurface (resp. a Gorenstein) singularity. The proposed Workshop will thus focus on the singularity theory of hypersurface, respectively Gorenstein, singularities but not restrict to those coming from Brieskorn singularities. A general feature for them is that the natural habitat for their investigation is the study of associated Cohen-Macaulay representations, which in the situation, we focus upon, is built by the category of graded Cohen-Macaulay modules, carrying a natural exact structure, which is Frobenius and leads to the triangulated stable category of Cohen-Macaulay modules cite{Buchweitz:1986},cite{Orlov:2009}. This category, in turn, is in the center of many investigations, and related to the strange duality of Arnold, more generally to mirror symmetry and mathematical physics (matrix factorizations and Landau-Ginzburg models).

In particular, we are going to investigate situations where the stable categories are (fractionally) Calabi-Yau; an instance of this is given by the graded coordinate algebras of weighted projective spaces, given by a single Brieskorn singularity. Further, and then assuming the base field of complex numbers, we will look into recent work by Ebeling and Takahashi giving interpretations of the Milnor lattice for certain isolated singularity, equipped with the Seifert form, to be isomorphic to the Grothendieck group of the stable category of Cohen-Macaulay modules, equipped with the homological Euler form, giving a new grip to attack mirror symmetry from a new angle.

Finally, and complementing a fundamental theorem by Orlov cite{Orlov:2009}, relating the derived category $mathrm{coh},mathbb{X}$ and the singularity category of $S$ by perpendicular constructions, we want to investigate another relationship, sofar only established for triangle singularities cite{Kussin:Lenzing:Meltzer:2013adv}, to investigate conditions, allowing to interpret the category of (graded) Cohen-Macaulay modules directly as a full subcategory of the category of vector bundles on $mathbb{X}$. In particular, the search for tilting objects will then conjecturally lead to interesting classes of vector bundles, not identified sofar. In the spirit of cite{Lenzing:Pena:2008}, we will also investigate the K-theoretic implications.

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