Grassmannian categories of infinite rank joint with Jenny August, - PowerPoint PPT Presentation

Grassmannian categories of infinite rank joint with Jenny August, Man-Wai Cheung, Eleonore Faber and Sira Gratz Sibylle Schroll University of Leicester Dimers in Combinatorics and Cluster Algebras 3-14 August 2020

Grassmannian categories of infinite rank Idea: Categorify Grassmannian cluster algebras of infinite rank Fomin-Zelevinsky 2002: A cluster algebra A is a subalgebra of Z [ X ± 1 , . . . , X ± n ] ◮ generators: cluster variables � clusters ◮ A generated by mutation of clusters

Grassmannian cluster algebras of finite rank Grassmannian of k -subspaces of C n Gr( k , n ) Theorem (Scott 2006) C [ Gr ( k , n )] has the structure of a cluster algebra. C [ p I | I ⊂ { 1 , . . . , n } , | I | = k ] / I P where k � ( − 1) r p J ′ ∪{ j r } p J \{ j r } | I P = � r =0 J , J ′ ⊂ [ n ] , | J | = k + 1 , | J ′ | = k − 1 , J = { j 0 , . . . , j k }�

Grassmannian cluster algebras of finite rank Definition Let I , J be two k -subsets of Z . ◮ I and J are crossing if there are i 1 , i 2 ∈ I \ J and j 1 , j 2 ∈ J \ I such that i 1 < j 1 < i 2 < j 2 or j 1 < i 1 < j 2 < j 2 ◮ The Pl¨ ucker coordinates p I and p J are compatible if I and J are non-crossing.

Grassmannian cluster algebras of finite rank Theorem (Scott 2006) Maximal sets of compatible Pl¨ ucker coordinates are (examples of) clusters. Example k = 2 1 − 1 Pl¨ ucker coordinates ← → cluster variables 1 − 1 Pl¨ ucker relations ← → exchange formulas

Grassmannian cluster categories of finite rank Jensen-King-Su 2016: Categorification of Grassmannian cluster algebras of finite rank: R n = C [ x , y ] / ( x k − y n − k ) Set The group µ n = { ξ ∈ C | ξ n = 1 } < SL 2 ( C ) acts on C [ x , y ] by x �→ ξ x and y �→ ξ − 1 y MCM µ n ( R n ) := µ n -equivariant maximal Cohen-Macaulay R n -modules

Grassmannian cluster categories of finite rank Theorem ( Jensen-King-Su 2016) MCM µ n ( R n ) is a Frobenius category and • rank 1 modules 1 − 1 ← → Pl¨ ucker coordinates M I ← → p I • Ext 1 ( M I , M J ) = 0 ⇐ ⇒ p I and p J are compatible • Maximal sets of compatible Pl¨ ucker coordinates correspond to cluster-tilting subcategories • define a cluster character (using the categorification of the affine open cell via the pre-projective algebra by Geiss-Leclerc-Schr¨ oer)

Grassmannian cluster algebras of infinite rank Set A k = C [ p I | I ⊂ Z , | I | = k ] / I P Theorem (Grabowski-Gratz 2014) A k can be endowed with the structure of an infinite rank cluster algebra in uncountably many ways. Theorem (Gratz 2015) A k is the colimt of cluster algebras of finite rank in the category of rooted cluster algebras. Theorem (Groechening 2014) Construction of A k as coordinate ring of an infinite rank Grassmannian � k = 2: A k is the homogeneous coordinate ring of an ’infinite’ Grassmannian, the 2-dimensional subspaces of a profinite-dimensional vector space

Grassmannian categories of infinite rank Idea: n → ∞ in Gr( k , n ) and x k − y n − k � Gr( k , ∞ ) and R := C [ x , y ] / x k G m = C ∗ acts on C [ x , y ] by x �→ ξ x and y �→ ξ − 1 y for ξ ∈ G m MCM G m R := G m -equivariant maximal Cohen-Macaulay modules Since Hom( G m , C ) ≃ Z , we have mod G m R ≃ gr R � MCM G m R ≃ MCM Z R The category of Z -graded maximal Cohen-Macaulay R -modules is a Grassmannian category of infinite rank.

Grassmannian categories of infinite rank MCM Z R is a Frobenius category Theorem (Buchweitz 1986) MCM Z R ≃ D sg ( gr R ) k = 2: Holm-Jørgensen 2012: The derived category with finite cohomology D f dg ( C [ y ]) of the differential graded algebra C [ y ] with deg ( y ) = − 1 has cluster combinatorics of type A . Remark: Set C = � generically free rank 1 MCM Z C [ x , y ] / x 2 modules � . Then C ≃ D f dg ( C [ y ]) . Yildirm-Paquette 2020: Completion of discrete cluster categories of infinite type by Igusa-Todorov (2015). � for k = 2 and with 1 accumulation point : Yildirim-Paquette completion ≃ MCM Z C [ x , y ] / x 2

Generically free modules Set F = C [ x , y ] / x k total ring of fractions Definition A module M in MCM Z R is generically free of rank n if M ⊗ R F is a graded free F -modules of rank n . Proposition 1. If M ∈ MCM Z R is generically free then M = Ω( N ) for some finite dimensional N ∈ gr R. 2. M ∈ MCM Z R is generically free of rank 1 ⇐ ⇒ M is a graded ideal of R and y n ∈ M for some n > 0 . 3. Every homogeneous ideal I of R can be generated by monomials.

Generically free rank 1 modules Theorem (August-Cheung-Faber-Gratz-S. 2020) A module M in MCM Z R is generically free of rank 1 ⇒ M = ( x k − 1 , x k − 2 y i 1 , x k − 3 y i 2 , . . . , xy i k − 2 , y i k − 1 )( i k ) ⇐ with 0 ≤ i 1 ≤ i 2 ≤ · · · ≤ i k − 1 and i k ∈ Z . Figure: Schematical view of a rank 1 module. Definition Define the strictly non-decreasing degree sequence to be ℓ I := ( − i k − 1 − i k , − i k − 2 − i k + 1 , . . . , − i k + k − 1)

Generically free rank 1 modules and Pl¨ ucker coordinates Corollary � generically free rank 1 � Pl¨ � � ucker coordinates 1 − 1 ← → modules in MCM Z R in A k I �− → p ℓ I I ( ℓ ) ← − � ℓ = ( ℓ 1 , . . . , ℓ k ) where I ( ℓ ) = ( x k − 1 , x k − 2 y i 1 , x k − 3 y i 2 , . . . , xy i k − 2 , y i k − 1 )( i k ) with i k = k − 1 − ℓ k and i k − r = ℓ k − ℓ r − k + r for 1 ≤ r ≤ k − 1 . Remark: This bijection is structure preserving.

Rigidity and compatibility Theorem (August-Cheung-Faber-Gratz-S. 2020) Let I , J ∈ MCM Z R generically free of rank 1. Then Ext 1 ( I , J ) = 0 ⇐ ⇒ p ℓ I and p ℓ J are compatible Ext 1 ( J , I ) = 0 ⇐ ⇒ Corollary Generically free rank 1 modules in MCM Z R are rigid.

Idea of Proof: I generically free MCM Z R module. The matrix factorisation of I → R k − M N R k → R k − − → I − → 0 gives a graded projective presentation of I . Apply graded Hom( − , J ) noting that Hom( R ( m ) , J ) = J ( − m ) J N ⊤ → J (1) M ⊤ − − → J ( k ) � where J is a direct sum of appropriately shifted copies of J . Ker M ⊤ / Im N ⊤ � 0 = Ker( M ⊤ ) 0 / Im( N ⊤ ) 0 Ext 1 ( I , J ) = � �

Dimension formula dim Ext 1 ( I , J ) = dim Ker( M ⊤ ) 0 − dim Im( N ⊤ ) 0 dim Ker( M ⊤ ) 0 = dim J (1) 0 − dim Im( M ⊤ ) 0 dim Im( N ⊤ ) 0 = dim J 0 − dim Ker( N ⊤ ) 0 We then show dim J 0 − dim J (1) 0 = | ℓ I ∩ ℓ J | dim Im( M ⊤ ) 0 = k − β ( ℓ I , ℓ J ) dim Ker( N ⊤ ) 0 = α ( ℓ I , ℓ J ) Theorem (August-Cheung-Faber-Gratz-S. 2020) dim Ext 1 ( I , J ) = α ( ℓ I , ℓ J ) + β ( ℓ I , ℓ J ) − k − | ℓ I ∩ ℓ J | = dim Ext 1 ( J , I )

New combinatorial tool: staircase paths Example of calculation of dim Ext 1 ( I , J ): k = 3 : ℓ I = ( − 5 , 1 , 3) = ( ℓ 1 , ℓ 2 , ℓ 3 ) = ℓ ℓ J = (0 , 1 , 4) = ( m 1 , m 2 , m 3 ) = m α ( ℓ, m ) = # diagonals strictly above A ( ℓ, m ) = 3 β ( ℓ, m ) = # diagonals strictly below B ( ℓ, m ) = 2 ⇒ dim Ext 1 ( I , J ) = 3 + 2 − 3 − 1 = 1 | ℓ ∩ m | = 1 = with I = ( x 2 , xy , y 6 )( − 1) and J = ( x 2 , xy 2 , y 2 )( − 2) in MCM Z R .

k=2 Proposition The indecomposable MCM Z ( C [ x , y ] / x 2 ) modules correspond to • ( x , y k )( − ℓ ) • C [ y ]( − ℓ ) Two arcs γ , δ corresponding to I ( γ ) , I ( δ ) ∈ MCM Z ( C [ x , y ] / x 2 ) dim Ext 1 ( I ( α ) , I ( β )) = 1 ⇐ ⇒ γ and δ cross (possibly at ∞ ). dim Ext 1 ( I ( α ) , I ( β )) = 0 ⇐ ⇒ γ and δ do not crossing.

Cluster tilting subcategories We can completely describe the Hom-spaces between indecomposables Theorem (August-Cheung-Faber-Gratz-S. 2020) MCM Z ( C [ x , y ] / x 2 ) has cluster tilting subcategories and they are of the form