Kn orrers periodicity for skew quadric hypersurfaces . Kenta - PowerPoint PPT Presentation

Introduction Results . Kn¨ orrer’s periodicity for skew quadric hypersurfaces . Kenta Ueyama and Izuru Mori Hirosaki University and Shizuoka University The 8th CJK International Symposium on Ring Theory Nagoya August 27 2019 Kenta Ueyama (Hirosaki) and Izuru Mori (Shizuoka) Kn¨ orrer’s periodicity for skew quadric hypersurfaces

Introduction Results k : an algebraically closed field of characteristic not 2. . Theorem 1 (Kn¨ orrer’s periodicity theorem) . deg x i ∈ N + , S = k [ x 1 , . . . , x n ] 0 ̸ = f ∈ S 2 e (homog. polynomial of even degree 2 e). Then = CM Z ( S [ u , v ] / ( f + u 2 + v 2 )) CM Z ( S / ( f )) ∼ where deg u = deg v = e. . . Theorem 2 . f = x 2 1 + · · · + x 2 S = k [ x 1 , . . . , x n ] deg x i = 1 , n ∈ S 2 . (1) If n is odd, then CM Z ( S / ( f )) ∼ 1 )) ∼ = CM Z ( k [ x 1 ] / ( x 2 = D b (mod k ) . (2) If n is even, then CM Z ( S / ( f )) ∼ = CM Z ( k [ x 1 , x 2 ] / ( x 2 1 + x 2 2 )) ∼ = D b (mod k 2 ) . . Kenta Ueyama (Hirosaki) and Izuru Mori (Shizuoka) Kn¨ orrer’s periodicity for skew quadric hypersurfaces

Introduction Results In this talk, we study a “skew version” of Theorem 2. . Setting . For ε := ( ε ij ) ∈ M n ( k ) s.t. ε ii = 1 and ε ij = ε ji = ± 1 , we fix the following notation: S ε := k ⟨ x 1 , . . . , x n ⟩ / ( x i x j − ε ij x j x i ) deg x i = 1 (( ± 1)-skew polynomial algebra generated in degree 1). f ε := x 2 1 + · · · + x 2 n ∈ S ε (cental element). A ε := S ε / ( f ε ). CM Z ( A ε ) := { M ∈ mod Z A ε | Ext i A ε ( M , A ε ) = 0 ( i > 0) } (the category of graded MCM modules). CM Z ( A ε ): stable category of CM Z ( A ε ) (triang. cat.). . . Aim . To study CM Z ( A ε ). . Kenta Ueyama (Hirosaki) and Izuru Mori (Shizuoka) Kn¨ orrer’s periodicity for skew quadric hypersurfaces

Introduction Results . Example S ε = k ⟨ x 1 , x 2 , x 3 ⟩ / ( x 1 x 2 + x 2 x 1 , x 1 x 3 + x 3 x 1 , x 2 x 3 + x 3 x 2 ) ( ε 12 = ε 13 = ε 23 = − 1 ) f ε = x 2 1 + x 2 2 + x 2 3 . Then we have f ε =( x 1 + x 2 + x 3 )( x 1 + x 2 + x 3 ) = ( x 1 − x 2 + x 3 )( x 1 − x 2 + x 3 ) =( x 1 + x 2 − x 3 )( x 1 + x 2 − x 3 ) = ( x 1 − x 2 − x 3 )( x 1 − x 2 − x 3 ) in S ε (matrix factorizations of f ε of rank 1). M 1 = A ε / ( x 1 + x 2 + x 3 ) A ε , M 2 = A ε / ( x 1 − x 2 + x 3 ) A ε M 3 = A ε / ( x 1 + x 2 − x 3 ) A ε , M 4 = A ε / ( x 1 − x 2 − x 3 ) A ε are non-isomorphic MCM modules over A ε (= S ε / ( f ε )) ． In fact, CM Z ( A ε ) ∼ = D b (mod k 4 ) . Kenta Ueyama (Hirosaki) and Izuru Mori (Shizuoka) Kn¨ orrer’s periodicity for skew quadric hypersurfaces

Introduction Results . Graphical methods for computation of CM Z ( A ε ) . Definition 3 . For ε := ( ε ij ) ∈ M n ( k ) s.t. ε ii = 1 and ε ij = ε ji = ± 1, we define the graph G ε by (vertices) V ( G ε ) := { 1 , 2 , . . . , n } (edges) E ( G ε ) := { ( i , j ) | ε ij = ε ji = 1 } . . Example . ( n = 4) ε 12 = ε 13 = ε 14 = +1 ε 23 = ε 24 = ε 34 = − 1 Then 1 G ε = 2 4 3 . Kenta Ueyama (Hirosaki) and Izuru Mori (Shizuoka) Kn¨ orrer’s periodicity for skew quadric hypersurfaces

Introduction Results . Two Mutations . Definition 4 . G : a simple graph, v ∈ V ( G ). def µ v ( G ) : the mutation of G at v ⇐ ⇒ µ v ( G ) is the graph such that V ( µ v ( G )) := V ( G ) and for u ̸ = v , ( v , u ) ∈ E ( µ v ( G )) : ⇔ ( v , u ) ̸∈ E ( G ), for u , u ′ ̸ = v , ( u , u ′ ) ∈ E ( µ v ( G )) : ⇔ ( u , u ′ ) ∈ E ( G ). . . Example . 1 1 G = = ⇒ µ 2 ( G ) = 2 4 2 4 3 3 . Kenta Ueyama (Hirosaki) and Izuru Mori (Shizuoka) Kn¨ orrer’s periodicity for skew quadric hypersurfaces

Introduction Results . Definition 5 . G : a simple graph, v , w ∈ V ( G ). def µ v ← w ( G ) : the relative mutation of G at v by w ⇐ ⇒ µ v ← w ( G ) is the graph such that V ( µ v ( G )) := V ( G ) and for u ̸ = v , w , ( v , u ) ∈ E ( µ v ← w ( G )) : ⇔ ( v , u ) ∈ E ( G ) , ( w , u ) ̸∈ E ( G ) or ( v , u ) ̸∈ E ( G ) , ( w , u ) ∈ E ( G ) , ( v , w ) ∈ E ( µ v ← w ( G )) : ⇔ ( v , w ) ∈ E ( G ), for u , u ′ ̸ = v , ( u , u ′ ) ∈ E ( µ v ← w ( G )) : ⇔ ( u , u ′ ) ∈ E ( G ). . . Example . 1 1 2 6 2 6 G = = ⇒ µ 6 ← 5 ( G ) = 3 5 3 5 4 4 . Kenta Ueyama (Hirosaki) and Izuru Mori (Shizuoka) Kn¨ orrer’s periodicity for skew quadric hypersurfaces

Introduction Results . Theorem 6 (Mutation [MU]) . If G ε ′ = µ v ( G ε ) , then CM Z ( A ε ) ∼ = CM Z ( A ε ′ ) . . . Theorem 7 (Relative Mutation [MU]) . Assume that G ε has an isolated vertex u. If G ε ′ = µ v ← w ( G ε ) ( v , w ̸ = u ) , then CM Z ( A ε ) ∼ = CM Z ( A ε ′ ) . . Kenta Ueyama (Hirosaki) and Izuru Mori (Shizuoka) Kn¨ orrer’s periodicity for skew quadric hypersurfaces

Introduction Results . Two Reductions . Theorem 8 (Kn¨ orrer Reduction [MU]) . Assume that G ε has an isolated segment [ v , w ] . If G ε ′ = G ε \ [ v , w ] , then CM Z ( A ε ) ∼ = CM Z ( A ε ′ ) . . . Example . 1 1 2 6 G ε = = ⇒ G ε \ [5 , 6] = 2 4 3 5 4 3 . . Remark 9 . Kn¨ orrer reduction is a consequence of noncommutative Kn¨ orrer’s periodicity theorem presented in Mori’s talk. . Kenta Ueyama (Hirosaki) and Izuru Mori (Shizuoka) Kn¨ orrer’s periodicity for skew quadric hypersurfaces

Introduction Results . Theorem 10 (Two Points Reduction [MU]) . Assume that G ε has two distinct isolated vertices v , w. If G ε ′ = G ε \ { v } , then CM Z ( A ε ) ∼ = CM Z ( A ε ′ ) × 2 . . . Theorem 11 ([MU]) . By using mutation, relative mutation, Kn¨ orrer reduction, and two points reduction, we can completely compute CM Z ( A ε ) up to n = 6 . . This result suggests that these methods are powerful! I plan to generalize for any n in future work. Kenta Ueyama (Hirosaki) and Izuru Mori (Shizuoka) Kn¨ orrer’s periodicity for skew quadric hypersurfaces

Introduction Results . Demonstration ( n = 6) S ε = k ⟨ x 1 , . . . , x 6 ⟩ / ( x i x j − ε ij x j x i ) where ε 12 = ε 14 = ε 23 = ε 25 = ε 35 = ε 36 = ε 46 = ε 56 = +1 ε 13 = ε 15 = ε 16 = ε 24 = ε 26 = ε 34 = ε 45 = − 1 f ε = x 2 1 + · · · + x 2 6 ∈ S ε A ε = S ε / ( f ε ) Then 1 2 6 G ε = 3 5 4 We can transform G ε to a disjoint union of two isolated segments and two isolated vertices by applying mutation and relative mutation several times. Hence we have . = CM Z ( k [ x ] / ( x 2 )) × 2 ∼ = D b (mod k ) × 2 ∼ . CM Z ( A ε ) ∼ = D b (mod k 2 ) . Kenta Ueyama (Hirosaki) and Izuru Mori (Shizuoka) Kn¨ orrer’s periodicity for skew quadric hypersurfaces

Introduction Results ε ij ε jh ε hi = − 1 V ( x i x j x h ) ⊂ P n − 1 (point scheme of S ε ) E ε := ∩ . Corollary 12 ([MU]) . Let ℓ be the number of irreducible components of E ε that are isomorphic to P 1 . Assume that n ≤ 6 . (1) If n is odd, then ℓ ≤ 10 and ⇒ CM Z ( A ε ) ∼ = D b (mod k ) , ℓ = 0 ⇐ ⇒ CM Z ( A ε ) ∼ = D b (mod k 4 ) , 0 < ℓ ≤ 3 ⇐ ⇒ CM Z ( A ε ) ∼ = D b (mod k 16 ) . 3 < ℓ ≤ 10 ⇐ (2) If n is even, then ℓ ≤ 15 and ⇒ CM Z ( A ε ) ∼ = D b (mod k 2 ) , 0 ≤ ℓ ≤ 1 ⇐ ⇒ CM Z ( A ε ) ∼ = D b (mod k 8 ) , 1 < ℓ ≤ 6 ⇐ ⇒ CM Z ( A ε ) ∼ = D b (mod k 32 ) . 6 < ℓ ≤ 15 ⇐ . Note that this corollary does not hold in the case n = 7. Kenta Ueyama (Hirosaki) and Izuru Mori (Shizuoka) Kn¨ orrer’s periodicity for skew quadric hypersurfaces