Trace Modules and Rigidity Haydee Lindo Williams College @CGMRT, - PowerPoint PPT Presentation

Trace Modules and Rigidity Haydee Lindo Williams College @CGMRT, November 2017 Haydee Lindo (Williams) Trace Ideals @CGMRT, November 2017 1 / 17

Notation In what follows: R is a local commutative Noetherian ring M , X are finitely generated R -modules Hom R ( M , X ) denotes the set of R -linear homomorphisms from M to X Haydee Lindo (Williams) Trace Ideals @CGMRT, November 2017 2 / 17

A History of Conjectures Conjecture (Generalized Nakayama Conjecture, 1975) Let Λ be an Artin Algebra. Then any indecomposable injective Λ -module appears as a direct summand in the minimal injective resolution of Λ . Haydee Lindo (Williams) Trace Ideals @CGMRT, November 2017 3 / 17

A History of Conjectures Conjecture (Generalized Nakayama Conjecture, 1975) Let Λ be an Artin Algebra. Then any indecomposable injective Λ -module appears as a direct summand in the minimal injective resolution of Λ . Equivalently: Conjecture (Auslander-Reiten Conjecture (ARC), 1975) Let Λ be an Artin Algebra and M a finitely generated Λ -module If Ext i Λ ( M , M ) = 0 = Ext i Λ ( M , Λ) for all i > 0 then M is projective Haydee Lindo (Williams) Trace Ideals @CGMRT, November 2017 3 / 17

ARC in Commutative Algebra Conjecture (ARC, 1993) Let R be a commutative Noetherian ring and M a fin. gen. R-module. If Ext i R ( M , M ) = 0 = Ext i R ( M , R ) , for all i > 0 then M is projective Haydee Lindo (Williams) Trace Ideals @CGMRT, November 2017 4 / 17

ARC in Commutative Algebra Conjecture (ARC, 1993) Let R be a commutative Noetherian ring and M a fin. gen. R-module. If Ext i R ( M , M ) = 0 = Ext i R ( M , R ) , for all i > 0 then M is projective Proved for : Complete Intersection rings (Auslander, Reiten, Solberg - 1993) Haydee Lindo (Williams) Trace Ideals @CGMRT, November 2017 4 / 17

ARC in Commutative Algebra Conjecture (ARC, 1993) Let R be a commutative Noetherian ring and M a fin. gen. R-module. If Ext i R ( M , M ) = 0 = Ext i R ( M , R ) , for all i > 0 then M is projective Proved for : Complete Intersection rings (Auslander, Reiten, Solberg - 1993) Gorenstein Normal domains (Huneke, Leuschke- 2004) Haydee Lindo (Williams) Trace Ideals @CGMRT, November 2017 4 / 17

ARC in Commutative Algebra Conjecture (ARC, 1993) Let R be a commutative Noetherian ring and M a fin. gen. R-module. If Ext i R ( M , M ) = 0 = Ext i R ( M , R ) , for all i > 0 then M is projective Proved for : Complete Intersection rings (Auslander, Reiten, Solberg - 1993) Gorenstein Normal domains (Huneke, Leuschke- 2004) Gorenstein rings if ARC holds in codimension 1 (Araya - 2009) Haydee Lindo (Williams) Trace Ideals @CGMRT, November 2017 4 / 17

ARC in Commutative Algebra Conjecture (ARC, 1993) Let R be a commutative Noetherian ring and M a fin. gen. R-module. If Ext i R ( M , M ) = 0 = Ext i R ( M , R ) , for all i > 0 then M is projective Proved for : Complete Intersection rings (Auslander, Reiten, Solberg - 1993) Gorenstein Normal domains (Huneke, Leuschke- 2004) Gorenstein rings if ARC holds in codimension 1 (Araya - 2009) Haydee Lindo (Williams) Trace Ideals @CGMRT, November 2017 4 / 17

Focus: Rigid Modules Definition We call M n -rigid if Ext i R ( M , M ) = 0 for all 1 � i � n . We call M rigid if Ext 1 R ( M , M ) = 0. Haydee Lindo (Williams) Trace Ideals @CGMRT, November 2017 5 / 17

Example: Rigid Module Let R = k [[ x , y ]] / ( xy ) and M = R / ( x ) . Haydee Lindo (Williams) Trace Ideals @CGMRT, November 2017 6 / 17

Example: Rigid Module Let R = k [[ x , y ]] / ( xy ) and M = R / ( x ) . y x x · · · − → R − → R − → R − → 0 . Haydee Lindo (Williams) Trace Ideals @CGMRT, November 2017 6 / 17

Example: Rigid Module Let R = k [[ x , y ]] / ( xy ) and M = R / ( x ) . y x x · · · − → R − → R − → R − → 0 . Apply Hom R ( − , R / ( x )) : y x � Hom R ( R , R / ( x )) � Hom R ( R , R / ( x )) � Hom R ( R , R / ( x )) 0 Haydee Lindo (Williams) Trace Ideals @CGMRT, November 2017 6 / 17

� � � Example: Rigid Module Let R = k [[ x , y ]] / ( xy ) and M = R / ( x ) . y x x · · · − → R − → R − → R − → 0 . Apply Hom R ( − , R / ( x )) : y x � Hom R ( R , R / ( x )) � Hom R ( R , R / ( x )) Hom R ( R , R / ( x )) 0 ∼ ∼ = = y � R / ( x ) R / ( x ) Haydee Lindo (Williams) Trace Ideals @CGMRT, November 2017 6 / 17

� � � Example: Rigid Module Let R = k [[ x , y ]] / ( xy ) and M = R / ( x ) . y x x · · · − → R − → R − → R − → 0 . Apply Hom R ( − , R / ( x )) : y x � Hom R ( R , R / ( x )) � Hom R ( R , R / ( x )) Hom R ( R , R / ( x )) 0 ∼ ∼ = = y � R / ( x ) R / ( x ) Mult by y is injective on R / ( x ) , so Ext 1 R ( R / ( x ) , R / ( x )) = 0 . Haydee Lindo (Williams) Trace Ideals @CGMRT, November 2017 6 / 17

Some Behaviour of Rigid Modules Haydee Lindo (Williams) Trace Ideals @CGMRT, November 2017 7 / 17

Some Behaviour of Rigid Modules R a 0-dimensional local ring, M � X finitely generated R -modules. Haydee Lindo (Williams) Trace Ideals @CGMRT, November 2017 7 / 17

Some Behaviour of Rigid Modules R a 0-dimensional local ring, M � X finitely generated R -modules. π 0 − → M − → X → X / M − → 0 − Haydee Lindo (Williams) Trace Ideals @CGMRT, November 2017 7 / 17

� Some Behaviour of Rigid Modules R a 0-dimensional local ring, M � X finitely generated R -modules. π 0 − → M − → X → X / M − → 0 − If M is rigid, apply Hom R ( M , − ) � Hom R ( M , X ) π ∗ � Hom R ( M , X / M ) � Ext 1 R ( M , M ) Haydee Lindo (Williams) Trace Ideals @CGMRT, November 2017 7 / 17

� � Some Behaviour of Rigid Modules R a 0-dimensional local ring, M � X finitely generated R -modules. π 0 − → M − → X → X / M − → 0 − If M is rigid, apply Hom R ( M , − ) ✿ 0 π ∗ � = 0 ✘ ✘✘✘✘✘✘ � Hom R ( M , X ) � Hom R ( M , X / M ) Ext 1 R ( M , M ) � = 0 Haydee Lindo (Williams) Trace Ideals @CGMRT, November 2017 7 / 17

� � � Some Behaviour of Rigid Modules α � = 0 ¯ M X / M α π X Observe : ∃ α : M → X such that Im ( α ) �⊆ M Haydee Lindo (Williams) Trace Ideals @CGMRT, November 2017 8 / 17

Compare Trace Modules For R -modules M and X Definition The trace (module) of M in X is � τ M ( X ) := α ( M ) α ∈ Hom R ( M , X ) We call τ M ( R ) the trace ideal of M . Haydee Lindo (Williams) Trace Ideals @CGMRT, November 2017 9 / 17

Compare Trace Modules For R -modules M and X Definition The trace (module) of M in X is � τ M ( X ) := α ( M ) α ∈ Hom R ( M , X ) We call τ M ( R ) the trace ideal of M . Note: τ M ( M ) = M and τ R ( M ) = M . Haydee Lindo (Williams) Trace Ideals @CGMRT, November 2017 9 / 17

Compare Trace Modules For R -modules M and X Definition The trace (module) of M in X is � τ M ( X ) := α ( M ) α ∈ Hom R ( M , X ) We call τ M ( R ) the trace ideal of M . Note: τ M ( M ) = M and τ R ( M ) = M . Focus : proper trace modules M = τ M ( X ) � X . Haydee Lindo (Williams) Trace Ideals @CGMRT, November 2017 9 / 17

Example: Trace Ideals Field k , R = k [ x , y , z ] / ( y 2 − xz , x 2 y − z 2 , x 3 − yz ) with I = ( x , y ) Haydee Lindo (Williams) Trace Ideals @CGMRT, November 2017 10 / 17

Example: Trace Ideals Field k , R = k [ x , y , z ] / ( y 2 − xz , x 2 y − z 2 , x 3 − yz ) with I = ( x , y ) � − z x 2 − z 2 � y τ ( x , y ) ( R ) = I 1 ( left ker of pres matrix of I : ) y − x − z yz Haydee Lindo (Williams) Trace Ideals @CGMRT, November 2017 10 / 17

Example: Trace Ideals Field k , R = k [ x , y , z ] / ( y 2 − xz , x 2 y − z 2 , x 3 − yz ) with I = ( x , y ) � − z x 2 − z 2 � y τ ( x , y ) ( R ) = I 1 ( left ker of pres matrix of I : ) y − x − z yz     y z = I 1 x y     x 2 z Haydee Lindo (Williams) Trace Ideals @CGMRT, November 2017 10 / 17

Example: Trace Ideals Field k , R = k [ x , y , z ] / ( y 2 − xz , x 2 y − z 2 , x 3 − yz ) with I = ( x , y ) � − z x 2 − z 2 � y τ ( x , y ) ( R ) = I 1 ( left ker of pres matrix of I : ) y − x − z yz     y z = I 1 x y     x 2 z = ( x , y , z ) Haydee Lindo (Williams) Trace Ideals @CGMRT, November 2017 10 / 17

Example: Trace Ideals Field k , R = k [ x , y , z ] / ( y 2 − xz , x 2 y − z 2 , x 3 − yz ) with I = ( x , y ) � − z x 2 − z 2 � y τ ( x , y ) ( R ) = I 1 ( left ker of pres matrix of I : ) y − x − z yz     y z = I 1 x y     x 2 z = ( x , y , z ) Classes of Examples: (1) All ideals of grade � 2 (2) All ideals when R is Artinian Gorenstein Haydee Lindo (Williams) Trace Ideals @CGMRT, November 2017 10 / 17

Unique Behaviour of Trace Modules � X α τ M ( X ) Haydee Lindo (Williams) Trace Ideals @CGMRT, November 2017 11 / 17

Unique Behaviour of Trace Modules � � τ M ( X ) � X M n α Haydee Lindo (Williams) Trace Ideals @CGMRT, November 2017 11 / 17

� Unique Behaviour of Trace Modules � X M n α � � τ M ( X ) ∈ ⊕ n i = 1 Hom R ( M , X ) Haydee Lindo (Williams) Trace Ideals @CGMRT, November 2017 11 / 17