Prime monomial ideals of subsemigroup algebras of free nilpotent - PowerPoint PPT Presentation

. . . . . . . . . . . . . . . Prime monomial ideals of subsemigroup algebras of free nilpotent groups Tomer Bauer joint work with Be’eri Greenfeld Department of Mathematics Bar-Ilan University Groups, Rings and Associated Structures 2019 T. Bauer (BIU) . . . . . . . . . . . . . . . . . . . . . . . . . Prime ideals of semigroup algebras of free nilpotent groups

. . . . . . . . . . . . . . . . . Background: Group algebras Problem T. Bauer (BIU) . . . . . . . . . . . . . . . . . . Prime ideals of semigroup algebras of free nilpotent groups . . . . . Let F be a field and G be a finitely generated nilpotent group. Study the prime spectrum of F [ G ] . F [ G ] is Noetherian with a finite Gelfand–Kirillov dimension. The classical Krull dimension of F [ G ] equals the Hirsch length of F [ G ] . What about semigroup algebras of subsemigroups of G ?

. . . . . . . . . . . . . . . . . Background: Group algebras Problem T. Bauer (BIU) . . . . . . . . . . . . . . . . . . Prime ideals of semigroup algebras of free nilpotent groups . . . . . Let F be a field and G be a finitely generated nilpotent group. Study the prime spectrum of F [ G ] . F [ G ] is Noetherian with a finite Gelfand–Kirillov dimension. The classical Krull dimension of F [ G ] equals the Hirsch length of F [ G ] . What about semigroup algebras of subsemigroups of G ?

. . . . . . . . . . . . . . . . . Background: Group algebras Problem T. Bauer (BIU) . . . . . . . . . . . . . . . . . . Prime ideals of semigroup algebras of free nilpotent groups . . . . . Let F be a field and G be a finitely generated nilpotent group. Study the prime spectrum of F [ G ] . F [ G ] is Noetherian with a finite Gelfand–Kirillov dimension. The classical Krull dimension of F [ G ] equals the Hirsch length of F [ G ] . What about semigroup algebras of subsemigroups of G ?

. . . . . . . . . . . . . . . . . Background: Group algebras Problem T. Bauer (BIU) . . . . . . . . . . . . . . . . . . Prime ideals of semigroup algebras of free nilpotent groups . . . . . Let F be a field and G be a finitely generated nilpotent group. Study the prime spectrum of F [ G ] . F [ G ] is Noetherian with a finite Gelfand–Kirillov dimension. The classical Krull dimension of F [ G ] equals the Hirsch length of F [ G ] . What about semigroup algebras of subsemigroups of G ?

. . . . . . . . . . . . . . . . . Background: Group algebras Problem T. Bauer (BIU) . . . . . . . . . . . . . . . . . . Prime ideals of semigroup algebras of free nilpotent groups . . . . . Let F be a field and G be a finitely generated nilpotent group. Study the prime spectrum of F [ G ] . F [ G ] is Noetherian with a finite Gelfand–Kirillov dimension. The classical Krull dimension of F [ G ] equals the Hirsch length of F [ G ] . What about semigroup algebras of subsemigroups of G ?

. . . . . . . . . . . . . . . . Semigroup algebras Problem necessarily Noetherian. There are semigroup algebras whose classical Krull dimension is not bounded by their Gelfand–Kirillov dimension. T. Bauer (BIU) . . . . . . . . . . . . . . . . . . . . Prime ideals of semigroup algebras of free nilpotent groups . . . . Let F be a field, G be a finitely generated nilpotent group, and S a finitely generated subsemigroup of G . Study the prime spectrum of F [ S ] . F [ S ] has a finite Gelfand–Kirillov dimension, but is not What can be said about the prime spectrum of F [ S ] from that of F [ G ] ?

. . . . . . . . . . . . . . . . Semigroup algebras Problem necessarily Noetherian. There are semigroup algebras whose classical Krull dimension is not bounded by their Gelfand–Kirillov dimension. T. Bauer (BIU) . . . . . . . . . . . . . . . . . . . . Prime ideals of semigroup algebras of free nilpotent groups . . . . Let F be a field, G be a finitely generated nilpotent group, and S a finitely generated subsemigroup of G . Study the prime spectrum of F [ S ] . F [ S ] has a finite Gelfand–Kirillov dimension, but is not What can be said about the prime spectrum of F [ S ] from that of F [ G ] ?

. . . . . . . . . . . . . . . . Semigroup algebras Problem necessarily Noetherian. There are semigroup algebras whose classical Krull dimension is not bounded by their Gelfand–Kirillov dimension. T. Bauer (BIU) . . . . . . . . . . . . . . . . . . . . Prime ideals of semigroup algebras of free nilpotent groups . . . . Let F be a field, G be a finitely generated nilpotent group, and S a finitely generated subsemigroup of G . Study the prime spectrum of F [ S ] . F [ S ] has a finite Gelfand–Kirillov dimension, but is not What can be said about the prime spectrum of F [ S ] from that of F [ G ] ?

. . . . . . . . . . . . . . . . Semigroup algebras Problem necessarily Noetherian. There are semigroup algebras whose classical Krull dimension is not bounded by their Gelfand–Kirillov dimension. T. Bauer (BIU) . . . . . . . . . . . . . . . . . . . . Prime ideals of semigroup algebras of free nilpotent groups . . . . Let F be a field, G be a finitely generated nilpotent group, and S a finitely generated subsemigroup of G . Study the prime spectrum of F [ S ] . F [ S ] has a finite Gelfand–Kirillov dimension, but is not What can be said about the prime spectrum of F [ S ] from that of F [ G ] ?

. . . . . . . . . . . . . . . . Semigroup algebras Problem necessarily Noetherian. There are semigroup algebras whose classical Krull dimension is not bounded by their Gelfand–Kirillov dimension. T. Bauer (BIU) . . . . . . . . . . . . . . . . . . . . Prime ideals of semigroup algebras of free nilpotent groups . . . . Let F be a field, G be a finitely generated nilpotent group, and S a finitely generated subsemigroup of G . Study the prime spectrum of F [ S ] . F [ S ] has a finite Gelfand–Kirillov dimension, but is not What can be said about the prime spectrum of F [ S ] from that of F [ G ] ?

. . . . . . . . . . . . . . . . Jespers and Okniński (2016) well behaved prime spectrum (Jespers and Okniński): The prime ideals are completely prime. The classical Krull dimension is bounded by the Hirsch length T. Bauer (BIU) . . . . . . . . . . . . . . . . . . . . . . . . Prime ideals of semigroup algebras of free nilpotent groups If G is nilpotent of class 2 , then the semigroup algebra F [ S ] have of the G . For nilpotency class 3 , the situation is more complicated. In F [ S ] , where S = ⟨ b, c ⟩ is the free nilpotent semigroup of class 3 , there exist prime non-completely prime ideals (of Gelfand–Kirillov codimension 1 ).

. . . . . . . . . . . . . . . . Jespers and Okniński (2016) well behaved prime spectrum (Jespers and Okniński): The prime ideals are completely prime. The classical Krull dimension is bounded by the Hirsch length T. Bauer (BIU) . . . . . . . . . . . . . . . . . . . . . . . . Prime ideals of semigroup algebras of free nilpotent groups If G is nilpotent of class 2 , then the semigroup algebra F [ S ] have of the G . For nilpotency class 3 , the situation is more complicated. In F [ S ] , where S = ⟨ b, c ⟩ is the free nilpotent semigroup of class 3 , there exist prime non-completely prime ideals (of Gelfand–Kirillov codimension 1 ).

. . . . . . . . . . . . . . . . Jespers and Okniński (2016) well behaved prime spectrum (Jespers and Okniński): The prime ideals are completely prime. The classical Krull dimension is bounded by the Hirsch length T. Bauer (BIU) . . . . . . . . . . . . . . . . . . . . . . . . Prime ideals of semigroup algebras of free nilpotent groups If G is nilpotent of class 2 , then the semigroup algebra F [ S ] have of the G . For nilpotency class 3 , the situation is more complicated. In F [ S ] , where S = ⟨ b, c ⟩ is the free nilpotent semigroup of class 3 , there exist prime non-completely prime ideals (of Gelfand–Kirillov codimension 1 ).

. . . . . . . . . . . . . . . . Jespers and Okniński (2016) well behaved prime spectrum (Jespers and Okniński): The prime ideals are completely prime. The classical Krull dimension is bounded by the Hirsch length T. Bauer (BIU) . . . . . . . . . . . . . . . . . . . . . . . . Prime ideals of semigroup algebras of free nilpotent groups If G is nilpotent of class 2 , then the semigroup algebra F [ S ] have of the G . For nilpotency class 3 , the situation is more complicated. In F [ S ] , where S = ⟨ b, c ⟩ is the free nilpotent semigroup of class 3 , there exist prime non-completely prime ideals (of Gelfand–Kirillov codimension 1 ).