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. . . . . . . . . . . . . . . Prime monomial ideals of subsemigroup algebras of free nilpotent groups Tomer Bauer joint work with Beeri Greenfeld Department of Mathematics Bar-Ilan University Groups, Rings and Associated

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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

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Background: Group algebras

Let F be a field and G be a finitely generated nilpotent group. Problem 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

f F[G].

What about semigroup algebras of subsemigroups of G?

T. Bauer (BIU)

Prime ideals of semigroup algebras of free nilpotent groups

SLIDE 3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background: Group algebras

Let F be a field and G be a finitely generated nilpotent group. Problem 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

f F[G].

What about semigroup algebras of subsemigroups of G?

T. Bauer (BIU)

Prime ideals of semigroup algebras of free nilpotent groups

SLIDE 4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background: Group algebras

Let F be a field and G be a finitely generated nilpotent group. Problem 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

f F[G].

What about semigroup algebras of subsemigroups of G?

T. Bauer (BIU)

Prime ideals of semigroup algebras of free nilpotent groups

SLIDE 5

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background: Group algebras

Let F be a field and G be a finitely generated nilpotent group. Problem 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

f F[G].

What about semigroup algebras of subsemigroups of G?

T. Bauer (BIU)

Prime ideals of semigroup algebras of free nilpotent groups

SLIDE 6

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background: Group algebras

Let F be a field and G be a finitely generated nilpotent group. Problem 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

f F[G].

What about semigroup algebras of subsemigroups of G?

T. Bauer (BIU)

Prime ideals of semigroup algebras of free nilpotent groups

SLIDE 7

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Semigroup algebras

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

f F[G]?
T. Bauer (BIU)

Prime ideals of semigroup algebras of free nilpotent groups

SLIDE 8

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Semigroup algebras

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

f F[G]?
T. Bauer (BIU)

Prime ideals of semigroup algebras of free nilpotent groups

SLIDE 9

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Semigroup algebras

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

f F[G]?
T. Bauer (BIU)

Prime ideals of semigroup algebras of free nilpotent groups

SLIDE 10

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Semigroup algebras

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

f F[G]?
T. Bauer (BIU)

Prime ideals of semigroup algebras of free nilpotent groups

SLIDE 11

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Semigroup algebras

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

f F[G]?
T. Bauer (BIU)

Prime ideals of semigroup algebras of free nilpotent groups

SLIDE 12

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Jespers and Okniński (2016)

If G is nilpotent of class 2, then the semigroup algebra F[S] have well behaved prime spectrum (Jespers and Okniński): The prime ideals are completely prime. The classical Krull dimension is bounded by the Hirsch length

f 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).

T. Bauer (BIU)

Prime ideals of semigroup algebras of free nilpotent groups

SLIDE 13

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Jespers and Okniński (2016)

If G is nilpotent of class 2, then the semigroup algebra F[S] have well behaved prime spectrum (Jespers and Okniński): The prime ideals are completely prime. The classical Krull dimension is bounded by the Hirsch length

f 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).

T. Bauer (BIU)

Prime ideals of semigroup algebras of free nilpotent groups

SLIDE 14

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Jespers and Okniński (2016)

If G is nilpotent of class 2, then the semigroup algebra F[S] have well behaved prime spectrum (Jespers and Okniński): The prime ideals are completely prime. The classical Krull dimension is bounded by the Hirsch length

f 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).

T. Bauer (BIU)

Prime ideals of semigroup algebras of free nilpotent groups

SLIDE 15

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Jespers and Okniński (2016)

If G is nilpotent of class 2, then the semigroup algebra F[S] have well behaved prime spectrum (Jespers and Okniński): The prime ideals are completely prime. The classical Krull dimension is bounded by the Hirsch length

f 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).

T. Bauer (BIU)

Prime ideals of semigroup algebras of free nilpotent groups

SLIDE 16

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Jespers and Okniński (2016)

If G is nilpotent of class 2, then the semigroup algebra F[S] have well behaved prime spectrum (Jespers and Okniński): The prime ideals are completely prime. The classical Krull dimension is bounded by the Hirsch length

f 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).

T. Bauer (BIU)

Prime ideals of semigroup algebras of free nilpotent groups

SLIDE 17

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Jespers and Okniński (2016)

S is the free 2-generated nilpotent semigroup of class 3. Problems (Jespers and Okniński) Do there exist prime homomorphic images of F[S] that are not Goldie? Could they be of the form F[S]/F[P] for a prime ideal P of S? Can infinite chains of primes exist in S? Can F[S] have an infinite classical Krull dimension?

T. Bauer (BIU)

Prime ideals of semigroup algebras of free nilpotent groups

SLIDE 18

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Jespers and Okniński (2016)

S is the free 2-generated nilpotent semigroup of class 3. Problems (Jespers and Okniński) Do there exist prime homomorphic images of F[S] that are not Goldie? Could they be of the form F[S]/F[P] for a prime ideal P of S? Can infinite chains of primes exist in S? Can F[S] have an infinite classical Krull dimension?

T. Bauer (BIU)

Prime ideals of semigroup algebras of free nilpotent groups

SLIDE 19

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Jespers and Okniński (2016)

S is the free 2-generated nilpotent semigroup of class 3. Problems (Jespers and Okniński) Do there exist prime homomorphic images of F[S] that are not Goldie? Could they be of the form F[S]/F[P] for a prime ideal P of S? Can infinite chains of primes exist in S? Can F[S] have an infinite classical Krull dimension?

T. Bauer (BIU)

Prime ideals of semigroup algebras of free nilpotent groups

SLIDE 20

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Jespers and Okniński (2016)

S is the free 2-generated nilpotent semigroup of class 3. Problems (Jespers and Okniński) Do there exist prime homomorphic images of F[S] that are not Goldie? Could they be of the form F[S]/F[P] for a prime ideal P of S? Can infinite chains of primes exist in S? Can F[S] have an infinite classical Krull dimension?

T. Bauer (BIU)

Prime ideals of semigroup algebras of free nilpotent groups

SLIDE 21

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Jespers and Okniński (2016)

S is the free 2-generated nilpotent semigroup of class 3. Problems (Jespers and Okniński) Do there exist prime homomorphic images of F[S] that are not Goldie? Could they be of the form F[S]/F[P] for a prime ideal P of S? Can infinite chains of primes exist in S? Can F[S] have an infinite classical Krull dimension?

T. Bauer (BIU)

Prime ideals of semigroup algebras of free nilpotent groups

SLIDE 22

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The monomial prime spectrum

Theorem (B, Greenfeld) If P ◁ F[S] is a prime monomial ideal, then F[S]/P is just-infinite of at most quadratic growth. It is either PI of linear growth or primitive of quadratic growth. The prime ideals of linear growth are precisely the ideals discovered by Jespers and Okniński. Corollary Every ascending chain of prime ideals in F[S] with at least one monomial member is finite.

T. Bauer (BIU)

Prime ideals of semigroup algebras of free nilpotent groups

SLIDE 23

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The monomial prime spectrum

Theorem (B, Greenfeld) If P ◁ F[S] is a prime monomial ideal, then F[S]/P is just-infinite of at most quadratic growth. It is either PI of linear growth or primitive of quadratic growth. The prime ideals of linear growth are precisely the ideals discovered by Jespers and Okniński. Corollary Every ascending chain of prime ideals in F[S] with at least one monomial member is finite.

T. Bauer (BIU)

Prime ideals of semigroup algebras of free nilpotent groups

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The monomial prime spectrum

Theorem (B, Greenfeld) If P ◁ F[S] is a prime monomial ideal, then F[S]/P is just-infinite of at most quadratic growth. It is either PI of linear growth or primitive of quadratic growth. The prime ideals of linear growth are precisely the ideals discovered by Jespers and Okniński. Corollary Every ascending chain of prime ideals in F[S] with at least one monomial member is finite.

T. Bauer (BIU)

Prime ideals of semigroup algebras of free nilpotent groups

SLIDE 25

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Proof idea

Suppose we have an infinite word w = w(b, c) which is not rewritable in S. If w is eventually periodic, we are done. Otherwise, prove that w is in fact a word in the new ‘letuers’ b1 = bci, c1 = bci+1. Iterate with b2 = b1cj

1, c2 = b1cj+1 1

,and so on… Show that w shares the same factors with b∞. Compute properties of b∞ implying the associated homomorphic image is primitive just-infinite of quadratic growth.

T. Bauer (BIU)

Prime ideals of semigroup algebras of free nilpotent groups

SLIDE 26

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Proof idea

Suppose we have an infinite word w = w(b, c) which is not rewritable in S. If w is eventually periodic, we are done. Otherwise, prove that w is in fact a word in the new ‘letuers’ b1 = bci, c1 = bci+1. Iterate with b2 = b1cj

1, c2 = b1cj+1 1

,and so on… Show that w shares the same factors with b∞. Compute properties of b∞ implying the associated homomorphic image is primitive just-infinite of quadratic growth.

T. Bauer (BIU)

Prime ideals of semigroup algebras of free nilpotent groups

SLIDE 27

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Proof idea

Suppose we have an infinite word w = w(b, c) which is not rewritable in S. If w is eventually periodic, we are done. Otherwise, prove that w is in fact a word in the new ‘letuers’ b1 = bci, c1 = bci+1. Iterate with b2 = b1cj

1, c2 = b1cj+1 1

,and so on… Show that w shares the same factors with b∞. Compute properties of b∞ implying the associated homomorphic image is primitive just-infinite of quadratic growth.

T. Bauer (BIU)

Prime ideals of semigroup algebras of free nilpotent groups

SLIDE 28

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Proof idea

Suppose we have an infinite word w = w(b, c) which is not rewritable in S. If w is eventually periodic, we are done. Otherwise, prove that w is in fact a word in the new ‘letuers’ b1 = bci, c1 = bci+1. Iterate with b2 = b1cj

1, c2 = b1cj+1 1

,and so on… Show that w shares the same factors with b∞. Compute properties of b∞ implying the associated homomorphic image is primitive just-infinite of quadratic growth.

T. Bauer (BIU)

Prime ideals of semigroup algebras of free nilpotent groups

SLIDE 29

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Proof idea

Suppose we have an infinite word w = w(b, c) which is not rewritable in S. If w is eventually periodic, we are done. Otherwise, prove that w is in fact a word in the new ‘letuers’ b1 = bci, c1 = bci+1. Iterate with b2 = b1cj

1, c2 = b1cj+1 1

,and so on… Show that w shares the same factors with b∞. Compute properties of b∞ implying the associated homomorphic image is primitive just-infinite of quadratic growth.

T. Bauer (BIU)

Prime ideals of semigroup algebras of free nilpotent groups

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Candidate

Consider the monomial algebra A associated with the infinite word bcbc2bcbc2bc2bcbc2bcbc2bc2bcbc2bc2 · · · This is φ∞(b), the fixed point of the morphism: φ: b → bc, c → bc2 Conjecture A is a homomorphic image of F[S].

Thank you for your atuention!

T. Bauer (BIU)

Prime ideals of semigroup algebras of free nilpotent groups

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Candidate

Consider the monomial algebra A associated with the infinite word bcbc2bcbc2bc2bcbc2bcbc2bc2bcbc2bc2 · · · This is φ∞(b), the fixed point of the morphism: φ: b → bc, c → bc2 Conjecture A is a homomorphic image of F[S].

Thank you for your atuention!

T. Bauer (BIU)

Prime ideals of semigroup algebras of free nilpotent groups

SLIDE 32

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Candidate

Consider the monomial algebra A associated with the infinite word bcbc2bcbc2bc2bcbc2bcbc2bc2bcbc2bc2 · · · This is φ∞(b), the fixed point of the morphism: φ: b → bc, c → bc2 Conjecture A is a homomorphic image of F[S].

Thank you for your atuention!

T. Bauer (BIU)

Prime ideals of semigroup algebras of free nilpotent groups