Prime ideals in rings of power series and polynomials Sylvia Wiegand (work of E. Celikbas, C. Eubanks-Turner, SW) Department of Mathematics University of Nebraska–Lincoln Honolulu 2019 Sylvia Wiegand (work of E. Celikbas, C. Eubanks-Turner, SW) Prime ideals, power series
Intro, history: For R a commutative ring, Spec ( R ) := { prime ideals of R } , a partially ordered set under ⊆ . Sylvia Wiegand (work of E. Celikbas, C. Eubanks-Turner, SW) Prime ideals, power series
Intro, history: For R a commutative ring, Spec ( R ) := { prime ideals of R } , a partially ordered set under ⊆ . Questions. Q1 1950, I. Kaplansky: [Work of Nagata ’50s; Hochster ’69; Lewis and Ohm ’71(?), McAdam ’77, Heitmann ’77,’79; Ratliff ’60s-70s] Sylvia Wiegand (work of E. Celikbas, C. Eubanks-Turner, SW) Prime ideals, power series
Intro, history: For R a commutative ring, Spec ( R ) := { prime ideals of R } , a partially ordered set under ⊆ . Questions. Q1 1950, I. Kaplansky: [Work of Nagata ’50s; Hochster ’69; Lewis and Ohm ’71(?), McAdam ’77, Heitmann ’77,’79; Ratliff ’60s-70s] Q2. What posets arise as Spec ( R ) for R a 2-dim Noetherian domain? Sylvia Wiegand (work of E. Celikbas, C. Eubanks-Turner, SW) Prime ideals, power series
Intro, history: For R a commutative ring, Spec ( R ) := { prime ideals of R } , a partially ordered set under ⊆ . Questions. Q1 1950, I. Kaplansky: [Work of Nagata ’50s; Hochster ’69; Lewis and Ohm ’71(?), McAdam ’77, Heitmann ’77,’79; Ratliff ’60s-70s] Q2. What posets arise as Spec ( R ) for R a 2-dim Noetherian domain? What is Spec ( R ) for a particular ring R ? i.e. Give a characterization of that poset? e.g. R a polynomial ring? Or a ring of power series? [Work of R. Wiegand, Heinzer, S. Wiegand, A.Li, Saydam 70s-90s] Sylvia Wiegand (work of E. Celikbas, C. Eubanks-Turner, SW) Prime ideals, power series
Intro, history: For R a commutative ring, Spec ( R ) := { prime ideals of R } , a partially ordered set under ⊆ . Questions. Q1 1950, I. Kaplansky: [Work of Nagata ’50s; Hochster ’69; Lewis and Ohm ’71(?), McAdam ’77, Heitmann ’77,’79; Ratliff ’60s-70s] Q2. What posets arise as Spec ( R ) for R a 2-dim Noetherian domain? What is Spec ( R ) for a particular ring R ? i.e. Give a characterization of that poset? e.g. R a polynomial ring? Or a ring of power series? [Work of R. Wiegand, Heinzer, S. Wiegand, A.Li, Saydam 70s-90s] Or a ring that has both? Sylvia Wiegand (work of E. Celikbas, C. Eubanks-Turner, SW) Prime ideals, power series
Setting/Goal Let E = k [[ x ]][ y , z ] , R [[ x ]][ y ] , or R [ y ][[ x ]] , a mixed poly-power series, where k = a field or R = a 1-dim Noetherian integral domain, Let Q ∈ Spec E , ht Q = 1, (usually) Q ̸ = xE . Goal Question: What is Spec ( E / Q ) ? • dim ( E / Q ) ≤ 2. Assume dim ( E / Q ) = 2 . (Dim 1 is easy.) Sylvia Wiegand (work of E. Celikbas, C. Eubanks-Turner, SW) Prime ideals, power series
Setting/Goal Let E = k [[ x ]][ y , z ] , R [[ x ]][ y ] , or R [ y ][[ x ]] , a mixed poly-power series, where k = a field or R = a 1-dim Noetherian integral domain, Let Q ∈ Spec E , ht Q = 1, (usually) Q ̸ = xE . Goal Question: What is Spec ( E / Q ) ? • dim ( E / Q ) ≤ 2. Assume dim ( E / Q ) = 2 . (Dim 1 is easy.) • E / Q catenary, Noetherian. A ring A is catenary provided for every pair P ⊊ Q in Spec ( A ) , the number of prime ideals in every maximal chain of form P = P 0 ⊊ P 1 ⊊ P 2 ⊊ . . . ⊊ P n = Q is the same. Example: What is Spec ( Z [ y ][[ x ]] / ( x − α ) , for α = 2 y ( y + 1 ) ∈ Z [ y ] ? Sylvia Wiegand (work of E. Celikbas, C. Eubanks-Turner, SW) Prime ideals, power series
What is Spec ( Z [ y ][[ x ]] / ( x − α ) ? Part 1 Here E = Z [ y ][[ x ]] , R = Z , Q = ( x − α ) , α = 2 y ( y + 1 ) ∈ Z [ y ] , B:=E/Q. Observations: •M ∈ max E & ht M = 3 = ⇒ M = ( m , h ( y ) , x ) , where m ∈ max R , and h ( y ) is irreducible in ( R / m )[ y ] . = ⇒ x ∈ M , ∀ M ∈ max B with ht M = 2. Sylvia Wiegand (work of E. Celikbas, C. Eubanks-Turner, SW) Prime ideals, power series
What is Spec ( Z [ y ][[ x ]] / ( x − α ) ? Part 1 Here E = Z [ y ][[ x ]] , R = Z , Q = ( x − α ) , α = 2 y ( y + 1 ) ∈ Z [ y ] , B:=E/Q. Observations: •M ∈ max E & ht M = 3 = ⇒ M = ( m , h ( y ) , x ) , where m ∈ max R , and h ( y ) is irreducible in ( R / m )[ y ] . = ⇒ x ∈ M , ∀ M ∈ max B with ht M = 2. • P ∈ Spec E , x / ∈ P , ht P = 2 & P NON-maximal = ⇒ P ⊆ UNIQUE ht-3 M . Sylvia Wiegand (work of E. Celikbas, C. Eubanks-Turner, SW) Prime ideals, power series
What is Spec ( Z [ y ][[ x ]] / ( x − α ) ? Part 1 Here E = Z [ y ][[ x ]] , R = Z , Q = ( x − α ) , α = 2 y ( y + 1 ) ∈ Z [ y ] , B:=E/Q. Observations: •M ∈ max E & ht M = 3 = ⇒ M = ( m , h ( y ) , x ) , where m ∈ max R , and h ( y ) is irreducible in ( R / m )[ y ] . = ⇒ x ∈ M , ∀ M ∈ max B with ht M = 2. • P ∈ Spec E , x / ∈ P , ht P = 2 & P NON-maximal = ⇒ P ⊆ UNIQUE ht-3 M . ( x , I ) = R [ y ] • B E E xB = ( x , Q ) = , where I = { f ( 0 , y ) | f ( x , y ) ∈ Q } . I Sylvia Wiegand (work of E. Celikbas, C. Eubanks-Turner, SW) Prime ideals, power series
What is Spec ( Z [ y ][[ x ]] / ( x − α ) ? Part 1 Here E = Z [ y ][[ x ]] , R = Z , Q = ( x − α ) , α = 2 y ( y + 1 ) ∈ Z [ y ] , B:=E/Q. Observations: •M ∈ max E & ht M = 3 = ⇒ M = ( m , h ( y ) , x ) , where m ∈ max R , and h ( y ) is irreducible in ( R / m )[ y ] . = ⇒ x ∈ M , ∀ M ∈ max B with ht M = 2. • P ∈ Spec E , x / ∈ P , ht P = 2 & P NON-maximal = ⇒ P ⊆ UNIQUE ht-3 M . ( x , I ) = R [ y ] • B E E xB = ( x , Q ) = , where I = { f ( 0 , y ) | f ( x , y ) ∈ Q } . I These items = ⇒ Spec ( Z [ y ] 2 y ( y + 1 ) ) is related to Spec ( Z [[ x ]][ y ] Z [ y ] I ) = Spec ( ( x − α ) ) . Sylvia Wiegand (work of E. Celikbas, C. Eubanks-Turner, SW) Prime ideals, power series
Spec ( Z [ y ] Z [ y ] I ) = Spec ( 2 y ( y + 1 ) ) Z [ y ] Z [[ x ]][ y ] ⇒ Spec ( 2 y ( y + 1 ) ) is part of Spec ( ( x − 2 y ( y + 1 )) ) . Previous slide = Z [ y ] U 0 = Spec ( 2 y ( y + 1 ) ) : Sylvia Wiegand (work of E. Celikbas, C. Eubanks-Turner, SW) Prime ideals, power series
Spec ( Z [ y ] Z [ y ] I ) = Spec ( 2 y ( y + 1 ) ) Z [ y ] Z [[ x ]][ y ] ⇒ Spec ( 2 y ( y + 1 ) ) is part of Spec ( ( x − 2 y ( y + 1 )) ) . Previous slide = Z [ y ] U 0 = Spec ( 2 y ( y + 1 ) ) : ℵ 0 ; ( y , 5 ) ∈ ( 2 , y ) ℵ 0 ( 2 , y + 1 ) ℵ 0 ; ( y + 1 , 3 ) ∈ ( y ) ( 2 ) ( y + 1 ) Let F = { ( y ) , ( 2 ) , ( y + 1 ); ( 2 , y ) , ( 2 , y + 1 ) } —a sort of skeleton for U 0 . We call it the determinator set for U 0 . Sylvia Wiegand (work of E. Celikbas, C. Eubanks-Turner, SW) Prime ideals, power series
What is Spec ( Z [ y ][[ x ]] / ( x − α ) ? part 2 Here Q = ( x − α ) , α = 2 y ( y + 1 ) ∈ Z [ y ] For Q ⊆ P ∈ Spec E , let P = π ( P ) , where π : E → E / Q . ℵ 0 ; ( x , y , 5 ) ∈ ( x , 2 , y ) ℵ 0 ( x , 2 , y + 1 ) ℵ 0 ; ( x , y + 1 , 3 ) ∈ ( x , y ) | R | ( x , 2 ) | R | ( x , y + 1 ) ( 0 ) = ( x − 2 y ( y + 1 ) Note: Every height-two element has a set of | R | elements below it and below no other height-two element (not shown). Sylvia Wiegand (work of E. Celikbas, C. Eubanks-Turner, SW) Prime ideals, power series
U = Spec ( R [ y ][[ x ]] / Q ) if | R | ≤ ℵ 0 , | max R | = ∞ . Theorem: Let E = R [ y ][[ x ]] , R [[ x ]][ y ] or k [[ z ]][ x , y ] ; | R | ≤ | N | , | max R | = ∞ , R = 1-dim Noetherian domain, or k a field, | R | , | k | ≤ ℵ 0 Let Q ∈ Spec E , ht Q = 1, Q ̸ = ( x ) with dim E / Q = 2. Let U = Spec ( E / Q ) , and let ε = |{ ht-1 max elements in U }| . Then: Sylvia Wiegand (work of E. Celikbas, C. Eubanks-Turner, SW) Prime ideals, power series
U = Spec ( R [ y ][[ x ]] / Q ) if | R | ≤ ℵ 0 , | max R | = ∞ . Theorem: Let E = R [ y ][[ x ]] , R [[ x ]][ y ] or k [[ z ]][ x , y ] ; | R | ≤ | N | , | max R | = ∞ , R = 1-dim Noetherian domain, or k a field, | R | , | k | ≤ ℵ 0 Let Q ∈ Spec E , ht Q = 1, Q ̸ = ( x ) with dim E / Q = 2. Let U = Spec ( E / Q ) , and let ε = |{ ht-1 max elements in U }| . Then: • ε = 0 or | R | , and •∃ F finite 1-dim poset ⊂ U \ { ( 0 ) } that determine U i.e. Every slot outside F and the ε slot has the same number of elements as for Z [ y ][[ x ]] / Q above. Notes: 1. E = R [ y ][[ x ]] = ⇒ ε = 0. 2. E = k [[ x ]][ z , y ] = ⇒ ε = | R | . 3. For E = R [[ x ]][ y ] , let ℓ y ( Q ) = { h t ( x ) | h t ( x ) y t + · · · h 0 ( x ) ∈ Q } . Then: ℓ y ( Q ) = ( 1 ) ⇐ ⇒ ε = 0; ℓ y ( Q ) ̸ = ( 1 ) ⇐ ⇒ ε = | R | . Sylvia Wiegand (work of E. Celikbas, C. Eubanks-Turner, SW) Prime ideals, power series
Properties of U and F What is the set F associated with U = Spec ( Z [ y ][[ x ]] / Q ) ? F 0 := { non-0, nonmax j -prime ideals } = { u ht-1 | | u ↑ | ≥ 2 } . Also F 0 corresponds to { nonmaximal prime ideals of E minimal over ( Q , x ) } and to {nonmaximal prime ideals of R [ y ] minimal over I }. f ̸ = g ∈ F 0 f ↑ ∩ g ↑ ) ∪ F 0 , a finite set by item 5 below. F := ( ∪ Sylvia Wiegand (work of E. Celikbas, C. Eubanks-Turner, SW) Prime ideals, power series
Recommend
More recommend
