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Prime ideals in rings of power series and polynomials Sylvia Wiegand (work of E. Celikbas, C. Eubanks-Turner, SW) Department of Mathematics University of NebraskaLincoln Honolulu 2019 Sylvia Wiegand (work of E. Celikbas, C. Eubanks-Turner,

SLIDE 1

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

SLIDE 2

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

SLIDE 3

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

SLIDE 4

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

SLIDE 5

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

SLIDE 6

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

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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

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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 = P0 ⊊ P1 ⊊ P2 ⊊ . . . ⊊ Pn = Q is the same. Example: What is Spec(Z[y][[x]]/(x − α), for α = 2y(y + 1) ∈ Z[y]?

Sylvia Wiegand (work of E. Celikbas, C. Eubanks-Turner, SW) Prime ideals, power series

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What is Spec(Z[y][[x]]/(x − α)? Part 1

Here E = Z[y][[x]], R = Z, Q = (x − α), α = 2y(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

SLIDE 10

What is Spec(Z[y][[x]]/(x − α)? Part 1

Here E = Z[y][[x]], R = Z, Q = (x − α), α = 2y(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

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What is Spec(Z[y][[x]]/(x − α)? Part 1

Here E = Z[y][[x]], R = Z, Q = (x − α), α = 2y(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.

xB = E (x,Q) = E (x,I) = R[y] I

, where I = {f(0, y) | f(x, y) ∈ Q}.

Sylvia Wiegand (work of E. Celikbas, C. Eubanks-Turner, SW) Prime ideals, power series

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What is Spec(Z[y][[x]]/(x − α)? Part 1

Here E = Z[y][[x]], R = Z, Q = (x − α), α = 2y(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.

xB = E (x,Q) = E (x,I) = R[y] I

, where I = {f(0, y) | f(x, y) ∈ Q}. These items = ⇒ Spec (Z[y]

I )= Spec ( Z[y] 2y(y+1)) is related to Spec (Z[[x]][y] (x−α) ).

Sylvia Wiegand (work of E. Celikbas, C. Eubanks-Turner, SW) Prime ideals, power series

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Spec (Z[y]

I )= Spec ( Z[y] 2y(y+1))

Previous slide = ⇒ Spec (

Z[y] 2y(y+1)) is part of Spec ( Z[[x]][y] (x−2y(y+1))).

U0 = Spec(

Z[y] 2y(y+1)):

Sylvia Wiegand (work of E. Celikbas, C. Eubanks-Turner, SW) Prime ideals, power series

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Spec (Z[y]

I )= Spec ( Z[y] 2y(y+1))

Previous slide = ⇒ Spec (

Z[y] 2y(y+1)) is part of Spec ( Z[[x]][y] (x−2y(y+1))).

U0 = Spec(

Z[y] 2y(y+1)):

(y) (2) (y + 1) ℵ0; (y, 5) ∈ (2, y) ℵ0 (2, y + 1) ℵ0; (y + 1, 3) ∈ Let F = {(y), (2), (y + 1); (2, y), (2, y + 1)}—a sort of skeleton for U0. We call it the determinator set for U0.

Sylvia Wiegand (work of E. Celikbas, C. Eubanks-Turner, SW) Prime ideals, power series

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What is Spec(Z[y][[x]]/(x − α)? part 2

Here Q = (x − α), α = 2y(y + 1) ∈ Z[y] For Q ⊆ P ∈ Spec E, let P = π(P), where π : E → E/Q. (x, y) |R| (x, 2) |R| (x, y + 1) ℵ0; (x, y, 5) ∈ (x, 2, y) ℵ0 (x, 2, y + 1) ℵ0; (x, y + 1, 3) ∈ (0) = (x − 2y(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

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U = Spec(R[y][[x]]/Q) if |R| ≤ ℵ0, |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| ≤ ℵ0 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

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U = Spec(R[y][[x]]/Q) if |R| ≤ ℵ0, |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| ≤ ℵ0 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 ℓy(Q) = {ht(x) | ht(x)yt + · · · h0(x) ∈ Q}.

Then: ℓy(Q) = (1) ⇐ ⇒ ε = 0; ℓy(Q) ̸= (1) ⇐ ⇒ ε = |R|.

Sylvia Wiegand (work of E. Celikbas, C. Eubanks-Turner, SW) Prime ideals, power series

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Properties of U and F

What is the set F associated with U = Spec(Z[y][[x]]/Q)? F0 := { non-0, nonmax j-prime ideals} = {u ht-1 | |u↑| ≥ 2}. Also F0 corresponds to { nonmaximal prime ideals of E minimal over (Q, x)} and to {nonmaximal prime ideals of R[y] minimal over I }. F := (∪

f̸=g∈F0 f ↑ ∩ g↑) ∪ F0, a finite set by item 5 below.

Sylvia Wiegand (work of E. Celikbas, C. Eubanks-Turner, SW) Prime ideals, power series

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Properties of U and F

What is the set F associated with U = Spec(Z[y][[x]]/Q)? F0 := { non-0, nonmax j-prime ideals} = {u ht-1 | |u↑| ≥ 2}. Also F0 corresponds to { nonmaximal prime ideals of E minimal over (Q, x)} and to {nonmaximal prime ideals of R[y] minimal over I }. F := (∪

f̸=g∈F0 f ↑ ∩ g↑) ∪ F0, a finite set by item 5 below.

|{ht-2s in U}| = |N| |{ht-0s in U}| = 1 |{ht-1s in U}| = |R|

∀t ∈ U, ht t = 2 = ⇒ |t↓e| = |R|. t↓e = {v ∈ U | v < s ⇐ ⇒ s = t}.

∪

f∈F0 f ↑ = {ht-2 ∈ U}.

Sylvia Wiegand (work of E. Celikbas, C. Eubanks-Turner, SW) Prime ideals, power series

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Properties of U and F

What is the set F associated with U = Spec(Z[y][[x]]/Q)? F0 := { non-0, nonmax j-prime ideals} = {u ht-1 | |u↑| ≥ 2}. Also F0 corresponds to { nonmaximal prime ideals of E minimal over (Q, x)} and to {nonmaximal prime ideals of R[y] minimal over I }. F := (∪

f̸=g∈F0 f ↑ ∩ g↑) ∪ F0, a finite set by item 5 below.

|{ht-2s in U}| = |N| |{ht-0s in U}| = 1 |{ht-1s in U}| = |R|

∀t ∈ U, ht t = 2 = ⇒ |t↓e| = |R|. t↓e = {v ∈ U | v < s ⇐ ⇒ s = t}.

∪

f∈F0 f ↑ = {ht-2 ∈ U}.

∀f ∈ F0, |f ↑e| = N. ( = ⇒ F0 ⊆ {j-primes}.) f ↑e = {u ∈ U | u > t ⇐ ⇒ v = t}.

∀f ̸= g ∈ F0, |f ↑ ∩ g↑| < ∞.

Sylvia Wiegand (work of E. Celikbas, C. Eubanks-Turner, SW) Prime ideals, power series

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Existence Theorem

Theorem: For every finite poset F of dim 1, ∃Q ∈ Spec(Z[y][[x]]) such that F “determines" Spec(Z[y][[x]]/Q). (Technically, want F such that every ht-1 element is above 2 ht-0 elements of F, to ensure distinct F’s determine different U .)

Sylvia Wiegand (work of E. Celikbas, C. Eubanks-Turner, SW) Prime ideals, power series

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Existence Theorem

Theorem: For every finite poset F of dim 1, ∃Q ∈ Spec(Z[y][[x]]) such that F “determines" Spec(Z[y][[x]]/Q). (Technically, want F such that every ht-1 element is above 2 ht-0 elements of F, to ensure distinct F’s determine different U .)

Sylvia Wiegand (work of E. Celikbas, C. Eubanks-Turner, SW) Prime ideals, power series

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Other examples of spectra, 1

Example: Let R = Z(2) and I = 2y(2y − 1)(y + 2)Z(2)[y]. Then Spec( Z(2)[y]/(2y(2y − 1) (y + 2)Z(2)[y]) ) is shown below: (2y − 1) (y) (y + 2) (2) (2, y) ℵ0 Spec (

Z(2)[y] 2y(2y−1)(y+2)Z(2)[y]

) The structure of Spec( Z(2)[y]/(2y(2y − 1)(y + 2)Z(2)[y]) ) is determined by the partially ordered sets F = {(2), (y), (y + 2), (2, y)} and G = {(2y − 1)}, and by the cardinalities |(2)↑e| = ℵ0, and |(y)↑e| = |(y + 2)↑e| = 0. Then | min(F)| = 3, |G| = 1 and the type is (1; F; ℵ0, 0, 0).

Sylvia Wiegand (work of E. Celikbas, C. Eubanks-Turner, SW) Prime ideals, power series

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Other examples of spectra, 2

Example: Let E = Z(2)[y][ [x] ], Q = (x − 2y(2y − 1)(y + 2))E, and B = Z(2)[y][ [x] ]/Q. Spec B is shown below except ∃ boxes of size |R| under every height-two element in the box of ℵ0 elements. As above, I = 2y(2y − 1)(y + 2)Z(2)[y].

(2y − 1, x)
Q

(2, x) ℵ0

(y, x)
(2, y, x)
(y + 2, x)

|R| Spec B for B =

Z(2)[y][ [x] ] (x−y(y−2)(y−3)(2y+1)(4y−1))

Sylvia Wiegand (work of E. Celikbas, C. Eubanks-Turner, SW) Prime ideals, power series

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THANKS!

Sylvia Wiegand (work of E. Celikbas, C. Eubanks-Turner, SW) Prime ideals, power series