Decompositions of Binomial Ideals Laura Felicia Matusevich Texas - - PowerPoint PPT Presentation

Decompositions of Binomial Ideals

Laura Felicia Matusevich

Texas A&M University

AMS Spring Central Sectional Meeting, April 17, 2016

Polynomial Ideals

❘ ❂ k❬①✶❀ ✿ ✿ ✿ ❀ ①♥❪ the polynomial ring over a field k. A monomial is a polynomial with one term, a binomial is a polynomial with at most two terms. Monomial ideals are generated by monomials, binomial ideals are generated by binomials. Monomial ideals: Algebra, Combinatorics, Topology. Toric Ideals: Prime binomial ideals. Algebra, Combinatorics, Geometry.

Binomial Ideals

Theorem (Eisenbud and Sturmfels, 1994)

■ ✚ ❘ a binomial ideal, k algebraically closed.

◮ Geometric Statement:

Var✭■✮ is a union of toric varieties.

◮ Algebraic Statement:

The associated primes and primary components of ■ can be chosen binomial.

Why are Noetherian rings called Noetherian?

❘ commutative ring with ✶, Noetherian (ascending chains of ideals stabilize). A proper ideal ■ ✚ ❘ is prime if ①② ✷ ■ implies ① ✷ ■ or ② ✷ ■. ■ is primary if ①② ✷ ■ and ①♥ ❂ ✷ ■ ✽♥ ✷ N, implies ② ✷ ■.

Theorem (Lasker 1905 (special cases), Noether 1921)

Every proper ideal ■ ❘ has a decomposition as a finite intersection of primary ideals. The radicals of the primary ideals appearing in the decomposition are the associated primes of ■.

Binomial Ideals

Theorem (Eisenbud and Sturmfels, 1994)

■ ✚ ❘ a binomial ideal, k algebraically closed.

◮ Geometric Statement:

Var✭■✮ is a union of toric varieties.

◮ Algebraic Statement:

The associated primes and primary components of ■ can be chosen binomial.

◮ Combinatorial Statement:

The subject of this talk. Need k algebraically closed; ❝❤❛r✭k✮ makes a difference. Example: In k❬②❪, consider ■ ❂ ❤②♣ ✶✐. No hope of nice combinatorics for trinomial ideals.

There is combinatorics! (Slide of joy)

■ ❂ ❤①✷ ②✸❀ ①✸ ②✹✐ ❂ ❤① ✶❀ ② ✶✐ ❭ ✭■ ✰ ❤①✹❀ ①✸②❀ ①✷②✷❀ ①②✹❀ ②✺✐✮ Works for binomial ideals over k ❂ k with ❝❤❛r✭k✮ ❂ ✵. But how to make sure we have all bounded components?

There is combinatorics! (Slide of joy)

■ ❂ ❤①✷ ②✸❀ ①✸ ②✹✐ ❂ ❤① ✶❀ ② ✶✐ ❭ ✭■ ✰ ❤①✹❀ ①✸②❀ ①✷②✷❀ ①②✹❀ ②✺✐✮ Works for binomial ideals over k ❂ k with ❝❤❛r✭k✮ ❂ ✵. But how to make sure we have all bounded components?

There is combinatorics! (Slide of joy)

■ ❂ ❤①✷ ②✸❀ ①✸ ②✹✐ ❂ ❤① ✶❀ ② ✶✐ ❭ ✭■ ✰ ❤①✹❀ ①✸②❀ ①✷②✷❀ ①②✹❀ ②✺✐✮ Works for binomial ideals over k ❂ k with ❝❤❛r✭k✮ ❂ ✵. But how to make sure we have all bounded components?

There is combinatorics! (Slide of joy)

■ ❂ ❤①✷ ②✸❀ ①✸ ②✹✐ ❂ ❤① ✶❀ ② ✶✐ ❭ ✭■ ✰ ❤①✹❀ ①✸②❀ ①✷②✷❀ ①②✹❀ ②✺✐✮ Works for binomial ideals over k ❂ k with ❝❤❛r✭k✮ ❂ ✵. But how to make sure we have all bounded components?

There is combinatorics! (Slide of joy)

■ ❂ ❤①✷ ②✸❀ ①✸ ②✹✐ ❂ ❤① ✶❀ ② ✶✐ ❭ ✭■ ✰ ❤①✹❀ ①✸②❀ ①✷②✷❀ ①②✹❀ ②✺✐✮ Works for binomial ideals over k ❂ k with ❝❤❛r✭k✮ ❂ ✵. But how to make sure we have all bounded components?

There is combinatorics! (Slide of joy)

■ ❂ ❤①✷ ②✸❀ ①✸ ②✹✐ ❂ ❤① ✶❀ ② ✶✐ ❭ ✭■ ✰ ❤①✹❀ ①✸②❀ ①✷②✷❀ ①②✹❀ ②✺✐✮ Works for binomial ideals over k ❂ k with ❝❤❛r✭k✮ ❂ ✵. But how to make sure we have all bounded components?

There is combinatorics! (Slide of joy)

■ ❂ ❤①✷ ②✸❀ ①✸ ②✹✐ ❂ ❤① ✶❀ ② ✶✐ ❭ ✭■ ✰ ❤①✹❀ ①✸②❀ ①✷②✷❀ ①②✹❀ ②✺✐✮ Works for binomial ideals over k ❂ k with ❝❤❛r✭k✮ ❂ ✵. But how to make sure we have all bounded components?

There is combinatorics! (Slide of joy)

■ ❂ ❤①✷ ②✸❀ ①✸ ②✹✐ ❂ ❤① ✶❀ ② ✶✐ ❭ ✭■ ✰ ❤①✹❀ ①✸②❀ ①✷②✷❀ ①②✹❀ ②✺✐✮ Works for binomial ideals over k ❂ k with ❝❤❛r✭k✮ ❂ ✵. But how to make sure we have all bounded components?

Switch gears: Lattice Ideals

If ▲ ✒ Z♥ is a lattice, and ✚ ✿ ▲ ✦ k✄ is a group homomorphism, ■✭✚✮ ❂ ❤①✉ ✚✭✉ ✈✮①✈ ❥ ✉❀ ✈ ✷ N♥❀ ✉ ✈ ✷ ▲✐ ✚ k❬①✶❀ ✿ ✿ ✿ ❀ ①♥❪ is a lattice ideal.

Theorem (Eisenbud–Sturmfels)

A binomial ideal ■ is a lattice ideal iff ♠❜ ✷ ■ for ♠ monomial, ❜ binomial ✮ ❜ ✷ ■. If k is algebraically closed, the primary decomposition of ■✭✚✮ can be explicitly determined in terms of extensions of ✚ to ❙❛t✭▲✮ ❂ ✭Q ✡Z ▲✮ ❭ Z♥.

Lattice Ideals are easy to decompose

Example ▲ ❂ s♣❛♥Z❢✭✶❀ ✵❀ ✸❀ ✷✮❀ ✭✷❀ ✸❀ ✵❀ ✶✮❣ ✚ Z✹. ✚ ✿ Z✹ ✦ k✄ the trivial character. ■✭✚✮ ❂ ❤①✇✷ ③✸❀ ①✷✇ ②✸✐✿ ❙❛t✭▲✮ ❂ s♣❛♥Z❢✭✶❀ ✷❀ ✶❀ ✵✮❀ ✭✵❀ ✶❀ ✷❀ ✶✮❣ and ❥❙❛t✭▲✮❂▲❥ ❂ ✸ If ❝❤❛r✭k✮ ✻❂ ✸, then ■ ❂ ■✶ ❭ ■✷ ❭ ■✸, where ■❥ ❂ ❤②③ ✦❥①✇❀ ①③ ✦❥②✷❀ ③✷ ✦✷❥②✇✐❀ ✦✸ ❂ ✶❀ ✦ ✻❂ ✶✿ If ❝❤❛r✭k✮ ❂ ✸, ■ is primary.

What next

The good: Relevant combinatorics: monoid congruences.

Laura, don’t forget to explain what congruences are.

The not so good: Field assumptions, computability issues. Take a deep breath: Stop decomposing at the level of lattice ideals. The choices:

◮ Finest possible

✦ Mesoprimary Decomposition [Kahle-Miller]

◮ Coarsest possible

✦ Unmixed Decomposition [Eisenbud-Sturmfels], [Ojeda-Piedra], [Eser-M]

Too many definitions

Colon ideal and saturation: ✭■ ✿ ①✮ ❂ ❢❢ ❥ ①❢ ✷ ■❣ and ✭■ ✿ ①✶✮ ❂ ❢❢ ❥ ✾❵ ❃ ✵❀ ①❵❢ ✷ ■❣ ■ binomial ideal, ♠ monomial ✮ ✭■ ✿ ♠✮❀ ✭■ ✿ ♠✶✮ binomial. Let ✛ ✒ ❢✶❀ ✿ ✿ ✿ ❀ ♥❣. ■ ✒ k❬①✶❀ ✿ ✿ ✿ ❀ ①♥❪ is ✛-cellular if ✽✐ ✷ ✛, ✭■ ✿ ①✐✮ ❂ ■, and ✽❥ ❂ ✷ ✛, ✾❵❥ ❃ ✵ such that ①

❵❥ ❥ ✷ ■.

■ a ✛-cellular binomial ideal.

◮ ■ is mesoprime if ■ ❂ ❤■lat✐ ✰ ❤①❥ ❥ ❥ ❂

✷ ✛✐ for some lattice ideal ■lat ❂ ■lat ✚ k❬①✐ ❥ ✐ ✷ ✛❪.

◮ ■ is mesoprimary if ❜ ✷ k❬①✐ ❥ ✐ ✷ ✛❪ binomial, ♠ monomial

and ❜♠ ✷ ■ ✮ ♠ ✷ ■ or ❜ ✷ ■lat ❂ ■ ❭ k❬①✐ ❥ ✐ ✷ ✛❪.

◮ ■ is unmixed if ❆ss✭■✮ ❂ ❆ss✭❤■lat✐ ✰ ❤①❥ ❥ ①❥ ❂

✷ ✛✐✮, where ■lat ❂ ■ ❭ k❬①✐ ❥ ①✐ ✷ ✛❪.

Cellular, Mesoprimary, Unmixed

■ a ✛-cellular binomial ideal, mesoprime.

◮ ■ is mesoprime if ■ ❂ ❤■lat✐ ✰ ❤①❥ ❥ ❥ ❂

✷ ✛✐ for some lattice ideal ■lat ✚ k❬①✐ ❥ ✐ ✷ ✛❪.

◮ ■ is mesoprimary if ❜ ✷ k❬①✐ ❥ ✐ ✷ ✛❪ binomial, ♠ monomial

and ❜♠ ✷ ■ ✮ ♠ ✷ ■ or ❜ ✷ ■lat ❂ ■ ❭ k❬①✐ ❥ ✐ ✷ ✛❪.

◮ ■ is unmixed if ❆ss✭■✮ ❂ ❆ss✭❤■lat✐ ✰ ❤①❥ ❥ ①❥ ❂

✷ ✛✐✮, where ■lat ❂ ■ ❭ k❬①✐ ❥ ①✐ ✷ ✛❪.

Example

■ ❂ ❤①✸ ✶❀ ②✭① ✶✮❀ ②✷✐ cellular, unmixed, not mesoprimary, with decomposition ■ ❂ ❤①✸ ✶❀ ②✐ ❭ ❤① ✶❀ ②✷✐✿ If ❝❤❛r✭k✮ ❂ ✸, ■ is primary. If ❝❤❛r✭k✮ ✻❂ ✸, the primary decomposition is ■ ❂ ❤① ✦❀ ②✐ ❭ ❤① ✦✷❀ ②✐ ❭ ❤① ✶❀ ②✷✐; ✦✸ ❂ ✶❀ ✦ ✻❂ ✶.

Cellular, Mesoprimary, Unmixed

■ a ✛-cellular binomial ideal, mesoprime.

◮ ■ is mesoprime if ■ ❂ ❤■lat✐ ✰ ❤①❥ ❥ ❥ ❂

✷ ✛✐ for some lattice ideal ■lat ✚ k❬①✐ ❥ ✐ ✷ ✛❪.

◮ ■ is mesoprimary if ❜ ✷ k❬①✐ ❥ ✐ ✷ ✛❪ binomial, ♠ monomial

and ❜♠ ✷ ■ ✮ ♠ ✷ ■ or ❜ ✷ ■lat ❂ ■ ❭ k❬①✐ ❥ ✐ ✷ ✛❪.

◮ ■ is unmixed if ❆ss✭■✮ ❂ ❆ss✭❤■lat✐ ✰ ❤①❥ ❥ ①❥ ❂

✷ ✛✐✮, where ■lat ❂ ■ ❭ k❬①✐ ❥ ①✐ ✷ ✛❪.

Example

■ ❂ ❤■lat✐ ✰ ❤■art✐ is always mesoprimary but converse is not

• true. For instance

❤①✷②✷ ✶❀ ①③ ②✇❀ ③✷❀ ✇✷✐ is mesoprimary.

At last

Theorem

Decompositions of binomial ideals into

◮ mesoprimary binomial ideals [Kahle-Miller] ◮ unmixed cellular binomial ideals [Eisenbud-Sturmfels]

[Ojeda-Piedra] [Eser-M] exist over any field. The punchline: Now primary decomposition is easy!

But how to do it? (Handwavy slide, we are all tired)

The easy case: ■ is ✛-cellular. For ♠ monomial in k❬①❥ ❥ ❥ ❂ ✷ ✛❪, ❏♠ ❂ ✭■ ✿ ♠✮ ❭ k❬①✐ ❥ ✐ ✷ ✛❪ is a lattice ideal. The unmixed/mesoprimary components of ■ are of the form

✭■ ✰ ❏♠✮ ✿ ❨

✐✷✛

①✶

✐

✁ ✰ "combinatorial" monomial ideal

Mesoprimary decomposition: largest possible monomial ideal. Unmixed decomposition: smallest possible monomial ideal. It is easy to produce mesoprimary/unmixed decompositions. Controlling the decompositions is hard.

Slide of shame

Binomial ideals do not in general have irreducible binomial decompositions [Kahle-Miller-O’Neill]. ■ a binomial ideal.

◮ When is k❬①❪❂■ Cohen–Macaulay? ◮ Gorenstein? ◮ What are the Betti numbers of k❬①❪❂■? ◮ Can a (minimal) free resolution be constructed? ◮ Is there something like the Ishida complex? ◮ Ask any interesting question here...

I do not know. The optimistic ending: An emerging area, with lots of interesting

• pen problems!

THANK YOU!

Proof of Noether’s theorem (slide of the second wind)

■ ❘ is reducible if ■ ❂ ❏✶ ❭ ❏✷ with ❏✶❀ ❏✷ ■.

• 1. Every proper ideal has an irreducible decomposition.

If ■ does not have an irreducible decomposition, can produce a non-stabilizing ascending chain of ideals.

• 2. Irreducible ideals are primary.

■ is primary iff every ① ✷ ❘ is either nilpotent or a nonzerodivisor modulo ■. Suppose ① ✷ ❘ is neither nilpotent nor a nonzerodivisor mod ■. Then: ✭■ ✿ ①✮ ✚ ✭■ ✿ ①✷✮ ✚ ✭■ ✿ ①✸✮ ✚ ✁ ✁ ✁ so ✾◆: ✭■ ✿ ①◆✮ ❂ ✭■ ✿ ①◆✰✶✮ ❂ ✁ ✁ ✁ Claim. ■ ❂

■ ✰ ❤①◆✐ ✁ ❭ ■ ✿ ①◆✁