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 ❵ ❢ ✷ ■ ❣ ✭ ■ ✿ ① ✮ ❂ ❢ ❢ ❥ ①❢ ✷ ■ ❣ and ■ 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 ✷ ✐ ; ✦ ✸ ❂ ✶ ❀ ✦ ✻ ❂ ✶ .