Generic Initial Ideals Lecture 5

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Again, since I is strongly stable, it follows that w = x l1 · · · x lc ∈ G(I) for some c ≤ d. Since w σ divides v σ = u 0 , one has u 0 ∈ I σ , as desired. ̌ Let ∆ be a simplicial complex on [n]. Since the base field K is of characteristic 0, we have that gin
Lemma 3: If I ⊂ S is a strongly stable ideal, then β ii+j (I) = β ii+j (I σ ) for all i and j.
Page 1 and 2: Generic Initial Ideals; Lecture 5 J
Page 3 and 4: Outline Shifting operations Kalai
Page 5 and 6: Shifting operations Definition: A s
Page 7 and 8: Shifting operations Definition: A s
Page 9 and 10: (S 1 ) Shift(∆) is shifted; (S 2
Page 11 and 12: (S 1 ) Shift(∆) is shifted; (S 2
Page 13 and 14: Let ∆ be a simplicial complex on
Page 15 and 16: Let ∆ be a simplicial complex on
Page 17 and 18: A shifted complex which is obtained
Page 19 and 20: A shifted complex which is obtained
Page 21 and 22: Kalai’s squarefree operator Let K
Page 23 and 24: Kalai’s squarefree operator Let K
Page 25 and 26: Proof: Since gin
Page 27 and 28: Since the Betti number with the hig
Page 29 and 30: In order to define symmetric algebr
Page 31 and 32: In order to define symmetric algebr
Page 33 and 34: Corollary 1: Let I be a squarefree
Page 35 and 36: Corollary 1: Let I be a squarefree
Page 37 and 38: Proof: First one shows that G(I σ
Page 39 and 40: Proof: First one shows that G(I σ
Page 41: Again, since I is strongly stable,
Page 45 and 46: Lemma 3: If I ⊂ S is a strongly s
Page 47 and 48: Proposition 2: Let I be a strongly
Page 53 and 54: The proof of the theorem follows fr
Page 59 and 60: (b) ⇔ (c): We first observe that
Page 61 and 62: (b) ⇔ (c): We first observe that
Page 63 and 64: Problems: (a) Is it true that (∆
Page 69 and 70: To avoid the assumption char K = 0,
Page 71 and 72: To avoid the assumption char K = 0,
Page 73 and 74: One expects the following inequalit

Generic Initial Ideals Lecture 5

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