Generic Initial Ideals Lecture 5

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Shifting operations Definition: A simplicial complex ∆ on [n] is shifted if, for F ∈ ∆, i ∈ F and j ∈ [n] with j > i, one has (F \ {i}) ∪ {j} ∈ ∆. Note that ∆ is shifted if and only if I ∆ is squarefree strongly stable.
Shifting operations Definition: A simplicial complex ∆ on [n] is shifted if, for F ∈ ∆, i ∈ F and j ∈ [n] with j > i, one has (F \ {i}) ∪ {j} ∈ ∆. Note that ∆ is shifted if and only if I ∆ is squarefree strongly stable. Definition: A shifting operation on [n] is a map which associates each simplicial complex ∆ on [n] with a simplicial complex Shift(∆) on [n] and which satisfies the following conditions:
Page 1 and 2: Generic Initial Ideals; Lecture 5 J
Page 3 and 4: Outline Shifting operations Kalai
Page 5: 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 and 42: Again, since I is strongly stable,
Page 43 and 44: Lemma 3: If I ⊂ S is a strongly s
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 55 and 56: The proof of the theorem follows fr
Page 57 and 58:
The proof of the theorem follows fr
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(b) ⇔ (c): We first observe that
Page 61 and 62:
(b) ⇔ (c): We first observe that
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Problems: (a) Is it true that (∆
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Page 67 and 68:
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To avoid the assumption char K = 0,
Page 71 and 72:
To avoid the assumption char K = 0,
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One expects the following inequalit
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Generic Initial Ideals Lecture 5

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