The Linear Syzygies of Generic Forms

Hochstes, Melvin; Laksov, Dan

doi:10.1080/00927872.1987.10487449

Cited by 46 publications

(50 citation statements)

References 5 publications

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“…In this section we strengthen Corollary 3.3 in a form that is analogous to the result of Hochster and Laksov [4]. To help the reader see the analogy we shall first state their result.…”

Section: Corollaries and Examplesmentioning

confidence: 75%

“…A result of Hochster and Laksov [4] says that if i 2 and V ⊂ R i is a general subspace then the natural multiplication map from V ⊗ R 1 to R i+1 has maximal rank, that is, it is either injective or surjective, and it is not known what happens if one replaces R 1 by R d for d > 1. One may wonder which other graded rings have a similar property.…”

Section: Introductionmentioning

confidence: 98%

See 1 more Smart Citation

Almost split morphisms, preprojective algebras and multiplication maps of maximal rank

Diaz¹,

Kleiner²

2007

Journal of Algebra

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With a grading previously introduced by the second-named author, the multiplication maps in the preprojective algebra satisfy a maximal rank property that is similar to the maximal rank property proven by Hochster and Laksov for the multiplication maps in the commutative polynomial ring. The result follows from a more general theorem about the maximal rank property of a minimal almost split morphism, which also yields a quadratic inequality for the dimensions of indecomposable modules involved.

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“…In this section we strengthen Corollary 3.3 in a form that is analogous to the result of Hochster and Laksov [4]. To help the reader see the analogy we shall first state their result.…”

Section: Corollaries and Examplesmentioning

confidence: 75%

Section: Introductionmentioning

confidence: 98%

Almost split morphisms, preprojective algebras and multiplication maps of maximal rank

Diaz¹,

Kleiner²

2007

Journal of Algebra

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“…In contrast, Corollary 4.4 and Theorem 5.4 at least suggest -and prove some cases -of the following conjecture: Note that the first syzygy case of this conjecture was proved by M. Hochster and D. Laksov [19].…”

Section: By the Weak Lefschetz Property The Hilbert Function Of A/la Ismentioning

confidence: 96%

“…The case n = 3 was solved by D. Anick [1]. M. Hochster and D. Laksov [19] showed that a generically chosen set of forms of the same degree span as much as possible in the next degree. Note that this gives the value of the Hilbert function in the next degree, and it also gives the number of linear syzygies of the forms.…”

Section: Introductionmentioning

confidence: 99%

On the minimal free resolution of $n+1$ general forms

Migliore

Miró‐Roig²

2002

Trans. Amer. Math. Soc.

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. . . , xn] and let I be the ideal of n + 1 generically chosen forms of degreesWe give the precise graded Betti numbers of R/I in the following cases:• n = 3;• n = 4 and• n is even and all generators have the same degree, a, which is even;0. We give very good bounds on the graded Betti numbers in many other cases. We also extend a result of M. Boij by giving the graded Betti numbers for a generic compressed Gorenstein algebra (i.e., one for which the Hilbert function is maximal, given n and the socle degree) when n is even and the socle degree is large. A recurring theme is to examine when and why the minimal free resolution may be forced to have redundant summands. We conjecture that if the forms all have the same degree, then there are no redundant summands, and we present some evidence for this conjecture.

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“…(Fröberg's Conjecture, 1985 Conjecture 2.2 has been proven in the following cases: for r ≤ n (easy exercise, since in this case I is a complete intersection); for n ≤ 2, Fröberg (1985); for n = 3, Anick (1986), for r = n + 1, which follows from Stanley (1978). Additionally, in Hochster and Laksov (1987) it has been proven that (6) is correct in the first nontrivial degree min r i=1 (d i + 1). There are also other special results in the case d 1 = · · · = d r , see Fröberg and Hollman (1994), Aubry (1995), Migliore and Miro-Roig (2003), Nicklasson (2017a), Nenashev (2017).…”

Section: Definition 21 Given a Homogeneous Ideal I ⊂ S We Call The mentioning

confidence: 99%