Lectures on the Geometry of Syzygies

Eisenbud, David; Sidman, Jessica

doi:10.1017/cbo9780511756382.005

Cited by 121 publications

(120 citation statements)

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“…The module of relations in turn must be finitely generated but may not be free, so a set of generators for it also obeys some relations, and so on. The Hilbert Syzygy Theorem [20,21], to be explained next, says that for the polynomial ring R this so-called chain of syzygies can be constructed so that it terminates in a bounded number of steps.…”

Section: Statement and Proof Of No-go Theoremmentioning

confidence: 99%

“…The Syzygy Theorem can be described in the following way [21] (the application to our problem is immediate): First, we note that a free module of any rank n say, is isomorphic (over the polynomial ring R) to the module consisting of all n-component row vectors with entries in R. Then giving a set of generators for a module M is equivalent to specifying a map (a homomorphism of modules over R) of a free module onto M . As our modules consist of spaces of row vectors with entries in R, this map φ 1 can be described as in the previous paragraph by a matrix that we also denote by φ 1 , whose rows are the generators.…”

Section: Statement and Proof Of No-go Theoremmentioning

confidence: 99%

“…The structure of such a free resolution of M is described further by the Buchsbaum-Eisenbud Theorem [21], which says that the rank (over R) of each map φ µ , which we write as rank R φ µ , obeys the usual rank-nullity theorem of linear algebra, which here reads…”

Section: Statement and Proof Of No-go Theoremmentioning

confidence: 99%

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Tensor network trial states for chiral topological phases in two dimensions and a no-go theorem in any dimension

2015

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Trial wavefunctions that can be represented by summing over locally-coupled degrees of freedom are called tensor network states (TNSs); they have seemed difficult to construct for two-dimensional topological phases that possess protected gapless edge excitations. We show it can be done for chiral states of free fermions, using a Gaussian Grassmann integral, yielding px ± ipy and Chern insulator states, in the sense that the fermionic excitations live in a topologically non-trivial bundle of the required type. We prove that any strictly short-range quadratic parent Hamiltonian for these states is gapless; the proof holds for a class of systems in any dimension of space. The proof also shows, quite generally, that sets of compactly-supported Wannier-type functions do not exist for band structures in this class. We construct further examples of TNSs that are analogs of fractional (including non-Abelian) quantum Hall phases; it is not known whether parent Hamiltonians for these are also gapless.

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Section: Statement and Proof Of No-go Theoremmentioning

confidence: 99%

Section: Statement and Proof Of No-go Theoremmentioning

confidence: 99%

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Tensor network trial states for chiral topological phases in two dimensions and a no-go theorem in any dimension

2015

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“…That is, P E ℓ is the subspace corresponding to the projective coordinates (X i,ℓ ) (i,ℓ)∈∆ (1) ∩Z 2 . Also consider the rational normal curves parameterized by ν ℓ : P 1 → P E ℓ as in (18), i.e. ∀x ∈ k * : ν ℓ (x) = (1 : x : .…”

Section: Scrollar Invariantsmentioning

confidence: 99%

Linear Pencils Encoded in the Newton Polygon

Castryck

Cools

2016

Int Math Res Notices

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Abstract. Let C be an algebraic curve defined by a sufficiently generic bivariate Laurent polynomial with given Newton polygon ∆. It is classical that the geometric genus of C equals the number of lattice points in the interior of ∆. In this paper we give similar combinatorial interpretations for the gonality, the Clifford index and the Clifford dimension, by removing a technical assumption from a recent result of Kawaguchi. More generally, the method shows that apart from certain well-understood exceptions, every base-point free pencil whose degree equals or slightly exceeds the gonality is combinatorial, in the sense that it corresponds to projecting C along a lattice direction. Along the way we prove various features of combinatorial pencils. For instance, we give an interpretation for the scrollar invariants associated to a combinatorial pencil, and show how one can tell whether the pencil is complete or not.Among the applications, we find that every smooth projective curve admits at most one Weierstrass semi-group of embedding dimension 2, and that if a non-hyperelliptic smooth projective curve C of genus g ≥ 2 can be embedded in the n th Hirzebruch surface H n , then n is actually an invariant of C.

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“…There are other equivalent definitions of local cohomology, which make some of its properties evident, for example viaČech complexes. For more information on local cohomology see [12,9,16,24]. Most authors do not treat the graded case explicitly, but the first two of these sources do.…”

Section: Castelnuovo-mumford Regularitymentioning

confidence: 99%

On the Castelnuovo-Mumford regularity of the cohomology ring of a group

Symonds

2010

J. Amer. Math. Soc.

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We record here one easily stated consequence. Proposition 0.3. For any non-trivial finite group, the cohomology ring His generated by elements in degree at most |G| − 1 and the relations between them (as a graded commutative algebra) are generated in degrees at most 2(|G| − 1). This bound is weak, although it can be improved somewhat at the cost of a more complicated formulation, but previously no such bound was known.We go on to prove some other conjectures of Benson on the regularity of the cohomology of other classes of groups [3], specifically for compact Lie groups and virtual Poincaré duality groups.The proof uses standard techniques in equivariant cohomology based on work of Quillen [30] and a paper of Duflot [14].David Green and Simon King have verified Benson's Conjecture computationally for all groups of order less than 256 [20].There is a survey of the properties of group cohomology from the point of view of commutative algebra by Benson [3]. Castelnuovo-Mumford regularityWe will work in the category of Z-graded rings and modules, so maps will respect the grading, elements will be homogeneous, etc., without specific mention. For a module M = i∈Z M i , we will write M ≥d = i≥d M i and similarly for other

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Lectures on the Geometry of Syzygies

Cited by 121 publications

References 44 publications

Tensor network trial states for chiral topological phases in two dimensions and a no-go theorem in any dimension

Tensor network trial states for chiral topological phases in two dimensions and a no-go theorem in any dimension

Linear Pencils Encoded in the Newton Polygon

On the Castelnuovo-Mumford regularity of the cohomology ring of a group

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