arXiv subjects

Gunnar Floystad

Publications and source records attributed to Gunnar Floystad.

At least 19 recordsLinked to original sources

arXiv2015

Zipping Tate resolutions and exterior coalgebras

We conjecture what the cone of hypercohomology tables of bounded complexes of coherent sheaves on projective spaces are, when we have specified regularity conditions on the cohomology sheaves of this complex and its dual. There is an injection from the this cone into the cone of homological data sets of squarefree modules over a polynomial ring $\kk[x_1, \ldots, x_n]$, and we conjecture that this is an isomorphism: The Tate resolutions of a complex of coherent sheaves and the exterior coalgebra on $\langle x_1, \ldots, x_n \rangle$ may be amalgamated together to form a complex of free $\Sym(\oplus_i x_i \te W^*)$-modules, a procedure introduced by Cox and Materov. Via a reduction $\oplus_i x_i \te W^* \pil \oplus_i x_i \te \kk$ we get a complex of free modules over $\kk[x_1, \ldots, x_n]$ The extremal rays in the cone of squarefree complexes are conjecturally given by triplets of pure free squarefree complexes introduced in \cite{FlTr}. We describe the corresponding classes of hypercohomology tables, a class which generalizes vector bundles with supernatural cohomology. We also show how various pure resolutions in the literature, like resolutions of modules supported on determinantal varieties, and tensor complexes, may be obtained by the first part of the procedure.

math.AC↗

arXiv2014

Borel Degenerations of Arithmetically Cohen-Macaulay curves in P^3

We investigate Borel ideals on the Hilbert scheme components of arithmetically Cohen-Macaulay (ACM) codimension two schemes in P^n. We give a basic necessary criterion for a Borel ideal to be on such a component. Then considering ACM curves in P^3 on a quadric we compute in several examples all the Borel ideals on their Hilbert scheme component. Based on this we conjecture which Borel ideals are on such a component, and for a range of Borel ideals we prove that they are on the component.

math.AG↗

arXiv2012

Triplets of pure free squarefree complexes

On the category of bounded complexes of finitely generated free squarefree modules over the polynomial ring S, there is the standard duality functor D = Hom_S(-, omega_S) and the Alexander duality functor A. The composition AD is an endofunctor on this category, of order three up to translation. We consider complexes F of free squarefree modules such that both F, AD(F) and (AD)^2(F) are pure, when considered as singly graded complexes. We conjecture i) the existence of such triplets of complexes for given triplets of degree sequences, and ii) the uniqueness of their Betti numbers, up to scalar multiple. We show that this uniqueness follows from the existence, and we construct such triplets if two of them are linear.

math.AC↗

arXiv2011

Edge ideals: algebraic and combinatorial properties

Let C be a clutter and let I(C) be its edge ideal. This is a survey paper on the algebraic and combinatorial properties of R/I(C) and C, respectively. We give a criterion to estimate the regularity of R/I(C) and apply this criterion to give new proofs of some formulas for the regularity. If C is a clutter and R/I(C) is sequentially Cohen-Macaulay, we present a formula for the regularity of the ideal of vertex covers of C and give a formula for the projective dimension of R/I(C). We also examine the associated primes of powers of edge ideals, and show that for a graph with a leaf, these sets form an ascending chain.

math.AC↗

arXiv2010

Artin-Schelter regular algebras of dimension five

We show that there are exactly three types of Hilbert series of Artin-Schelter regular algebras of dimension five with two generators. One of these cases (the most extreme) may not be realized by an enveloping algebra of a graded Lie algebra. This is a new phenomenon compared to lower dimensions, where all resolution types may be realized by such enveloping algebras.

math.RA↗

arXiv2010

The Auslander-Reiten translate on monomial quotient rings

For a multidegree t in N^n, E.Miller has defined a category of positively t-determined modules over the polynomial ring S in n variables. We consider the Auslander-Reiten translate, Na_t, on the (derived) category of such modules. A monomial ideal I is positively t-determined if every generator x^a has a \leq t. We compute the multigraded cohomology- and betti spaces of Na_t^k(S/I) for every iterate k, and also the S-module structure of these cohomology modules. This comprehensively generalizes results of Hochster and Gräbe on local cohomology of Stanley-Reisner rings.

math.AC↗

arXiv2010

The colorful Helly theorem and colorful resolutions of ideals

We demonstrate that the topological Helly theorem and the algebraic Auslander-Buchsbaum may be viewed as different versions of the same phenomenon. Using this correspondence we show how the colorful Helly theorem of I.Barany and its generalizations by G.Kalai and R.Meshulam translates to the algebraic side. Our main results are algebraic generalizations of these translations, which in particular gives a syzygetic version of Hellys theorem.

math.AC↗

arXiv2010

The linear space of Betti diagrams of multigraded artinian modules

We study the linear space generated by the multigraded Betti diagrams of Z^n-graded artinian modules of codimension n whose resolutions become pure of a given type when taking total degrees. We show that the multigraded Betti diagram of the equivariant resolution constructed by D.Eisenbud, J.Weyman, and the author, and all its twists, form a basis for this linear space.

math.AC↗

arXiv2010

Gröbner bases of syzygies and Stanley depth

Let F. be a any free resolution of a Z^n-graded submodule of a free module over the polynomial ring K[x_1, ..., x_n]. We show that for a suitable term order on F., the initial module of the p'th syzygy module Z_p is generated by terms m_ie_i where the m_i are monomials in K[x_{p+1}, ..., x_n]. Also for a large class of free resolutions F., encompassing Eliahou-Kervaire resolutions, we show that a Gröbner basis for Z_p is given by the boundaries of generators of F_p. We apply the above to give lower bounds for the Stanley depth of the syzygy modules Z_p, in particular showing it is at least p+1. We also show that if I is any squarefree ideal in K[x_1, ..., x_n], the Stanley depth of I is at least of order the square root of 2n.

math.AC↗

arXiv2010

The cone of Betti diagrams of bigraded artinian modules of codimension two

We describe the positive cone generated by bigraded Betti diagrams of artinian modules of codimension two, whose resolutions become pure of a given type when taking total degrees. If the differences of these total degrees, p and q, are relatively prime, the extremal rays are parametrised by order ideals in N^2 contained in the region px + qy < (p-1)(q-1). We also consider some examples concerning artinian modules of codimension three.

math.AC↗

arXiv2010

The Existence of Pure Free Resolutions

Let d1,...,dn be a strictly increasing sequence of integers. Boij and Söderberg [arXiv:math/0611081] have conjectured the existence of a graded module M of finite length over any polynomial ring K[x_1,..., x_n], whose minimal free resolution is pure of type (d1,...,dn), in the sense that its i-th syzygies are generated in degree di. In this paper we prove a stronger statement, in characteristic zero: Such modules not only exist, but can be taken to be GL(n)-equivariant. In fact, we give two different equivariant constructions, and we construct pure resolutions over exterior algebras and Z/2-graded algebras as well. The constructions use the combinatorics of Schur functors and Bott's Theorem on the direct images of equivariant vector bundles on Grassmann varieties.

math.AC↗

arXiv2009

Cellular resolutions of Cohen-Macaulay monomial quotient rings

We investigate monomial labellings on cell complexes, giving a minimal cellular resolution of the ideal generated by these monomials, and such that the associated quotient ring is Cohen-Macaulay. We introduce a notion of such a labelling being maximal. There is only a finite number of maximal labellings for each cell complex, and we classify these for trees, partly for subdivisions of polygons, and for some classes of selfdual polytopes.

math.AC↗

arXiv2009

Geometric properties derived from generic initial spaces

For a vector space V of homogeneous forms of the same degree in a polynomial ring, we investigate what can be said about the generic initial ideal of the ideal generated by V, from the form of the generic initial space gin(V) for the revlex order. Our main result is a considerable generalisation of a previous result by the first author.

math.AC↗

arXiv2007

Enriched homology and cohomology modules of simplicial complexes

For a simplicial complex X on {1,2, ..., n} we define enriched homology and cohomology modules. They are graded modules over k[x_1, ..., x_n] whose ranks are equal to the dimensions of the reduced homology and cohomology groups. We characterize Cohen-Macaulay, l-Cohen-Macaulay, Buchsbaum, and Gorenstein* complexes X, and also orientable manifolds in terms of the enriched modules. We introduce the notion of girth for simplicial complexes and make a conjecture relating the girth to invariants of the simplicial complex. We also put strong vanishing conditions on the enriched homology modules and describe the simplicial complexes we then get. They are block designs and include Steiner systems S(c,d,n) and cyclic polytopes of even dimension.

math.CO↗

arXiv2006

Cohen-Macaulay cell complexes

We show that a finite regular cell complex with the intersection property is a Cohen-Macaulay space iff the top enriched cohomology module is the only nonvanishing one. We prove a comprehensive generalization of Balinski's theorem on convex polytopes. Also we show that for any Cohen-Macaulay cell complex as above, although there is no generalization of the Stanley-Reisner ring of simplicial complexes, there is a generalization of its canonical module.

math.CO↗

arXiv2004

(Bi-)Cohen-Macaulay simplicial complexes and their associated coherent sheaves

Via the BGG correspondence a simplicial complex Delta on [n] is transformed into a complex of coherent sheaves on P^n-1. We show that this complex reduces to a coherent sheaf F exactly when the Alexander dual Delta^* is Cohen-Macaulay. We then determine when both Delta and Delta^* are Cohen-Macaulay. This corresponds to F being a locally Cohen-Macaulay sheaf. Lastly we conjecture for which range of invariants of such Delta it must be a cone.

math.AG↗

arXiv2003

Hierarchies of simplicial complexes via the BGG-correspondence

Via the BGG-correspondence a simplicial complex D on [n] is transformed into a complex of coherent sheaves L(D) on the projective space n-1-space. In general we compute the support of each of its cohomology sheaves. When the Alexander dual D* is Cohen-Macaulay there is only one such non-zero cohomology sheaf. We investigate when this sheaf can be an a'th syzygy sheaf in a locally free resolution and show that this corresponds exactly to the case of D* being a+1-Cohen-Macaulay as defined by K.Baclawski. By putting further conditions on the sheaves we get nice subclasses of a+1- Cohen-Macaulay simplicial complexes whose f-vector depends only on a and the invariants n,d, and c. When a=0 these are the bi-Cohen-Macaulay simplicial complexes, when a=1 and d=2c cyclic polytopes are examples, and when a=c we get Alexander duals of the Steiner systems S(c,d,n). We also show that D* is Gorenstein* iff the associated coherent sheaf of D is an ideal sheaf.

math.CO↗

arXiv2003

Koszul duality and equivalences of categories

Let A and A! be dual Koszul algebras. By Positselski a filtered algebra U with gr U = A is Koszul dual to differential graded algebra (A!,d). We relate the module categories of this dual pair by a tensor-Hom adjunction. This descends to give an equivalence of suitable quotient categories and generalizes work of Beilinson, Ginzburg and Soergel.

math.RA↗