results for au:Floystad_G in:math

- We relate composition and substitution in pre- and post-Lie algebras to algebraic geometry. The Connes-Kreimer Hopf algebras, and MKW Hopf algebras are then coordinate rings of the infinite-dimensional affine varieties consisting of series of trees, resp.\ Lie series of ordered trees. Furthermore we describe the Hopf algebras which are coordinate rings of the automorphism groups of these varieties, which govern the substitution law in pre- and post-Lie algebras.
- We compute the deformation space of quadratic letterplace ideals $L(2,P)$ of finite posets $P$ when its Hasse diagram is a rooted tree. These deformations are unobstructed. The deformed family has a polynomial ring as the base ring. The ideal $J(2,P)$ defining the full family of deformations is a rigid ideal and we compute it explicitly. In simple example cases $J(2,P)$ is the ideal of maximal minors of a generic matrix, the Pfaffians of a skew-symmetric matrix, and a ladder determinantal ideal.
- The Chow form of the essential variety in computer vision is calculated. Our derivation uses secant varieties, Ulrich sheaves and representation theory. Numerical experiments show that our formula can detect noisy point correspondences between two images.
- We give the resolutions of co-letterplace ideals of posets in a completely explicit, very simple form. This generalizes and simplifies a number of linear resolutions in the literature, among them the Eliahou-Kervaire resolutions of strongly stable ideals generated in a single degree. Our method is based on a general result of K. Yanagawa using the canonical module of a Cohen-Macaulay Stanley-Reisner ring. We discuss in detail how the canonical module may effectively be computed, and from this derive directly the resolutions. A surprising consequence is that we obtain a large class of simplicial spheres comprehensively generalizing Bier spheres.
- Jan 13 2016 math.AC arXiv:1601.02792v2We investigate resolutions of letterplace ideals of posets. We develop topological results to compute their multigraded Betti numbers, and to give structural results on these Betti numbers. If the poset is a union of no more than $c$ chains, we show that the Betti numbers may be computed from simplicial complexes of no more than $c$ vertices. We also give a recursive procedure to compute the Betti diagrams when the Hasse diagram of $P$ has tree structure.
- This note surveys how the exterior algebra and deformations or quotients of it, gives rise to centrally important notions in five domains of mathematics: Combinatorics, Topology, Lie theory, Mathematical physics, and Algebraic geometry.
- Jan 20 2015 math.AC arXiv:1501.04523v3To a natural number $n$, a finite partially ordered set $P$ and a poset ideal ${\mathcal J}$ in the poset $Hom(P,[n])$ of isotonian maps from $P$ to the chain on $n$ elements, we associate two monomial ideals, the letterplace ideal $L(n,P;{\mathcal J})$ and the co-letterplace ideal $L(P,n;{\mathcal J})$. These ideals give a unified understanding of a number of ideals studied in monomial ideal theory in recent years. By cutting down these ideals by regular sequences of variable differences we obtain: multichain ideals and generalized Hibi type ideals, initial ideals of determinantal ideals, strongly stable ideals, $d$-partite $d$-uniform ideals, Ferrers ideals, edge ideals of cointerval $d$-hypergraphs, and uniform face ideals.
- Dec 18 2012 math.AC arXiv:1212.3675v3We 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 \citeFlTr. 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.
- Assume that $X= {x_1,...,x_g}$ is a finite alphabet and $K$ is a field. We study monomial algebras $A= K <X> /(W)$, where $W$ is an antichain of Lyndon words in $X$ of arbitrary cardinality. We find a Poincaré-Birkhoff-Witt type basis of $A$ in terms of its \emphLyndon atoms $N$, but, in general, $N$ may be infinite. We prove that if $A$ has polynomial growth of degree $d$ then $A$ has global dimension $d$ and is standard finitely presented, with $d-1 \leq |W| \leq d(d-1)/2$. Furthermore, $A$ has polynomial growth iff the set of Lyndon atoms $N$ is finite. In this case $A$ has a $K$-basis $\mathfrak{N} = {l_1^{\alpha_{1}}l_2^{\alpha_{2}}... l_d^{\alpha_{d}} \mid \alpha_{i} \geq 0, 1 \leq i \leq d}$, where $N = {l_1, ...,l_d}$. We give an extremal class of monomial algebras, the Fibonacci-Lyndon algebras, $F_n$, with global dimension $n$ and polynomial growth, and show that the algebra $F_6$ of global dimension 6 cannot be deformed, keeping the multigrading, to an Artin-Schelter regular algebra.
- Jul 10 2012 math.AC arXiv:1207.2071v3On 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.
- 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.
- Jun 03 2011 math.AC arXiv:1106.0381v2Boij-Söderberg theory describes the Betti diagrams of graded modules over the polynomial ring up to multiplication by a rational number. Analog Eisenbud-Schreyer theory also describes the cohomology tables of vector bundles on projective spaces up to rational multiple. We give an introduction and survey of these newly developed areas.
- Mar 24 2010 math.AC arXiv:1003.4495v1Let 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.
- We investigate the analogy between squarefree Cohen-Macaulay modules supported on a graph and line bundles on a curve. We prove a Riemann-Roch theorem, we study the Jacobian and gonality of a graph, and we prove Clifford's theorem.
- Jan 20 2010 math.AC arXiv:1001.3238v1We 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.
- Jan 20 2010 math.AC arXiv:1001.3235v2We 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.
- Nov 27 2009 math.RA arXiv:0911.5129v3We 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.
- 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.
- 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 ≤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.
- Oct 18 2007 math.AC arXiv:0710.3271v2For 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.
- 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.
- 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.
- May 27 2005 math.RA arXiv:math/0505570v2For a quotient algebra $U$ of the tensor algebra we give explicit conditions on its relations for $U$ being a PBW-deformation of an $N$-Koszul algebra $A$. We show there is a one-one correspondence between such deformations and a class of $A_\infty$-structures on the Yoneda algebra $Ext_A^*(k,k)$ of $A$. We compute the PBW-deformations of the algebra whose relations are the anti-symmetrizers of degree $N$ and also of cubic Artin-Schelter algebras.
- 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.
- 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.
- 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.
- 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.
- For the exterior algebra E over a vector space, we consider general maps E^a -> E(1)^b and general symmetric and skew-symmetric maps E^a -> E(1)^a and describe the associated exterior algebra resolutions. Using the theory of homogeneous bundles on homogeneous varieties we also describe the exterior algebra resolutions of much wider classes of maps E^a -> E(r)^b.
- In this paper we study the Bernstein-Gel'fand-Gel'fand (BGG) correspondence linking sheaves on a projective space to graded modules over an exterior algebra. We give an explicit construction of a Beilinson monad for a sheaf on projective space. The explicitness allows us to to prove two conjectures about the morphisms in the monad. We also construct all the monads for a sheaf that can be built from sums of line bundles. A large subclass, containing many monads introduced by others, are uniquely characterized by simple numerical data. Our methods also yield an efficient method for machine computation of the cohomology of sheaves. Along the way we study minimal free resolutions over an exterior algebra. For example, we show that such resolutions are eventually dominated by their "linear parts" in the sense that erasing all terms of degree $>1$ in the complex yields a new complex which is eventually exact. This paper is a joint, extended version of papers previously written by Eisenbud-Schreyer and by Fløystad seperately.
- Jan 01 2001 math.AG arXiv:math/0012263v2A new section on projections of coherent sheaves from a projective space to a lower-dimensional projective space has been added. Also some of the notation has been altered to bring it into line with the joint paper with Eisenbud and Schreyer.
- 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.
- This is a little investigation into the classification of complexes of direct sums of line bundles on projective spaces. We consider complexes on projective k-space Pk : O_Pk(-1)^a --> O_Pk^b --> O_Pk(1)^c, with the first map injective and the second map surjective. This is called a monad. We classify completely when such monads exist. Furthermore, whenever it exists we show that the first map may be assumed to degenerate in expected codimension, which is b-a-c+1.
- Let $S_d$ be the vector space of monomials of degree $d$ in the variables $x_1, ..., x_s$. For a subspace $V \sus S_d$ which is in general coordinates, consider the subspace $\gin V \sus S_d$ generated by initial monomials of polynomials in $V$ for the revlex order. We address the question of what properties of $V$ may be deduced from $\gin V$. % This is an approach for understanding what algebraic or geometric properties of a homogeneous ideal $I \sus k[x_1, ..., x_s]$ that may be deduced from its generic initial ideal $\gin I$.