5,364 research outputs found

    An Alexandrov topology for maximal Cohen-Macaulay modules

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    Using the theory of cohomology annihilators, we define a family of topologies on the set of isomorphism classes of maximal Cohen-Macaulay modules over a Gorenstein ring. We study compactness of these topologies

    Classical and quantum algorithms for scaling problems

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    This thesis is concerned with scaling problems, which have a plethora of connections to different areas of mathematics, physics and computer science. Although many structural aspects of these problems are understood by now, we only know how to solve them efficiently in special cases.We give new algorithms for non-commutative scaling problems with complexity guarantees that match the prior state of the art. To this end, we extend the well-known (self-concordance based) interior-point method (IPM) framework to Riemannian manifolds, motivated by its success in the commutative setting. Moreover, the IPM framework does not obviously suffer from the same obstructions to efficiency as previous methods. It also yields the first high-precision algorithms for other natural geometric problems in non-positive curvature.For the (commutative) problems of matrix scaling and balancing, we show that quantum algorithms can outperform the (already very efficient) state-of-the-art classical algorithms. Their time complexity can be sublinear in the input size; in certain parameter regimes they are also optimal, whereas in others we show no quantum speedup over the classical methods is possible. Along the way, we provide improvements over the long-standing state of the art for searching for all marked elements in a list, and computing the sum of a list of numbers.We identify a new application in the context of tensor networks for quantum many-body physics. We define a computable canonical form for uniform projected entangled pair states (as the solution to a scaling problem), circumventing previously known undecidability results. We also show, by characterizing the invariant polynomials, that the canonical form is determined by evaluating the tensor network contractions on networks of bounded size

    Slopes of modular forms and geometry of eigencurves

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    Under a stronger genericity condition, we prove the local analogue of ghost conjecture of Bergdall and Pollack. As applications, we deduce in this case (a) a folklore conjecture of Breuil--Buzzard--Emerton on the crystalline slopes of Kisin's crystabelian deformation spaces, (b) Gouvea's ⌊k−1p+1⌋\lfloor\frac{k-1}{p+1}\rfloor-conjecture on slopes of modular forms, and (c) the finiteness of irreducible components of the eigencurve. In addition, applying combinatorial arguments by Bergdall and Pollack, and by Ren, we deduce as corollaries in the reducible and strongly generic case, (d) Gouvea--Mazur conjecture, (e) a variant of Gouvea's conjecture on slope distributions, and (f) a refined version of Coleman's spectral halo conjecture.Comment: 97 pages; comments are welcom

    Constructible sheaves on schemes

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    We present a uniform theory of constructible sheaves on arbitrary schemes with coefficients in topological or even condensed rings. This is accomplished by defining lisse sheaves to be the dualizable objects in the derived infinity-category of pro\'etale sheaves, while constructible sheaves are those that are lisse on a stratification. We show that constructible sheaves satisfy pro\'etale descent. We also establish a t-structure on constructible sheaves in a wide range of cases. We finally provide a toolset to manipulate categories of constructible sheaves with respect to the choices of coefficient rings, and use this to prove that our notions reproduce and extend the various approaches to, say, constructible ell-adic sheaves in the literature.Comment: This paper has been split off arXiv:2012.02853. Comments welcome

    Deformation theory of G-valued pseudocharacters and symplectic determinant laws

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    We give an introduction to the theory of pseudorepresentations of Taylor, Rouquier, Chenevier and Lafforgue. We refer to Taylor’s and Rouquier’s pseudorepresentations as pseudocharacters. They are very closely related, the main difference being that Taylor’s pseudocharacters are defined for a group, where as Rouquier’s pseudocharacters are defined for algebras. Chenevier’s pseudorepresentations are so-called polynomial laws and will be called determinant laws. Lafforgue’s pseudorepresentations are a generalization of Taylor’s pseudocharacters to other reductive groups G, in that the corresponding notion of representation is that of a G-valued representation of a group. We refer to them as G-pseudocharacters. We survey the known comparison theorems, notably Emerson’s bijection between Chenevier’s determinant laws and Lafforgue’s GL(n)-pseudocharacters and the bijection with Taylor’s pseudocharacters away from small characteristics. We show, that duals of determinant laws exist and are compatible with duals of representations. Analogously, we obtain that tensor products of determinant laws exist and are compatible with tensor products of representations. Further the tensor product of Lafforgue’s pseudocharacters agrees with the tensor product of Taylor’s pseudocharacters. We generalize some of the results of [Che14] to general reductive groups, in particular we show that the (pseudo)deformation space of a continuous Lafforgue G-pseudocharacter of a topologically finitely generated profinite group Γ with values in a finite field (of characteristic p) is noetherian. We also show, that for specific groups G it is sufficient, that Γ satisfies Mazur’s condition Ω_p. One further goal of this thesis was to generalize parts of [BIP21] to other reductive groups. Let F/Qp be a finite extension. In order to carry this out for the symplectic groups Sp2d, we obtain a simple and concrete stratification of the special fiber of the pseudodeformation space of a residual G-pseudocharater of Gal(F) into obstructed subloci Xdec(Θ), Xpair(Θ), Xspcl(Θ) of dimension smaller than the expected dimension n(2n + 1)[F : Qp]. We also prove that Lafforgue’s G-pseudocharacters over algebraically closed fields for possibly nonconnected reductive groups G come from a semisimple representation. We introduce a formal scheme and a rigid analytic space of all G-pseudocharacters by a functorial description and show, building on our results of noetherianity of pseudodeformation spaces, that both are representable and admit a decomposition as a disjoint sum indexed by continuous pseudocharacters with values in a finite field up to conjugacy and Frobenius automorphisms. At last, in joint work with Mohamed Moakher, we give a new definition of determinant laws for symplectic groups, which is based on adding a ’Pfaffian polynomial law’ to a determinant law which is invariant under an involution. We prove the expected basic properties in that we show that symplectic determinant laws over algebraically closed fields are in bijection with conjugacy classes of semisimple representation and that Cayley-Hamilton lifts of absolutely irreducible symplectic determinant laws to henselian local rings are in bijection with conjugacy classes of representations. We also give a comparison map with Lafforgue’s pseudocharacters and show that it is an isomorphism over reduced rings

    Periodicity of ideals of minors in free resolutions

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    We study the asymptotic behavior of the ideals of minors in minimal free resolutions over local rings. In particular, we prove that such ideals are eventually 2-periodic over complete intersections and Golod rings. We also establish general results on the stable behavior of ideals of minors in any infinite minimal free resolution. These ideals have intimate connections to trace ideals and cohomology annihilators. Constraints on the stable values attained by the ideals of minors in many situations are obtained, and they can be explicitly computed in certain cases.Comment: 28 page

    The local Picard group of a ring extension

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    Given an integral domain DD and a DD-algebra RR, we introduce the local Picard group LPic(R,D)\mathrm{LPic}(R,D) as the quotient between the Picard group Pic(R)\mathrm{Pic}(R) and the canonical image of Pic(D)\mathrm{Pic}(D) in Pic(R)\mathrm{Pic}(R), and its subgroup LPicu(R,D)\mathrm{LPic}_u(R,D) generated by the the integral ideals of RR that are unitary with respect to DD. We show that, when D⊆RD\subseteq R is a ring extension that satisfies certain properties (for example, when RR is the ring of polynomial D[X]D[X] or the ring of integer-valued polynomials Int(D)\mathrm{Int}(D)), it is possible to decompose LPic(R,D)\mathrm{LPic}(R,D) as the direct sum ⹁LPic(RT,T)\bigoplus\mathrm{LPic}(RT,T), where TT ranges in a Jaffard family of DD. We also study under what hypothesis this isomorphism holds for pre-Jaffard families of DD

    Diagonal F-splitting and Symbolic Powers of Ideals

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    Let JJ be any ideal in a strongly FF-regular, diagonally FF-split ring RR essentially of finite type over an FF-finite field. We show that Js+t⊆τ(Js−ϔ)τ(Jt−ϔ)J^{s+t} \subseteq \tau(J^{s - \epsilon}) \tau(J^{t-\epsilon}) for all s,t,Ï”>0s, t, \epsilon > 0 for which the formula makes sense. We use this to show a number of novel containments between symbolic and ordinary powers of prime ideals in this setting, which includes all determinantal rings and a large class of toric rings in positive characteristic. In particular, we show that P(2hn)⊆PnP^{(2hn)} \subseteq P^n for all prime ideals PP of height hh in such rings.Comment: Many small changes. Notably, I added missing Noetherianity and reducedness assumptions to section 2 and corrected an error in lemma 2.2. Upon reflection, the assumptions on AA and BB in prop 5.3 were just slightly more general than assuming AA and BB were fields, so I went ahead and said AA and BB should be field

    Interpolation Problems and the Characterization of the Hilbert Function

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    In mathematics, it is often useful to approximate the values of functions that are either too awkward and difficult to evaluate or not readily differentiable or integrable. To approximate its values, we attempt to replace such functions with more well-behaving examples such as polynomials or trigonometric functions. Over the algebraically closed field C, a polynomial passing through r distinct points with multiplicities m1, ..., mr on the affine complex line in one variable is determined by its zeros and the vanishing conditions up to its mi − 1 derivative for each point. A natural question would then be to consider the case in higher dimensions corresponding to polynomials in several variables. For this thesis, we will classify polynomials in three variables passing through a set of discrete points using an abstract algebraic structure known as an ideal. Then, we will analyze these ideals and specifically provide structural and numerical information. That is, we characterize their Hilbert Functions, which in our setting, are functions describing the number of linearly independent polynomials passing through the set of up to six points with a given multiplicity. Specifically, we will also see that in these cases, there is an expected ”maximal” Hilbert Function value, and the main goal is to determine whether these ideals have the ”maximal” Hilbert Function or not

    Stabilisation for varieties in polynomial functors

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