Structure Morphism Research Articles

On the bottleneck stability of rank decompositions of multi-parameter persistence modules

A significant part of modern topological data analysis is concerned with the design and study of algebraic invariants of poset representations—often referred to as persistence modules. One such invariant is the minimal rank decomposition, which encodes the ranks of all the structure morphisms of the persistence module by a single ordered pair of rectangle-decomposable modules, interpreted as a signed barcode. This signed barcode generalizes the concept of persistence barcode from one-parameter persistence to any number of parameters, raising the question of its bottleneck stability. We show in this paper that the minimal rank decomposition is not stable under the natural notion of signed bottleneck matching between signed barcodes. We remedy this by turning our focus to the rank exact decomposition, a related signed barcode induced by the minimal projective resolution of the module relative to the so-called rank exact structure, which we prove to be bottleneck stable under signed matchings. As part of our proof, we obtain two intermediate results of independent interest: we compute the global dimension of the rank exact structure on the category of finitely presentable multi-parameter persistence modules, and we prove a bottleneck stability result for hook-decomposable modules. We also give a bound for the size of the rank exact decomposition that is polynomial in the size of the usual minimal projective resolution, we prove a universality result for the dissimilarity function induced by the notion of signed matching, and we compute, in the two-parameter case, the global dimension of a different exact structure related to the upsets of the indexing poset. This set of results combines concepts from topological data analysis and from the representation theory of posets, and we believe is relevant to both areas.

Mixed Hodge Structures on Alexander Modules

Motivated by the limit mixed Hodge structure on the Milnor fiber of a hypersurface singularity germ, we construct a natural mixed Hodge structure on the torsion part of the Alexander modules of a smooth connected complex algebraic variety. More precisely, let U U be a smooth connected complex algebraic variety and let f : U → C ∗ f\colon U\to \mathbb {C}^* be an algebraic map inducing an epimorphism in fundamental groups. The pullback of the universal cover of C ∗ \mathbb {C}^* by f f gives rise to an infinite cyclic cover U f U^f of U U . The action of the deck group Z \mathbb {Z} on U f U^f induces a Q [ t ± 1 ] \mathbb {Q}[t^{\pm 1}] -module structure on H ∗ ( U f ; Q ) H_*(U^f;\mathbb {Q}) . We show that the torsion parts A ∗ ( U f ; Q ) A_*(U^f;\mathbb {Q}) of the Alexander modules H ∗ ( U f ; Q ) H_*(U^f;\mathbb {Q}) carry canonical Q \mathbb {Q} -mixed Hodge structures. We also prove that the covering map U f → U U^f \to U induces a mixed Hodge structure morphism on the torsion parts of the Alexander modules. As applications, we investigate the semisimplicity of A ∗ ( U f ; Q ) A_*(U^f;\mathbb {Q}) , as well as possible weights of the constructed mixed Hodge structures. Finally, in the case when f : U → C ∗ f\colon U\to \mathbb {C}^* is proper, we prove the semisimplicity and purity of A ∗ ( U f ; Q ) A_*(U^f;\mathbb {Q}) , and we compare our mixed Hodge structure on A ∗ ( U f ; Q ) A_*(U^f;\mathbb {Q}) with the limit mixed Hodge structure on the generic fiber of f f .

Para-Abelian Varieties and Albanese Maps

We construct for every proper algebraic space over a ground field an Albanese map to a para-abelian variety, which is unique up to unique isomorphism. This holds in the absence of rational points or ample sheaves, and also for reducible or non-reduced spaces, under the mere assumption that the structure morphism is in Stein factorization. It also works under suitable assumptions in families. In fact the treatment of the relative setting is crucial, even to understand the situation over ground fields. This also ensures that Albanese maps are equivariant with respect to actions of group schemes. Our approach depends on the notion of families of para-abelian varieties, where each geometric fiber admits the structure of an abelian variety, and representability of tau-parts in relative Picard groups, together with structure results on algebraic groups.

Group-theoreticity of numerical invariants and distinguished subgroups of configuration space groups

Let $\Sigma$ be a set of prime numbers which is either of cardinality one or equal to the set of all prime numbers. In this paper, we prove that various objects that arise from the geometry of the configuration space of a hyperbolic curve over an algebraically closed field of characteristic zero may be reconstructed group-theoretically from the pro-$\Sigma$ fundamental group of the configuration space. Let $X$ be a hyperbolic curve of type $(g,r)$ over a field $k$ of characteristic zero. Thus, $X$ is obtained by removing from a proper smooth curve of genus $g$ over $k$ a closed subscheme [i.e., the "divisor of cusps"] of $X$ whose structure morphism to Spec$(k)$ is finite étale of degree $r$; $2g-2+r>0$. Write $X_n$ for the $n$-th configuration space associated to $X$, i.e., the complement of the various diagonal divisors in the fiber product over $k$ of $n$ copies of $X$. Then, when $k$ is algebraically closed, we show that the triple $(n,g,r)$ and the generalized fiber subgroups—i.e., the subgroups that arise from the various natural morphisms $X_n \to X_m$ $[m$ < $n]$, which we refer to as generalized projection morphisms—of the pro-$\Sigma$ fundamental group $\Pi_n$ of $X_n$ may be reconstructed group-theoretically from $\Pi_n$ whenever $n \geq 2$. This result generalizes results obtained previously by the first and third authors and A. Tamagawa to the case of arbitrary hyperbolic curves [i.e., without restrictions on $(g,r)$]. As an application, in the case where $(g,r)= (0,3)$ and $n \geq 2$, we conclude that there exists a direct product decomposition Out$(\Pi_n) = {\mathrm{GT}^{\Sigma}} \times \mathfrak{S}_{n + 3}$ —where we write "Out$(-)$" for the group of outer automorphisms [i.e., without any auxiliary restrictions!] of the profinite group in parentheses and ${\mathrm{GT}^{\Sigma}}$ (respectively, $\mathfrak{S}_{n + 3}$) for the pro-$\Sigma$ Grothendieck-Teichmüller group (respectively, symmetric group on $n + 3$ letters). This direct product decomposition may be applied to obtain a simplified purely group-theoretic equivalent definition—i.e., as the centralizer in ${\rm Out}(\Pi_n)$ of the union of the centers of the open subgroups of ${\rm Out}(\Pi_n)$—of ${\mathrm{GT}^{\Sigma}}$. One of the key notions underlying the theory of the present paper is the notion of a pro-$\Sigma$ log-full subgroup—which may be regarded as a sort of higher-dimensional analogue of the notion of a pro-$\Sigma$ cuspidal inertia subgroup of a surface group—of $\Pi_n$. In the final section of the present paper, we show that, when $X$ and $k$ satisfy certain conditions concerning "weights", the pro-$l$ log-full subgroups may be reconstructed group-theoretically from the natural outer action of the absolute Galois group of $k$ on the geometric pro-$l$ fundamental group of $X_n$.

Stated skein algebras of surfaces

We study the algebraic and geometric properties of stated skein algebras of surfaces with punctured boundary. We prove that the skein algebra of the bigon is isomorphic to the quantum group $\mathcal{O}{{q^2}}(\operatorname{SL}(2))$ thus providing a topological interpretation for its structure morphisms. We also show that its stated skein algebra lifts in a suitable sense the Reshetikhin–Turaev functor, and in particular, we recover the dual $R$-matrix for $\mathcal{O}{{q^2}}(\operatorname{SL}(2)$ in a topological way. We deduce that the skein algebra of a surface with $n$ boundary components is a comodule algebra over $\mathcal{O}{{q^2}}(\operatorname{SL}(2))^{\otimes n}$ and prove that cutting along an ideal arc corresponds to Hochshild cohomology of bicomodules. We give a topological interpretation of braided tensor product of stated skein algebras of surfaces as “gluing on a triangle”; then we recover topologically some bialgebras in the category of $\mathcal{O}{{q^2}}(\operatorname{SL}(2))$-comodules, among which the “transmutation” of $\mathcal{O}\_{{q^2}}(\operatorname{SL}(2))$. We also provide an operadic interpretation of stated skein algebras as an example of a “geometric non-symmetric modular operad”. In the last part of the paper, we define a reduced version of stated skein algebras and prove that it allows to recover Bonahon–Wong's quantum trace map and interpret skein algebras in the classical limit when $q\to 1$ as regular functions over a suitable version of moduli spaces of twisted bundles.

On the standard conjecture for a -dimensional variety fibred by curves with a non-injective Kodaira–Spencer map

We prove that the Grothendieck standard conjecture of Lefschetz type holds for a complex projective 3-dimensional variety fibred by curves (possibly with degeneracies) over a smooth projective surface provided that the endomorphism ring of the Jacobian variety of some smooth fibre coincides with the ring of integers and the corresponding Kodaira–Spencer map has rank on some non-empty open subset of the surface. When the generic fibre of the structure morphism is of genus , the condition on the endomorphisms of the Jacobian may be omitted.

Formula omitted]-action for ribbon quasi-Hopf algebras

We study the universal Hopf algebra L of Majid and Lyubashenko in the case that the underlying ribbon category is the category of representations of a finite dimensional ribbon quasi-Hopf algebra A. We show that L=A⁎ with coadjoint action and compute the Hopf algebra structure morphisms of L in terms of the defining data of A. We give explicitly the condition on A which makes Rep A factorisable and compute Lyubashenko's projective SL(2,Z)-action on the centre of A in this case.The point of this exercise is to provide the groundwork for the applications to ribbon categories arising in logarithmic conformal field theories – in particular symplectic fermions and Wp-models – and to test a conjectural non-semisimple Verlinde formula.

Biwreaths: A self-contained system in a 2-category that encodes different known algebraic constructions and gives rise to new ones

We introduce bimonads in a 2-category K and define biwreaths as bimonads in the 2-category bEM(K) of bimonads, in the analogous fashion as Lack and Street defined wreaths. A biwreath is then a system containing a wreath, a cowreath and their mixed versions, but also a 2-cell λ in bEM(K) governing the compatibility of the monad and the comonad structure of the biwreath. We deduce that the monad laws encode 2-(co)cycles and the comonad laws so called 3-(co)cycles, while the 2-cell conditions of the (co)monad structure 2-cells of the biwreath encode (co)actions twisted by these 2- and 3-(co)cycles. The compatibilities of λ deliver concrete expressions of the latter structure 2-cells. We concentrate on the examples of biwreaths in the 2-category induced by a braided monoidal category C and take for the distributive laws in a biwreath the braidings of the different categories of Yetter–Drinfel'd modules in C. We prove that the before-mentioned properties of a biwreath specified to the latter setting recover on the level of C different algebraic constructions known in the category MR of modules over a commutative ring R, such as Radford biproduct, Sweedler's crossed product algebra, comodule algebras over a quasi-bialgebra and the Drinfel'd twist. In this way we obtain that the known examples of (mixed) wreaths coming from MR are not merely examples, rather they are consequences of the structure of a biwreath, and that the form of their structure morphisms originates in the laws inside of a biwreath. Choosing different distributive laws and different 2-cells λ in a biwreath, leads to different and possibly new algebraic constructions.

On an inductive approach to the standard conjecture for a fibred complex variety with strong semistable degeneracies

We prove that Grothendieck's standard conjecture of Lefschetz type on the algebraicity of the operators and of Hodge theory holds for a 4-dimensional smooth projective complex variety fibred over a smooth projective curve provided that every degenerate fibre is a union of smooth irreducible components of multiplicity with normal crossings, the standard conjecture holds for a generic geometric fibre , there is at least one degenerate fibre and the rational cohomology rings and of the irreducible components of every degenerate fibre are generated by classes of algebraic cycles. We obtain similar results for -dimensional fibred varieties with algebraic invariant cycles (defined by the smooth part of the structure morphism ) or with a degenerate fibre all of whose irreducible components possess the property .

Elliptic curves with complex multiplication and {u039B}-structures

This thesis examines the relationship between elliptic curves with complex multiplication and Lambda structures. Our main result is to show that the moduli stack of elliptic curves with complex multiplication, and the universal elliptic curve with complex multiplication over it, both admit Lambda structures and that the structure morphism is a Lambda morphism. This implies that elliptic curves with complex multiplication can be canonically lifted to the Witt vectors of the base (these are big and global Witt vectors). We also show that elliptic curves with complex multiplication of Shimura type are precisely those admitting Lambda structures and that a large class of these elliptic curves admit global minimal models. Along the way, we present a detailed study of families of elliptic curves with complex multiplication over arbitrary bases, give new derivations of the reciprocity maps associated to local fields and imaginary quadratic fields, construct a new flat, affine and pro-smooth rigidification of the moduli stack of elliptic curves with complex multiplication and exhibit a relationship between perfect Lambda schemes and periods, both $p$-adic and analytic.

A Category of Multiplier Bimonoids

The central object studied in this paper is a multiplier bimonoid in a braided monoidal category C. Adapting the philosophy of Janssen and Vercruysse, and making some mild assumptions on the category C, we consider a category M whose objects are certain semigroups in C and whose morphisms from A to B can be regarded as suitable multiplicative morphisms from A to the multiplier monoid of B. We equip this category M with a monoidal structure and describe multiplier bimonoids in C (whose structure morphisms belong to a distinguished class of regular epimorphisms) as certain comonoids in M. This provides us with one possible notion of morphism between such multiplier bimonoids.

Nonarchimedean geometry of Witt vectors

AbstractLetRbe a perfect 𝔽-algebra equipped with the trivial norm. LetW(R) be the ring ofp-typical Witt vectors overRequipped with thep-adic norm. At the level of nonarchimedean analytic spaces (in the sense of Berkovich), we demonstrate a close analogy betweenW(R) and the polynomial ringR[T] equipped with the Gauss norm, in which the role of the structure morphism fromRtoR[T] is played by the Teichmüller map. For instance, we show that the analytic space associated toRis a strong deformation retract of the space associated toW(R). We also show that each fiber forms a tree under the relation of pointwise comparison, and we classify the points of fibers in the manner of Berkovich’s classification of points of a nonarchimedean disk. Some results pertain to the study ofp-adic representations of étale fundamental groups of nonarchimedean analytic spaces (i.e., relativep-adic Hodge theory).

A geometric construction for permutation equivariant categories from modular functors

Let G be a finite group. Given a finite G-set $\cal{X}$ and a modular tensor category $\cal{C}$ , we construct a weak G-equivariant fusion category $\cal{C}^{\cal{X}}$ , called the permutation equivariant tensor category. The construction is geometric and uses the formalism of modular functors. As an application, we concretely work out a complete set of structure morphisms for $\mathbb{Z}/2$ -permutation equivariant categories, finishing thereby a program we initiated in an earlier paper.

NEW SYMPLECTIC V-MANIFOLDS OF DIMENSION FOUR VIA THE RELATIVE COMPACTIFIED PRYMIAN

Three new examples of 4-dimensional irreducible symplectic V-manifolds are constructed. Two of them are relative compactified Prymians of a family of genus-3 curves with involution, and the third one is obtained from a Prymian by Mukai's flop. They have the same singularities as two of Fujiki's examples, namely, 28 isolated singular points analytically equivalent to the Veronese cone of degree 8, but a different Euler number. The family of curves used in this construction forms a linear system on a K3 surface with involution. The structure morphism of both Prymians to the base of the family is a Lagrangian fibration in abelian surfaces with polarization of type (1,2). No example of such fibration is known on nonsingular irreducible symplectic varieties.

Hopfish algebras

We introduce a notion of structure on an associative algebra, allowing the structure morphisms (coproduct, counit, antipode) to be bimodules rather than algebra homomorphisms. We prove that quasi-Hopf algebras are examples of hopfish algebras. We find that a hopfish structure on the commutative algebra of functions on a finite set G is closely related to a hypergroupoid structure on G. The Morita theory of hopfish algebras is also discussed.

BRAUER GROUP INVARIANTS ASSOCIATED TO ORTHOGONAL EPSILON-CONSTANTS

In this paper, the theory of e-constants associated to tame finite group actions on arithmetic surfaces is used to define a Brauer group invariant µ(X ,G , V) associated to certain symplectic motives of weight one. The relationship between this invariant and w2(π) (the Galois-theoretic invariant associated to tame covers of surfaces defined by Cassou-Nogues, Erez and Taylor) is also discussed. In his paper (4), Deligne used elements in the Brauer group of Q and their relation- ship with certain e-constants to give a proof of the Frohlich-Queyrut theorem. In particular, he showed that certain global orthogonal root numbers are equal to one, by interpreting the associated local orthogonal root numbers as Stiefel-Whitney classes and then using the local root numbers to define an element of order two in the Brauer group of Q. This idea was furthered by Saito (in (13), for example) and others who defined Brauer group invariants associated to situations which can be interpreted as motives that are orthogonal and of even weight. In this paper, we define a Brauer group invariant associated to certain motives that are symplectic and have weight one. In order to construct the relevant motives, we first define X to be an arithmetic surface of dimension two which is flat, regular, and projective over Z. Throughout this paper, we assume that f : X− →Spec(Z) is the structure morphism. Let G be a finite group that acts tamely on X. In other words, for each closed point x ∈ X, the order of the inertia group of x is relatively prime to the residue characteristic of x .L etY be the quotient scheme X /G, which we assume is regular, and assume that for all finite places v, the fiber Yv =( Xv)/G = Y⊗ Z (Z/p(v)) has normal crossings and smooth irreducible components with multiplicities relatively prime to the residue characteristic of v. Finally, let V be a virtual representation of G over

Line bundles, rational points and ideal classes

In this note, we will use the term “arithmetic variety” for a normal scheme X for which the structure morphism f : X → Spec(Z) is proper and flat. Let V be a proper, normal (not necessarily geometrically connected) variety over Q. Let us choose a normal model for V over Z, that is an arithmetic variety X whose generic fiber is identified with V . Suppose that F is a number field and consider the F -rational points of V . These correspond bijectively to R-valued points of X, with R the ring of integers of F .I fP is an F -rational point of V , we will also denote by P : Spec(R) → X the corresponding R-valued point of X. Suppose that L is a line bundle on the arithmetic variety X. We say that L is trivial, when it is isomorphic to the structure sheaf OX. We will denote by P ∗ L the pull-back of L to Spec(R) via the morphism P ; then P ∗ L is a line bundle on Spec(R). It gives an element (P ∗ L) in the class group Pic(R )o fR. In what follows, we will identify Pic(R) with the ideal class group Cl(F ). This paper is motivated by the following question of the second named author: Question. Suppose that the line bundle L on X is not trivial. Is there a number field F and an F -rational point P of V such that the ideal class (P ∗ L) is not trivial? As a variant of this question, we could also ask: Is there a scheme Z which is finite and flat over Spec(Z) and a morphism P : Z → X such that (P ∗ L )i s not trivial in Pic(Z)? L. Szpiro has informed us that he independently raised this question earlier. If the answer to the question is always positive, then line bundles on arithmetic varieties are characterized by their restrictions to integral points. Here are some interesting facts about this question: 1. The answer is positive when X = P n . Indeed, restricting along any linear morphism P 1 → P n gives an isomorphism on Picard groups. Therefore, it is enough to show the statement for X = P 1 . Take L = OX(n), n � 0. There is a number field F with an ideal A whose ideal class is not n-torsion. We can always write A = aOF + bOF with a, b in OF . Consider the F -rational point (a; b )o fP 1 . This gives a morphism P :S pec(OF ) → P 1F → P 1 .

P-cyclic coverings of the affine space

Let A be a Noetherian unique factorization domain containing the prime field F, with p elements. We assume below that every finitely generated, projective A-module of rank p, p 1, or p 2 is free. Let S = Spec(A). We denote by G (or Z/pZ) a cyclic group of order p. A finite &ale covering X of S (with the structure morphism 9: X -+ S) is said to be p-cyclic if there is given a free action of the group G on X (acting as S-automorphisms) and the quotient scheme of X with respect to the G-action is isomorphic to S (cf. Mumford [l] for definitions). A p-cyclic covering 4: X + S is said to be of Artin-Schreier type if the coordinate ring R of X is A-isomorphic to an A-algebra A[y], where

Finite Subsghemes of Group Schemes

If G is an ordinary group and H is a non-empty subset of G, then there are two elementary criteria for H to be a subgroup of G. The first and more general is that the mapping H × H → G × G → G, via 〈x, y〉 ⟼ xy–1 factor through H. The second is that H be finite and closed under multiplication.In the category of group schemes, if one writes down the hypotheses for the first criterion in diagram form, one can supply the proof by a suitable translation of the classical arguments. The only point that causes any difficulty whatsoever is that one must assume that the structure morphism πH: H → S (S is the base scheme) is an epimorphism in order to factor the identity section through H. The second criterion is also true for group schemes under a mild finite presentation hypothesis. It is our aim to provide a simple proof for the following theorem.