Stable Torsion Research Articles

Stable Torsion Length

Abstract The stable torsion length in a group is the stable word length with respect to the set of all torsion elements. We show that the stable torsion length vanishes in crystallographic groups. We then give a linear programming algorithm to compute a lower bound for stable torsion length in free products of groups. Moreover, we obtain an algorithm that exactly computes stable torsion length in free products of finite groups. The nature of the algorithm shows that stable torsion length is rational in this case. As applications, we give the 1st exact computations of stable torsion length for nontrivial examples.

Stable Torsion Torque Control Based on Friction-Backlash Analogy for Two-Inertia Systems with Backlash

Stable Torsion Torque Control Based on Friction-Backlash Analogy for Two-Inertia Systems with Backlash

Toric Sheaves on Hirzebruch Orbifolds

We provide a stacky fan description of the total space of certain split vector bundles, as well as their projectivization, over toric Deligne-Mumford stacks. We then specialize to the case of Hirzebruch orbifold $\mathcal{H}_{r}^{ab}$ obtained by projectivizing $\mathcal{O} \oplus \mathcal{O}(r)$ over the weighted projective line $\mathbb{P}(a,b)$. Next, we give a combinatorial description of toric sheaves on $\mathcal{H}_{r}^{ab}$ and investigate their basic properties. With fixed choice of polarization and a generating sheaf, we describe the fixed point locus of the moduli scheme of $\mu$-stable torsion free sheaves of rank $1$ and $2$ on $\mathcal{H}_{r}^{ab}$. As an example, we obtain explicit formulas for generating functions of Euler characteristics of locally free sheaves of rank 2 on $\mathbb{P}(1,2) \times \mathbb{P}^1$.

Higher rank sheaves on threefolds and functional equations

We consider the moduli space of stable torsion free sheaves of any rank on a smooth projective threefold. The singularity set of a torsion free sheaf is the locus where the sheaf is not locally free. On a threefold it has dimension $\leq 1$. We consider the open subset of moduli space consisting of sheaves with empty or 0-dimensional singularity set. For fixed Chern classes $c_1,c_2$ and summing over $c_3$, we show that the generating function of topological Euler characteristics of these open subsets equals a power of the MacMahon function times a Laurent polynomial. This Laurent polynomial is invariant under $q \leftrightarrow q^{-1}$ (upon replacing $c_1 \leftrightarrow -c_1$). For some choices of $c_1,c_2$ these open subsets equal the entire moduli space. The proof involves wall-crossing from Quot schemes of a higher rank reflexive sheaf to a sublocus of the space of Pandharipande-Thomas pairs. We interpret this sublocus in terms of the singularities of the reflexive sheaf.

Birkhoff's theorem for stable torsion theories

We present a novel approach to establish the Birkhoff's theorem validity in the so-called quadratic Poincaré Gauge theories of gravity. By obtaining the field equations via the Palatini formalism, we find paradigmatic scenarios where the theorem applies neatly. For more general and physically relevant situations, a suitable decomposition of the torsion tensor also allows us to establish the validity of the theorem. Our analysis shows rigorously how for all stable cases under consideration, the only solution of the vacuum field equations is a torsionless Schwarzschild spacetime, although it is possible to find non-Schwarzschild metrics in the realm of unstable Lagrangians. Finally, we study the weakened formulation of the Birkhoff's theorem where an asymptotically flat metric is assumed, showing that the theorem also holds.

Weakly Stable Torsion Classes

Weakly stable torsion classes were introduced by the author and Yekutieli to provide a torsion theoretic characterisation of the notion of weak proregularity from commutative algebra. In this paper we investigate weakly stable torsion classes, with a focus on aspects related to localisation and completion. We characterise when torsion classes arising from left denominator sets and idempotent ideals are weakly stable. We show that every weakly stable torsion class $\operatorname{\mathsf{T}}$ can be associated with a dg ring $A_{\operatorname{\mathsf{T}}}$; in well behaved situations there is a homological epimorphism $A\to A_{\operatorname{\mathsf{T}}}$. We end by studying torsion and completion with respect to a single regular and normal element.

Generalized twisted cubics on a cubic fourfold as a moduli space of stable objects

We revisit the work of Lehn–Lehn–Sorger–van Straten on twisted cubic curves in a cubic fourfold not containing a plane in terms of moduli spaces. We show that the blow-up Z′ along the cubic of the irreducible holomorphic symplectic eightfold Z, described by the four authors, is isomorphic to an irreducible component of a moduli space of Gieseker stable torsion sheaves or rank three torsion free sheaves.For a very general such cubic fourfold, we show that Z is isomorphic to a connected component of a moduli space of tilt-stable objects in the derived category and to a moduli space of Bridgeland stable objects in the Kuznetsov component. Moreover, the contraction between Z′ and Z is realized as a wall-crossing in tilt-stability.Finally, Z is birational to an irreducible component of a moduli space of Gieseker stable aCM bundles of rank six.

Rank 2 Sheaves on Toric 3-Folds: Classical and Virtual Counts

Let $\mathcal{M}$ be the moduli space of rank 2 stable torsion free sheaves with Chern classes $c_i$ on a smooth 3-fold $X$. When $X$ is toric with torus $T$, we describe the $T$-fixed locus of the moduli space. Connected components of $\mathcal{M}^T$ with constant reflexive hulls are isomorphic to products of $\mathbb{P}^1$. We mainly consider such connected components, which typically arise for any $c_1$, "low values" of $c_2$, and arbitrary $c_3$. In the classical part of the paper, we introduce a new type of combinatorics called double box configurations, which can be used to compute the generating function $\mathsf{Z}(q)$ of topological Euler characteristics of $\mathcal{M}$ (summing over all $c_3$). The combinatorics is solved using the double dimer model in a companion paper. This leads to explicit formulae for $\mathsf{Z}(q)$ involving the MacMahon function. In the virtual part of the paper, we define Donaldson-Thomas type invariants of toric Calabi-Yau 3-folds by virtual localization. The contribution to the invariant of an individual connected component of the $T$-fixed locus is in general not equal to its signed Euler characteristic due to $T$-fixed obstructions. Nevertheless, the generating function of all invariants is given by $\mathsf{Z}(q)$ up to signs.

Stable reflexive sheaves and localization

We study moduli spaces N of rank 2 stable reflexive sheaves on P3. Fixing Chern classes c1, c2, and summing over c3, we consider the generating function Zrefl(q) of Euler characteristics of such moduli spaces. The action of the torus T on P3 lifts to N and we classify all sheaves in NT. This leads to an explicit expression for Zrefl(q). Since c3 is bounded below and above, Zrefl(q) is a polynomial. We find a simple formula for its leading term when c1=−1.Next, we study moduli spaces of rank 2 stable torsion free sheaves on P3 and consider the generating function of Euler characteristics of such moduli spaces. We give an expression for this generating function in terms of Zrefl(q) and Euler characteristics of Quot schemes of certain T-equivariant reflexive sheaves, which are studied elsewhere. Many techniques of this paper apply to any toric 3-fold. In general, Zrefl(q) depends on the choice of polarization which leads to wall-crossing phenomena. We briefly illustrate this in the case of P2×P1.

Stable Sheave Moduli of Rank 2 with Chern Classes c 1 = -1; c2 = 2; c3 = 0 on Q3

Рассматривается схема MQ( 2;-1; 2; 0 ) модулей стабильных пучков ранга 2 без кручения с классами Черна c1 = -1; c2 = 2; c3 = 0 на гладкой трехмерной проективной квадрике Q. МногообразиеMQ(-1; 2) модулей расслоений ранга 2 с классами Черна c1 = -1, c2 = 2 на Q было изучено Оттавиани и Шуреком в 1994 г. В 2007 г. автором было получено описание замыкания многообразия MQ (-1; 2) в схеме MQ(2;-1; 2; 0). В настоящей статье доказывается, что в MQ(2;-1; 2; 0) существует единственная неприводимая компонента, отличная от MQ(-1; 2), являющаяся рациональным многообразием размерности 10.

Quasi-static micromirror with enlarged deflection based on aluminum nitride thin film springs

Uniaxial electrostatically driven micromirrors with very large non-resonant rotation angles are presented. The mirrors achieve an analog deflection of about ±12° at voltages between 200 and 320V. At pull-in, a digital tilt angle of approx. ±21° is found. A mirror control for a nearly linear characteristic curve, by using a counter torque is approved. The key for large quasi-static deflections are novel aluminum nitride based thin film springs. The high mechanical strength of AlN enables the fabrication of thin but stable torsion springs. The mirror has a size of 1mm×1.2mm with a bending of less than 0.2μm and high surface quality (Ra<2nm). The rotational eigenfrequencies are found to be between 37 and 73Hz and the eigenfrequencies of the vertical mirror movement are between 1235 and 1751Hz.

Flips of moduli of stable torsion free sheaves with $c_1=1$ on $P^2$

Flips of moduli of stable torsion free sheaves with $c_1=1$ on $P^2$

Stability of Gieseker stable sheaves on K3 surfaces in the sense of Bridgeland and some applications

We show that some Gieseker stable sheaves on a projective K3 surface $X$ are stable with respect to a stability condition of Bridgeland on the derived category of $X$ if the stability condition is in explicit subsets of the space of stability conditions depending on the sheaves. Furthermore we shall give two applications of the result. As a part of these applications, we show that the fine moduli space of Gieseker stable torsion free sheaves on a K3 surface with Picard number one is the moduli space of $\mu$-stable locally free sheaves if the rank of the sheaves is not a square number.

DECOMPOSITIONS OF MODULES SUPPLEMENTED RELATIVE TO A TORSION THEORY

Let R be a ring, M a right R-module and a hereditary torsion theory in Mod-R with associated torsion functor τ for the ring R. Then M is called τ-supplemented when for every submodule N of M there exists a direct summand K of M such that K ≤ N and N/K is τ-torsion module. In [4], M is called almost τ-torsion if every proper submodule of M is τ-torsion. We present here some properties of these classes of modules and look for answers to the following questions posed by the referee of the paper [4]: (1) Let a module M = M′ ⊕ M″ be a direct sum of a semisimple module M′ and τ-supplemented module M″. Is M τ-supplemented? (2) Can one find a non-stable hereditary torsion theory τ and τ-supplemented modules M′ and M″ such that M′ ⊕ M″ is not τ-supplemented? (3) Can one find a stable hereditary torsion theory τ and a τ-supplemented module M such that M/N is not τ-supplemented for some submodule N of M? (4) Let τ be a non-stable hereditary torsion theory and the module M be a finite direct sum of almost τ-torsion submodules. Is M τ-supplemented? (5) Do you know an example of a torsion theory τ and a τ-supplemented module M with τ-torsion submodule τ(M) such that M/τ(M) is not semisimple?

On stable torsion length of a Dehn twist

On stable torsion length of a Dehn twist

Hereditary torsion theory of pseudo-regular S-systems

The concept of pseudo-regular S-system is introduced. A torsion theory \(\tau _u = \left( {U_S ,\bar U_S } \right)\) with its torsion class US consisting of pseudo-regular S-systems, is constructed. Its corresponding quasi-filter is completely described. A number of results on the relations between τu and two special torsion theories, the stable and Lambek torsion theories, are obtained.

Degenerations of the moduli spaces of vector bundles on curves II (generalized Gieseker moduli spaces)

LetX 0 be a projective curve whose singularity is one ordinary double point. We construct a birational modelG(n, d) of the moduli spaceU(n, d) of stable torsion free sheaves in the case (n, d)= 1, such that G(n, d) has normal crossing singularities and behaves well under specialization i.e. if a smooth projective curve specializes toX 0, then the moduli space of stable vector bundles of rankn and degreed onX specializes toG(n, d). This generalizes an earlier work of Gieseker in the rank two case.

Stable sheaves on reduced projective curves

Let X be a reduced projective curve; assume X either irreducible or stable. Here we study some geometric properties of stable vector bundles and stable torsion free sheaves on X (essentially, the existence of stable objects with a prescribed order of stability).

Stable torsion radicals over noetherian rings

Stable torsion radicals over noetherian rings

Stable torsion radicals over FBN rings

Gabriel [9] called a localizing subcategory stable if it is closed under injective envelopes. (In this case, the associated torsion radical or torsion theory is also said to be stable.) He showed that over a commutative Noetherian ring R every localizing subcategory of R-Mod is stable. He noted the connections between stability and the Artin-Rees property for ideals, and considered the stability of the localizing subcategory generated by the class of simple modules. Papp [18] studied Noetherian rings for which every localizing subcategory is stable, and Louden [16] utilized stable torsion radicals to define a sheaf over the spectrum of an FBN ring. This paper gives a characterization of stable torsion radicals over left FBN rings. The equivalent conditions stated in Theorem 1.2 exhibit the connections with the Artin-Rees property and with the notion of ideal invariance introduced by Robson [19]. Over an FBN ring, there is a natural correspondence between torsion radicals in R-Mod and those in Mod-R. In Theorem 1.6 it is shown that corresponding torsion radicals are both stable if and only if they are both ideal invariant, and that such torsion radicals correspond to the biradicals defined by Jategaonkar [12], and thus to the link-closed hereditary sets of prime ideals in the sense of Miiller [17]. Finally, it is shown that every torsion radical over an FBN ring is stable if and only if every prime ideal is localizable. In the second section of the paper, applications are given to the study of certain specific torsion radicals. For a left FBN ring, necessary and sufficient conditions are given under which any finitely generated essential extension of a module of Krull dimension at most a again has Krull dimension at most a. These conditions can be checked easily for an FBN ring or for a left Noetherian ring integral over its center, yielding a unified approach to certain results of Jategaonkar [13] and Chamarie and Hudry [5]. Finally, the results in the first section are applied to determine several conditions under which finitely generated essential extensions of Artinian modules are again Artinian, extending a result of Ginn and Moss [lo].