Variation of geometric invariant theory quotients

Geometric Invariant Theory gives a method for constructing quotients for group actions on algebraic varieties which in many cases appear as moduli spaces parameterizing isomorphism classes of geometric objects (vector bundles, polarized varieties, etc.). The quotient depends on a choice of an ample linearized line bundle. Two choices are equivalent if they give rise to identical quotients. A priori, there are infinitely many choices since there are infinitely many isomorphism classes of linearized ample line bundles. Hence several natural questions arise. Is the set of equivalence classes, and hence the set of non-isomorphic quotients, finite? How does the quotient vary under change of the equivalence class? In this paper we give partial answers to these questions in the case of actions of reductive algebraic groups on nonsingular projective algebraic varieties. We shall show that among ample line bundles which give projective geometric quotients there are only finitely many equivalence classes. These classes span certain convex subsets (chambers) in a certain convex cone in Euclidean space, and when we cross a wall separating one chamber from another, the corresponding quotient undergoes a birational transformation which is similar to a Mori flip.     

On Drinfeld's DG quotient

The possibility of constructing quotients of differential graded (= dg) categories is essential in non-commutative algebraic geometry. The first construction of dg quotients appeared in Keller's work (Keller (1994) [21]) and it was recently followed by Drinfeld's elegant approach (Drinfeld (2004) [9]). Although Drinfeld's dg quotient admits a very simple construction, it didn't seem to be intrinsically defined. In this article we complete this aspect of Drinfeld's work by providing three different characterizations of Drinfeld's dg quotient in terms of simple universal properties.     

Free operated monoids and rewriting systems

The construction of bases for quotients is an important problem. In this paper, applying the method of rewriting systems, we give a unified approach to construct sections—an alternative name for bases in semigroup theory—for quotients of free operated monoids. As applications, we capture sections of free \(*\)-monoids and free groups, respectively.     

Continued fractions for certain algebraic power series

Let F be an arbitrary field and let K = F((x⁻¹)) be the field of formal Laurent series in x⁻¹ over F. The usual theory of continued fractions carries over to K, with the polynomials in x playing the role of the integers. We study the continued fraction expansions of elements of K which are algebraic over F(x), the field of rational functions of x.We give the first explicit expansions of algebraic elements of degree greater than 2 for which the degrees of the partial quotients are bounded. In particular we give explicitly the continued fraction expansion for the solution f in K of the cubic equation xf³ + f + x = 0 when F = GF(2). This cubic was studied by Baum and Sweet. We give examples, for every field F of characteristic greater than 2, of algebraic elements of degree greater than 2 whose partial quotients are all linear, and we give these expansions explicitly. These are the first known examples with partial quotients of bounded degree when F has characteristic greater than 2.     

The Poincaré series of multiplier ideals of a simple complete ideal in a local ring of a smooth surface

For a simple complete ideal ℘ of a local ring at a closed point on a smooth complex algebraic surface, we introduce an algebraic object, named Poincaré series P_℘, that gathers in a unified way the jumping numbers and the dimensions of the vector space quotients given by consecutive multiplier ideals attached to ℘. This paper is devoted to prove that P_℘ is a rational function giving an explicit expression for it.     

Algebraic quantum hypergroups

An algebraic quantum group is a regular multiplier Hopf algebra with integrals. In this paper we will develop a theory of algebraic quantum hypergroups. It is very similar to the theory of algebraic quantum groups, except that the comultiplication is no longer assumed to be a homomorphism. We still require the existence of a left and of a right integral. There is also an antipode but it is characterized in terms of these integrals. We construct the dual, just as in the case of algebraic quantum groups and we show that the dual of the dual is the original quantum hypergroup. We define algebraic quantum hypergroups of compact type and discrete type and we show that these types are dual to each other. The algebraic quantum hypergroups of compact type are essentially the algebraic ingredients of the compact quantum hypergroups as introduced and studied (in an operator algebraic context) by Chapovsky and Vainerman.We will give some basic examples in order to illustrate different aspects of the theory. In a separate note, we will consider more special cases and more complicated examples. In particular, in that note, we will give a general construction procedure and show how known examples of these algebraic quantum hypergroups fit into this framework.     

Equivariant primary decomposition and toric sheaves

We study global primary decompositions in the category of sheaves on a scheme which are equivariant under the action of an algebraic group. We show that equivariant primary decompositions exist if the group is connected. As main application we consider the case of varieties which are quotients of a quasi-affine variety by the action of a diagonalizable group and thus admit a homogeneous coordinate ring, such as toric varieties. Comparing these decompositions with primary decompositions of graded modules over the homogeneous coordinate ring, we show that these are equivalent if the action of the diagonalizable group is free. We give some specific examples for the case of toric varieties.     

Jacobian quotients, an algebraic proof

We give an algebraic proof of a theorem of H. Maugendre showing how the jacobian quotients of a pair of germs of plane curve may be computed from their simultaneous immerged resolution, thus proving in particular their topological invariance.     

A Method of Representing Rough Sets System Determined by Quasi Orders

In this paper, we study about the ordered structure of rough sets determined by a quasi order. A characterization theorem for rough sets of an approximation space (U, R) based on a quasi order R is given in Nagarajan and Umadevi (2010). Then using the characterization of rough sets determined by a quasi order, its rough sets system is represented by a new construction. This construction is generalized and abstracted into a new method of constructing Kleene based algebraic structures from dually isomorphic distributive lattices. Then by using different varieties of distributive lattices, we obtain various Kleene based algebraic structures. By this construction, we give various algebraic structures to the rough sets system determined by a quasi order R.     

Algebraic lattices and locally finitely presentable categories

We show that subobjects and quotients respectively of any object K in a locally finitely presentable category form an algebraic lattice. The same holds for the internal equivalence relations on K. In fact, these results turn out to be??at least in the case of subobjects??nothing but simple consequences of well known closure properties of the classes of locally finitely presentable categories and accessible categories, respectively. We thus get a completely categorical explanation of the well known fact that the subobject- and congruence lattices of algebras in finitary varieties are algebraic. Moreover we also obtain new natural examples: in particular, for any (not necessarily finitary) polynomial set-functor F, the subcoalgebras of an F-coalgebra form an algebraic lattice; the same holds for the lattices of regular congruences and quotients of these F-coalgebras.     

On identities in orthocomplemented difference lattices

In this note we continue the investigation of algebraic properties of orthocomplemented (symmetric) difference lattices (ODLs) as initiated and previously studied by the authors. We take up a few identities that we came across in the previous considerations. We first see that some of them characterize, in a somewhat non-trivial manner, the ODLs that are Boolean. In the second part we select an identity peculiar for set-representable ODLs. This identity allows us to present another construction of an ODL that is not set-representable. We then give the construction a more general form and consider algebraic properties of the ‘orthomodular support’.     

Residues of a Pfaff system relative to an invariant subscheme

In this paper we give a purely algebraic construction of the theory of residues of a Pfaff system relative to an invariant subscheme. This construction is valid over an arbitrary base scheme of any characteristic.

     

Proving the triviality of rational points on Atkin–Lehner quotients of Shimura curves

In this paper we give a method for studying global rational points on certain quotients of Shimura curves by Atkin–Lehner involutions. We obtain explicit conditions on such quotients for rational points to be “trivial” (coming from CM points only) and exhibit an explicit infinite family of such quotients satisfying these conditions.     

Additive Hilbert's Theorem 90 in the ring of algebraic integers

For a number field K, we give a complete characterization of algebraic numbers which can be expressed by a difference of two K-conjugate algebraic integers. These turn out to be the algebraic integers whose Galois group contains an element, acting as a cycle on some collection of conjugates which sum to zero. Hence there are no algebraic integers which can be written as a difference of two conjugate algebraic numbers but cannot be written as a difference of two conjugate algebraic integers. A generalization of the construction to a commutative ring is also given. Furthermore, we show that for n ?_ 3 there exist algebraic integers which can be written as a linear form in n K-conjugate algebraic numbers but cannot be written by the same linear form in K-conjugate algebraic integers.     

A Note on Reflected Dirichlet Forms

Potential Analysis - In this paper we give an algebraic construction of the (active) reflected Dirichlet form. We prove that it is the maximal Silverstein extension whenever the given form does not...     

Directed complete poset congruences

The study of algebraic properties of ordered structures has shown that their behavior in many cases is different from algebraic structures. For example, the analogues of the fundamental mapping theorem for sets which characterizes surjective maps as quotient sets modulo their kernel relations, is not true for order-preserving maps between posets (partially ordered sets). The main objective of this paper is to study the quotients of dcpos (directed complete partially ordered sets), and their relations with surjective dcpo maps (directed join preserving maps). The motivation of studying such infinitary ordered structures is their importance in domain theory, a theory on the borderline of mathematics and theoretical computer science.In this paper, introducing the notion of a pre-congruence on dcpos (directed complete partially ordered sets), we give a characterization of dcpo congruences. Also, it is proved that unlike natural dcpo congruences, the dcpo congruences are precisely kernels of surjective dcpo maps. Also, while it is known that the image of a dcpo map is not necessarily a subdcpo of its codomain, we find equivalent conditions on a dcpo map to satisfy this property. Moreover, we prove the Decomposition Theorem and its consequences for dcpo maps.     

Affine Quotients of Supergroups

In this paper we consider sheaf quotients of affine superschemes by affine supergroups that act on them freely. The necessary and sufficient conditions for such quotients to be affine are given. If G is an affine supergroup and H is its normal supersubgroup, then we prove that a dur K-sheaf is again an affine supergroup. Additionally, if G is algebraic, then a K-sheaf is also an algebraic supergroup and it coincides with . In particular, any normal supersubgroup of an affine supergroup is faithfully exact. Supported by RFFI 07-01-00392 and by FAPESP (proc. 07/54834-9).     

Peripheral fillings of relatively hyperbolic groups

In this paper a group theoretic version of Dehn surgery is studied. Starting with an arbitrary relatively hyperbolic group G we define a peripheral filling procedure, which produces quotients of G by imitating the effect of the Dehn filling of a complete finite volume hyperbolic 3-manifold M on the fundamental group π₁(M). The main result of the paper is an algebraic counterpart of Thurston’s hyperbolic Dehn surgery theorem. We also show that peripheral subgroups of G ‘almost’ have the Congruence Extension Property and the group G is approximated (in an algebraic sense) by its quotients obtained by peripheral fillings. Mathematics Subject Classification (2000) 20F65, 20F67, 20F06, 57M27, 20E26     

The heart of the Banach spaces

Consider an exact category in the sense of Quillen. Assume that in this category every morphism has a kernel and that every kernel is an inflation. In their seminal 1982 paper, Beĭlinson, Bernstein and Deligne consider in this setting a t-structure on the derived category and remark that its heart can be described as a category of formal quotients. They further point out that the category of Banach spaces is an example, and that here a similar category of formal quotients was studied by Waelbroeck already in 1962. In the current article, we give a direct and rigorous construction of the latter category by considering first the monomorphism category. Then we localize with respect to a multiplicative system. Our approach gives rise to a heart-like category not only for the Banach spaces. In particular, the main results apply to categories in which the set of all kernel–cokernel pairs does not form an exact structure. Such categories arise frequently in functional analysis.