scholarly journals TILTING CHAINS OF NEGATIVE CURVES ON RATIONAL SURFACES

2017 ◽  
Vol 235 ◽  
pp. 26-41 ◽  
Author(s):  
LUTZ HILLE ◽  
DAVID PLOOG
Keyword(s):  
Abelian Category ◽  
Rational Surface ◽  
Rational Curves ◽  
Endomorphism Algebra ◽  
Exceptional Sequence ◽  
Universal Extension ◽  
Tilting Object ◽  
Category Of Modules ◽  
A Chain ◽  
Tilting Objects

We introduce the notion of exact tilting objects, which are partial tilting objects $T$ inducing an equivalence between the abelian category generated by $T$ and the category of modules over the endomorphism algebra of  $T$ . Given a chain of sufficiently negative rational curves on a rational surface, we construct an exceptional sequence whose universal extension is an exact tilting object. For a chain of $(-2)$ -curves, we obtain an equivalence with modules over a well-known algebra.

scholarly journals Torsion pairs and filtrations in abelian categories with tilting objects

2015 ◽  
Vol 14 (08) ◽  
pp. 1550121
Author(s):  
Jason Lo
Keyword(s):  
Abelian Category ◽  
Projective Dimension ◽  
Explicit Description ◽  
Homological Dimension ◽  
Abelian Categories ◽  
Dimension 2 ◽  
Tilting Object ◽  
Category Of Modules ◽  
Torsion Pairs ◽  
Tilting Objects

Given a noetherian abelian k-category [Formula: see text] of finite homological dimension, with a tilting object T of projective dimension 2, the abelian category [Formula: see text] and the abelian category of modules over End (T) op are related by a sequence of two tilts; we give an explicit description of the torsion pairs involved. We then use our techniques to obtain a simplified proof of a theorem of Jensen–Madsen–Su, that [Formula: see text] has a three-step filtration by extension-closed subcategories. Finally, we generalize Jensen–Madsen–Su's filtration to the case where T has any finite projective dimension.

scholarly journals Non Kählerian surfaces with a cycle of rational curves

2021 ◽  
Vol 8 (1) ◽  
pp. 208-222
Author(s):  
Georges Dloussky
Keyword(s):  
Deformation Theory ◽  
Complex Surface ◽  
Rational Curves ◽  
Intersection Form ◽  
Connected Component ◽  
Hirzebruch Surface ◽  
Compact Complex Surface ◽  
A Chain

Abstract Let S be a compact complex surface in class VII0 + containing a cycle of rational curves C = ∑Dj . Let D = C + A be the maximal connected divisor containing C. If there is another connected component of curves C ′ then C ′ is a cycle of rational curves, A = 0 and S is a Inoue-Hirzebruch surface. If there is only one connected component D then each connected component Ai of A is a chain of rational curves which intersects a curve Dj of the cycle and for each curve Dj of the cycle there at most one chain which meets Dj . In other words, we do not prove the existence of curves other those of the cycle C, but if some other curves exist the maximal divisor looks like the maximal divisor of a Kato surface with perhaps missing curves. The proof of this topological result is an application of Donaldson theorem on trivialization of the intersection form and of deformation theory. We apply this result to show that a twisted logarithmic 1-form has a trivial vanishing divisor.

scholarly journals Extension of matrix pencil reduction to abelian categories

2018 ◽  
Vol 17 (04) ◽  
pp. 1850062
Author(s):  
Olivier Verdier
Keyword(s):  
Normal Form ◽  
Vector Space ◽  
Decomposition Theorem ◽  
Abelian Category ◽  
Jordan Canonical Form ◽  
Main Technique ◽  
Abelian Categories ◽  
Category Of Modules ◽  
Space Case ◽  
And Control

Matrix pencils, or pairs of matrices, are used in a variety of applications. By the Kronecker decomposition theorem, they admit a normal form. This normal form consists of four parts, one part based on the Jordan canonical form, one part made of nilpotent matrices, and two other dual parts, which we call the observation and control part. The goal of this paper is to show that large portions of that decomposition are still valid for pairs of morphisms of modules or abelian groups, and more generally in any abelian category. In the vector space case, we recover the full Kronecker decomposition theorem. The main technique is that of reduction, which extends readily to the abelian category case. Reductions naturally arise in two flavors, which are dual to each other. There are a number of properties of those reductions which extend remarkably from the vector space case to abelian categories. First, both types of reduction commute. Second, at each step of the reduction, one can compute three sequences of invariant spaces (objects in the category), which generalize the Kronecker decomposition into nilpotent, observation and control blocks. These sequences indicate whether the system is reduced in one direction or the other. In the category of modules, there is also a relation between these sequences and the resolvent set of the pair of morphisms, which generalizes the regular pencil theorem. We also indicate how this allows to define invariant subspaces in the vector space case, and study the notion of strangeness as an example.

Homological Dimensions Relative to Special Subcategories

2021 ◽  
Vol 28 (01) ◽  
pp. 131-142
Author(s):  
Weiling Song ◽  
Tiwei Zhao ◽  
Zhaoyong Huang
Keyword(s):  
Abelian Category ◽  
Projective Dimension ◽  
Homological Dimensions ◽  
Category Of Modules ◽  
The Right

Let [Formula: see text] be an abelian category, [Formula: see text] an additive, full and self-orthogonal subcategory of [Formula: see text] closed under direct summands, [Formula: see text] the right Gorenstein subcategory of [Formula: see text] relative to [Formula: see text], and [Formula: see text] the left orthogonal class of [Formula: see text]. For an object [Formula: see text] in [Formula: see text], we prove that if [Formula: see text] is in the right 1-orthogonal class of [Formula: see text], then the [Formula: see text]-projective and [Formula: see text]-projective dimensions of [Formula: see text] are identical; if the [Formula: see text]-projective dimension of [Formula: see text] is finite, then the [Formula: see text]-projective and [Formula: see text]-projective dimensions of [Formula: see text] are identical. We also prove that the supremum of the [Formula: see text]-projective dimensions of objects with finite [Formula: see text]-projective dimension and that of the [Formula: see text]-projective dimensions of objects with finite [Formula: see text]-projective dimension coincide. Then we apply these results to the category of modules.

scholarly journals The Tilting–Cotilting Correspondence

2019 ◽  
Author(s):  
Leonid Positselski ◽  
Jan Šťovíček
Keyword(s):  
Abelian Category ◽  
Derived Categories ◽  
Module Category ◽  
Topological Ring ◽  
Projective Generator ◽  
Derived Equivalences ◽  
Wide Range ◽  
Injective Cogenerator ◽  
Tilting Object

Abstract To a big $n$-tilting object in a complete, cocomplete abelian category ${\textsf{A}}$ with an injective cogenerator we assign a big $n$-cotilting object in a complete, cocomplete abelian category ${\textsf{B}}$ with a projective generator and vice versa. Then we construct an equivalence between the (conventional or absolute) derived categories of ${\textsf{A}}$ and ${\textsf{B}}$. Under various assumptions on ${\textsf{A}}$, which cover a wide range of examples (for instance, if ${\textsf{A}}$ is a module category or, more generally, a locally finitely presentable Grothendieck abelian category), we show that ${\textsf{B}}$ is the abelian category of contramodules over a topological ring and that the derived equivalences are realized by a contramodule-valued variant of the usual derived Hom functor.

On Küneth’s correlation and its applications

2019 ◽  
Vol 26 (2) ◽  
pp. 295-301
Author(s):  
Leonard Mdzinarishvili
Keyword(s):  
Exact Sequence ◽  
Hausdorff Space ◽  
Chain Complex ◽  
Abelian Category ◽  
Derived Functor ◽  
Direct System ◽  
Arbitrary Complex ◽  
Formula Group ◽  
Čech Homology ◽  
A Chain

Abstract Let {\mathcal{K}} be an abelian category that has enough injective objects, let {T\colon\mathcal{K}\to A} be any left exact covariant additive functor to an abelian category A and let {T^{(i)}} be a right derived functor, {u\geq 1} , [S. Mardešić, Strong Shape and Homology, Springer Monogr. Math., Springer, Berlin, 2000]. If {T^{(i)}=0} for {i\geq 2} and {T^{(i)}C_{n}=0} for all {n\in\mathbb{Z}} , then there is an exact sequence 0\longrightarrow T^{(1)}H_{n+1}(C_{*})\longrightarrow H_{n}(TC_{*})% \longrightarrow TH_{n}(C_{*})\longrightarrow 0, where {C_{*}=\{C_{n}\}} is a chain complex in the category {\mathcal{K}} , {H_{n}(C_{*})} is the homology of the chain complex {C_{*}} , {TC_{*}} is a chain complex in the category A, and {H_{n}(TC_{*})} is the homology of the chain complex {TC_{*}} . This exact sequence is the well known Künneth’s correlation. In the present paper Künneth’s correlation is generalized. Namely, the conditions are found under which the infinite exact sequence \displaystyle\cdots\longrightarrow T^{(2i+1)}H_{n+i+1}\longrightarrow\cdots% \longrightarrow T^{(3)}H_{n+2}\longrightarrow T^{(1)}H_{n+1}\longrightarrow H_% {n}(TC_{*}) \displaystyle\longrightarrow TH_{n}(C_{*})\longrightarrow T^{(2)}H_{n+1}% \longrightarrow T^{(4)}H_{n+2}\longrightarrow\cdots\longrightarrow T^{(2i)}H_{% n+i}\longrightarrow\cdots holds, where {T^{(2i+1)}H_{n+i+1}=T^{(2i+1)}H_{n+i+1}(C_{*})} , {T^{(2i)}H_{n+i}=T^{(2i)}H_{n+i}(C_{*})} . The formula makes it possible to generalize Milnor’s formula for the cohomologies of an arbitrary complex, relatively to the Kolmogorov homology to the Alexandroff–Čech homology for a compact space, to a generative result of Massey for a local compact Hausdorff space X and a direct system {\{U\}} of open subsets U of X such that {\overline{U}} is a compact subset of X.

scholarly journals FILTRATIONS IN ABELIAN CATEGORIES WITH A TILTING OBJECT OF HOMOLOGICAL DIMENSION TWO

2012 ◽  
Vol 12 (02) ◽  
pp. 1250149 ◽  
Author(s):  
BERNT TORE JENSEN ◽  
DAG OSKAR MADSEN ◽  
XIUPING SU
Keyword(s):  
Abelian Category ◽  
Derived Categories ◽  
Module Category ◽  
Homological Dimension ◽  
Derived Equivalence ◽  
Torsion Pair ◽  
Abelian Categories ◽  
Tilting Object ◽  
Dimension One

We consider filtrations of objects in an abelian category [Formula: see text] induced by a tilting object T of homological dimension at most two. We define three extension closed subcategories [Formula: see text] and [Formula: see text] with [Formula: see text] for j > i, such that each object in [Formula: see text] has a unique filtration with factors in these categories. In dimension one, this filtration coincides with the classical two-step filtration induced by the torsion pair. We also give a refined filtration, using the derived equivalence between the derived categories of [Formula: see text] and the module category of [Formula: see text].

On isolated singularities of surfaces which do not affect the conditions of adjunction (Part III.)

1934 ◽  
Vol 30 (4) ◽  
pp. 483-491 ◽  
Author(s):  
Patrick Du Val
Keyword(s):  
Singular Point ◽  
Direct Product ◽  
Finite Groups ◽  
Rational Surface ◽  
Rational Curves ◽  
Singular Points ◽  
Isolated Singularities ◽  
Projective Model ◽  
Base Points ◽  
System 2

In the second part of this paper I pointed out how the presence of any singular point or points of the kind considered in the first part, on a rational surface, corresponds to a subgroup of the group of symmetry of the polytope which represents the properties of the system of base points in the plane representation of the surface. The subgroup is generated by reflexions, and may be the direct product of one or more factors (all the primes of symmetry in one factor being perpendicular to all those in any other factor). Each factor corresponds to a singular point on the surface, namely (in Coxeter's notation), a factor [ ] to a conic node C2, a factor [3n] to a binode Bn+2 (n ≥ 1), a factor [3n,1,1] to a unode Un+5, (n ≥ 1), and finally a factor [3n,2,1] to a unode The possible subgroups in the finite groups that arise have been enumerated by Coxeter; and we shall find that every subgroup generated by reflexions in the group of symmetry of the polytope in ε dimensions which represents ε base points corresponds to a possible configuration of the base points, in which just those rational curves of grade − 2 are actual which correspond to primes of symmetry belonging to the subgroup; without exception, for ε ≤ 6; with one exception—the subgroup [ ]7—for ε = 7; and with three exceptions—the subgroups [ ]7, [ ]8, and [31,1,1 × [ ]4—for ε = 8. Since moreover the system |k| of cubics passing simply through all the base points is in all these cases an actually existing system, for which all the rational curves of grade − 2 are fundamental, its projective model (or in the case ε = 8, in which |k| is only a pencil, the projective model of the system |2k|) provides a rational surface on which all the sets of curves corresponding to the subgroups in question actually appear as singular points.

scholarly journals Tilting and Cluster Tilting for Preprojective Algebras and Coxeter Groups

2017 ◽  
Vol 2019 (18) ◽  
pp. 5597-5634 ◽  
Author(s):  
Yuta Kimura
Keyword(s):  
Coxeter Groups ◽  
Cluster Category ◽  
Preprojective Algebra ◽  
Endomorphism Algebra ◽  
Factor Algebra ◽  
Stable Category ◽  
Preprojective Algebras ◽  
Tilting Object ◽  
Cluster Tilting ◽  
Silting Object

AbstractWe study the stable category of the graded Cohen–Macaulay modules of the factor algebra of the preprojective algebra associated with an element $w$ of the Coxeter group of a quiver. We show that there exists a silting object $M(\boldsymbol{w})$ of this category associated with each reduced expression $\boldsymbol{w}$ of $w$ and give a sufficient condition on $\boldsymbol{w}$ such that $M(\boldsymbol{w})$ is a tilting object. In particular, the stable category is triangle equivalent to the derived category of the endomorphism algebra of $M(\boldsymbol{w})$. Moreover, we compare it with a triangle equivalence given by Amiot–Reiten–Todorov for a cluster category.

scholarly journals On the Hochschild coho-mology groups of endomorphism algebras of exceptional sequences

2004 ◽  
Vol 35 (3) ◽  
pp. 189-196
Author(s):  
Hailou Yao ◽  
Lihong Huang
Keyword(s):  
Associative Algebra ◽  
Hochschild Cohomology ◽  
Closed Field ◽  
Cohomology Groups ◽  
Endomorphism Algebra ◽  
Exceptional Sequence ◽  
Algebraically Closed Field ◽  
Finite Dimensional ◽  
Endomorphism Algebras ◽  
Exceptional Sequences

Let $ A $ be a finite dimensional associative algebra over an algebraically closed field $ k $, and $\mod A$ be the category of finite dimensional left $ A $-module and $ X_1,X_2,\ldots,X_n$ in $\mod A$ be a complete exceptional sequence, then we investigate the Hochschild Cohomology groups of endomorphism algebra of exceptional sequence $ {X_1,X_2, \ldots,X_n}$ in this paper.

Sign in / Sign up

Export Citation Format

Share Document