scholarly journals Lower-Estimates on the Hochschild (Co)Homological Dimension of Commutative Algebras and Applications to Smooth Affine Schemes and Quasi-Free Algebras

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 251
Author(s):  
Anastasis Kratsios
Keyword(s):  
Vector Bundle ◽  
Upper Bound ◽  
Cohomological Dimension ◽  
Global Dimension ◽  
Homological Dimension ◽  
Free Algebras ◽  
Commutative Algebras ◽  
Flat Dimension ◽  
Smooth Affine ◽  
Affine Scheme

The Hochschild cohomological dimension of any commutative k-algebra is lower-bounded by the least-upper bound of the flat-dimension difference and its global dimension. Our result is used to show that for a smooth affine scheme X satisfying Pointcaré duality, there must exist a vector bundle with section M and suitable n which the module of algebraic differential n-forms Ωn(X,M). Further restricting the notion of smoothness, we use our result to show that most k-algebras fail to be smooth in the quasi-free sense. This consequence, extends the currently known results, which are restricted to the case where k=C.

scholarly journals Continuous CM-regularity of semihomogeneous vector bundles

2020 ◽  
Vol 20 (3) ◽  
pp. 401-412
Author(s):  
Alex Küronya ◽  
Yusuf Mustopa
Keyword(s):  
Vector Bundle ◽  
Abelian Variety ◽  
Upper Bound ◽  
Vector Bundles ◽  
Projective Variety ◽  
Constant Function ◽  
Piecewise Constant Function ◽  
Piecewise Constant ◽  
Generic Vanishing

AbstractWe ask when the CM (Castelnuovo–Mumford) regularity of a vector bundle on a projective variety X is numerical, and address the case when X is an abelian variety. We show that the continuous CM-regularity of a semihomogeneous vector bundle on an abelian variety X is a piecewise-constant function of Chern data, and we also use generic vanishing theory to obtain a sharp upper bound for the continuous CM-regularity of any vector bundle on X. From these results we conclude that the continuous CM-regularity of many semihomogeneous bundles — including many Verlinde bundles when X is a Jacobian — is both numerical and extremal.

scholarly journals Global dimensions of right coherent rings with left Krull dimension

1989 ◽  
Vol 39 (2) ◽  
pp. 215-223 ◽  
Author(s):  
Mark L. Teply
Keyword(s):  
Noetherian Ring ◽  
Global Dimension ◽  
Krull Dimension ◽  
Prime Ideals ◽  
Homological Dimension ◽  
Isomorphism Classes ◽  
Coherent Ring ◽  
Gabriel Dimension ◽  
Coherent Rings

The weak global dimension of a right coherent ring with left Krull dimension α ≥ 1 is found to be the supremum of the weak dimensions of the β-critical cyclic modules, where β < α. If, in addition, the mapping I → assl gives a bijection between isomorphism classes on injective left R-modules and prime ideals of R, then the weak global dimension of R is the supremum of the weak dimensions of the simple left R-modules. These results are used to compute the left homological dimension of a right coherent, left noetherian ring. Some analogues of our results are also given for rings with Gabriel dimension.

Upper bound for the length of commutative algebras

2009 ◽  
Vol 200 (12) ◽  
pp. 1767-1787 ◽  
Author(s):  
Ol'ga V Markova
Keyword(s):  
Upper Bound ◽  
Commutative Algebras

The Krull and global dimension of the tensor product of quantum tori

2016 ◽  
Vol 15 (09) ◽  
pp. 1650174
Author(s):  
Ashish Gupta
Keyword(s):  
Tensor Product ◽  
Upper Bound ◽  
Tensor Products ◽  
Global Dimension ◽  
Base Field ◽  
Quantum Torus ◽  
Common Value ◽  
Quantum Tori ◽  
Special Cases ◽  
The Common

An [Formula: see text]-dimensional quantum torus is defined as the [Formula: see text]-algebra generated by variables [Formula: see text] together with their inverses satisfying the relations [Formula: see text], where [Formula: see text]. The Krull and global dimensions of this algebra are known to coincide and the common value is equal to the supremum of the rank of certain subgroups of [Formula: see text] that can be associated with this algebra. In this paper we study how these dimensions behave with respect to taking tensor products of quantum tori over the base field. We derive a best possible upper bound for the dimension of such a tensor product and from this special cases in which the dimension is additive with respect to tensoring.

Rigidity dimension of algebras

2019 ◽  
pp. 1-27
Author(s):  
HONGXING CHEN ◽  
MING FANG ◽  
OTTO KERNER ◽  
STEFFEN KOENIG ◽  
KUNIO YAMAGATA
Keyword(s):  
Hochschild Cohomology ◽  
Global Dimension ◽  
Homological Dimension ◽  
Dominant Dimension ◽  
Finite Global Dimension ◽  
Derived Equivalences ◽  
Finite Dimensional ◽  
Stable Equivalences ◽  
Morita Type

Abstract A new homological dimension, called rigidity dimension, is introduced to measure the quality of resolutions of finite dimensional algebras (especially of infinite global dimension) by algebras of finite global dimension and big dominant dimension. Upper bounds of the dimension are established in terms of extensions and of Hochschild cohomology, and finiteness in general is derived from homological conjectures. In particular, the rigidity dimension of a non-semisimple group algebra is finite and bounded by the order of the group. Then invariance under stable equivalences is shown to hold, with some exceptions when there are nodes in case of additive equivalences, and without exceptions in case of triangulated equivalences. Stable equivalences of Morita type and derived equivalences, both between self-injective algebras, are shown to preserve rigidity dimension as well.

An Upper Bound for the w-Weak Global Dimension of Pullbacks

2021 ◽  
Vol 28 (04) ◽  
pp. 689-700
Author(s):  
Jin Xie ◽  
Gaohua Tang
Keyword(s):  
Commutative Ring ◽  
Upper Bound ◽  
Global Dimension ◽  
Factor Ring ◽  
Dimension Formula ◽  
Milnor Square

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] an ideal of [Formula: see text]. We introduce and study the [Formula: see text]-weak global dimension [Formula: see text] of the factor ring [Formula: see text]. Let [Formula: see text] be a [Formula: see text]-linked extension of [Formula: see text], and we also introduce the [Formula: see text]-weak global dimension [Formula: see text] of [Formula: see text]. We show that the ring [Formula: see text] with [Formula: see text] is exactly a field and the ring [Formula: see text] with [Formula: see text] is exactly a [Formula: see text]. As an application, we give an upper bound for the [Formula: see text]-weak global dimension of a Cartesian square [Formula: see text]. More precisely, if [Formula: see text] is [Formula: see text]-linked over [Formula: see text], then [Formula: see text]. Furthermore, for a Milnor square [Formula: see text], we obtain [Formula: see text].

scholarly journals Lagrangian subbundles of symplectic bundles over a curve

2012 ◽  
Vol 153 (2) ◽  
pp. 193-214 ◽  
Author(s):  
INSONG CHOE ◽  
GEORGE H. HITCHING
Keyword(s):  
Vector Bundle ◽  
Moduli Spaces ◽  
Upper Bound ◽  
Algebraic Curve ◽  
Maximal Degree ◽  
Full Picture

AbstractA symplectic bundle over an algebraic curve has a natural invariantsLagdetermined by the maximal degree of its Lagrangian subbundles. This can be viewed as a generalization of the classical Segre invariants of a vector bundle. We give a sharp upper bound onsLagwhich is analogous to the Hirschowitz bound on the classical Segre invariants. Furthermore, we study the stratifications induced bysLagon moduli spaces of symplectic bundles, and get a full picture for the case of rank four.

Equivariant Maps from Stiefel Bundles to Vector Bundles

2016 ◽  
Vol 60 (1) ◽  
pp. 231-250 ◽  
Author(s):  
Mahender Singh
Keyword(s):  
Vector Bundle ◽  
Lower Bound ◽  
Vector Bundles ◽  
Fibre Bundle ◽  
Cohomological Dimension ◽  
Stiefel Manifold ◽  
Zero Section ◽  
Zero Set ◽  
Compact Lie Group ◽  
Equivariant Maps

AbstractLet E → B be a fibre bundle and let Eʹ → B be a vector bundle. Let G be a compact Lie group acting fibre preservingly and freely on both E and Eʹ – 0, where 0 is the zero section of Eʹ → B. Let f : E → Eʹ be a fibre-preserving G-equivariant map and let Zf = {x ∈ E | f(x) = 0} be the zero set of f. In this paper we give a lower bound for the cohomological dimension of the zero set Zf when a fibre of E → B is a real Stiefel manifold with a free ℤ/2-action or a complex Stiefel manifold with a free 𝕊1-action. This generalizes a well-known result of Dold for sphere bundles equipped with free involutions.

Sheaves over infinite posets

2017 ◽  
Vol 16 (02) ◽  
pp. 1750026 ◽  
Author(s):  
J. Asadollahi ◽  
P. Bahiraei ◽  
R. Hafezi ◽  
R. Vahed
Keyword(s):  
Upper Bound ◽  
Partially Ordered Set ◽  
Topological Structure ◽  
Ordered Set ◽  
Global Dimension ◽  
Partially Ordered ◽  
Acyclic Complexes ◽  
Gorenstein Injective

In this paper, we study the category of sheaves over an infinite partially ordered set with its natural topological structure. Totally acyclic complexes in this category will be characterized in terms of their stalks. This leads us to describe Gorenstein projective, injective and flat sheaves. As an application, we get an analogue of a formula due to Mitchell, giving an upper bound on the Gorenstein global dimension of such categories. Based on these results, we present situations in which the class of Gorenstein projective sheaves is precovering as well as situations in which the class of Gorenstein injective sheaves is preenveloping.

Homological properties of the enveloping algebra U(Sl2)

1982 ◽  
Vol 91 (1) ◽  
pp. 29-37 ◽  
Author(s):  
J. T. Stafford
Keyword(s):  
Global Dimension ◽  
Homological Dimension ◽  
Enveloping Algebra

In this paper we will study the homological properties of the enveloping algebra U = U (Sl2(ℂ)), with particular reference to the homological dimension of simple U-modules and the global dimension of the primitive factor rings of U.

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