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On the Ext Ring of Koszul Rings  [PDF]
Adrian Manea
Mathematics , 2015,
Abstract: The aim of this article is to study the Ext ring associated to a Koszul $R$-ring and to use it to provide further characterisations of the former. As such, for $R$ being a semisimple ring and $A$ a graded Koszul $R$-ring, we will prove that there is an isomorphism of DG rings between $\mathcal{E}(A):=\mathrm{Ext}^\bullet_A(R,R)$ and $^{\ast-\text{gr}}\mathrm{T}(A) \simeq \mathrm{E}(^{\ast-\text{gr}\!} A)$. Also, the Ext $R$-ring will prove to be isomorphic to the shriek ring of the left graded dual of $A$, namely $\mathcal{E}(A) \simeq (^{\ast-\text{gr}\!} A)^!$. As an application, these isomorphisms will be studied in the context of incidence $R$-(co)rings for Koszul posets. Thus, we will obtain a description and method of computing the shriek ring for $\Bbbk^c[\mathcal{P}]$, the incidence $R$-coring of a Koszul poset. Another application is provided for monoid rings associated to submonoids of $\mathbb{Z}^n$.
The Ext--Group of Unitary Equivalence Classes of Unital Extensions  [PDF]
Yifeng Xue
Mathematics , 2010,
Abstract: Let $\A$ be a unital separable nuclear $C^*$--algebra which belongs to the bootstrap category $\N$ and $\B$ be a separable stable $C^*$--algebra. In this paper, we consider the group $\Ext_u(\A,\B)$ consisting of the unitary equivalence classes of unital extensions $\tau\colon\A\rightarrow Q(\B)$. The relation between $\Ext_u(\A,\B)$ and $\Ext(\A,\B)$ is established. Using this relation, we show the half--exactness of $\Ext_u(\cdot,\B)$ and the (UCT) for $\Ext_u(\A,\B)$. Furthermore, under certain conditions, we obtain the half--exactness and Bott periodicity of $\Ext_u(\A,\cdot)$.
Postnikov extensions of ring spectra  [PDF]
Daniel Dugger,Brooke Shipley
Mathematics , 2006, DOI: 10.2140/agt.2006.6.1785
Abstract: We give a functorial construction of k-invariants for ring spectra and use these to classify extensions in the Postnikov tower of a ring spectrum.
Computing finite presentations of Tor and Ext over skew PBW extensions and some applications  [PDF]
Oswaldo Lezama,Melisa Paiba
Mathematics , 2015,
Abstract: In this paper we compute the $Tor$ and $Ext$ modules over skew $PBW$ extensions. If $A$ is a bijective skew $PBW$ extension of a ring $R$, we give presentations of $Tor_r^{A}(M,N)$, where $M$ is a finitely generated centralizing subbimodule of $A^m$, $m\geq 1$, and $N$ is a left $A$-submodule of $A^l$, $l\geq 1$. In the case of $Ext_A^{r}(M,N)$, $M$ is a left $A$-submodule of $A^m$ and $N$ is a finitely generated centralizing subbimodule of $A^l$. As application of these computations, we test stably-freeness, reflexiveness, and we will compute also the torsion, the dual and the grade of a given submodule of $A^m$. Skew $PBW$ extensions include many important classes of non-commutative rings and algebras arising in quantum mechanics, for example, Weyl algebras, enveloping algebras of finite-dimensional Lie algebras (and its quantization), Artamonov quantum polynomials, diffusion algebras, Manin algebra of quantum matrices, among many others.
Duality for Ext-groups and extensions of discrete series for graded Hecke algebras  [PDF]
Kei Yuen Chan
Mathematics , 2014,
Abstract: In this paper, we study extensions of graded affine Hecke algebra modules. In particular, based on an explicit projective resolution on graded affine Hecke algebra modules, we prove a duality result for Ext-groups. This duality result with an Ind-Res resolution gives an algebraic proof of the fact that all higher Ext-groups between discrete series vanish.
Gorenstein dimensions in trivial ring extensions  [PDF]
Najib Mahdou,Khalid Ouarghi
Mathematics , 2008,
Abstract: In this paper, we show that the Gorenstein global dimension of trivial ring extensions is often infinite. Also we study the transfer of Gorenstein properties between a ring and its trivial ring extensions. We conclude with an example showing that, in general, the transfer of the notion of Gorenstein projective module does not carry up to pullback constructions.
Indices of inseparability in towers of field extensions  [PDF]
Kevin Keating
Mathematics , 2014,
Abstract: Let $K$ be a local field whose residue field has characteristic $p$ and let $L/K$ be a finite separable totally ramified extension of degree $n=ap^{\nu}$. The indices of inseparability $i_0,i_1,...,i_{\nu}$ of $L/K$ were defined by Fried in the case char$(K)=p$ and by Heiermann in the case char$(K)=0$; they give a refinement of the usual ramification data for $L/K$. The indices of inseparability can be used to construct "generalized Hasse-Herbrand functions" $\phi_{L/K}^j$ for $0\le j\le\nu$. In this paper we give an interpretation of the values $\phi_{L/K}^j(c)$ for natural numbers $c$. We use this interpretation to study the behavior of generalized Hasse-Herbrand functions in towers of field extensions.
Extensions of Picard 2-Stacks and the cohomology groups Ext^i of length 3 complexes  [PDF]
Cristiana Bertolin,Ahmet Emin Tatar
Mathematics , 2012, DOI: 10.1007/s10231-013-0347-5
Abstract: The aim of this paper is to define and study the 3-category of extensions of Picard 2-stacks over a site S and to furnish a geometrical description of the cohomology groups Ext^i of length 3 complexes of abelian sheaves. More precisely, our main Theorem furnishes (1) a parametrization of the equivalence classes of objects, 1-arrows, 2-arrows, and 3-arrows of the 3-category of extensions of Picard 2-stacks by the cohomology groups Ext^i, and (2) a geometrical description of the cohomology groups Ext^i of length 3 complexes of abelian sheaves via extensions of Picard 2-stacks. To this end, we use the triequivalence between the 3-category of Picard 2-stacks and the tricategory T^[-2,0](S) of length 3 complexes of abelian sheaves over S introduced by the second author in arXiv:0906.2393, and we define the notion of extension in this tricategory T^[-2,0](S), getting a pure algebraic analogue of the 3-category of extensions of Picard 2-stacks. The calculus of fractions that we use to define extensions in the tricategory T^[-2,0](S) plays a central role in the proof of our Main Theorem.
The Duality between Corings and Ring Extensions  [PDF]
Florin F. Nichita,Bartosz Zielinski
Axioms , 2012, DOI: 10.3390/axioms1020173
Abstract: We study the duality between corings and ring extensions. We construct a new category with a self-dual functor acting on it, which extends that duality. This construction can be seen as the non-commutative case of another duality extension: the duality between finite dimensional algebras and coalgebra. Both these duality extensions have some similarities with the Pontryagin-van Kampen duality theorem.
On free ring extensions of degree n  [cached]
George Szeto
International Journal of Mathematics and Mathematical Sciences , 1981, DOI: 10.1155/s0161171281000537
Abstract: Nagahara and Kishimoto [1] studied free ring extensions B(x) of degree n for some integer n over a ring B with 1, where xn=b, cx=x (c) for all c and some b in B( =automophism of ¢ € ‰ ¢ € ‰B), and {1,x ¢ € |,xn ¢ ’1} is a basis. Parimala and Sridharan [2], and the author investigated a class of free ring extensions called generalized quaternion algebras in which b= ¢ ’1 and is of order 2. The purpose of the present paper is to generalize a characterization of a generalized quaternion algebra to a free ring extension of degree n in terms of the Azumaya algebra. Also, it is shown that a one-to-one correspondence between the set of invariant ideals of B under and the set of ideals of B(x) leads to a relation of the Galois extension B over an invariant subring under to the center of B.
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