Local homology and dimensional full-valuedness

A natural criterion for dimensional full-valuedness of locally compact spaces with finitely generated local homology is given.     

Homological Dimension of Coalgebras and Crossed Coproducts

Let H be a Hopf k-algebra. We study the global homological dimension of the underlying coalgebra structure of H. We show that gl.dim(H) is equal to the injective dimension of the trivial right H-comodule k. We also prove that if D = C H is a crossed coproduct with invertible , then gl.dim(D) gl.dim(C) + gl.dim(H). Some applications of this result are obtained. Moreover, if C is a cocommutative coalgebra such that C ^* is noetherian, then the global dimension of the coalgebra C coincides with the global dimension of the algebra C ^*.     

Borel Subalgebras Redux with Examples from Algebraic and Quantum Groups

Both building upon and revising previous literature, this paper formulates the general notion of a Borel subalgebra B of a quasi-hereditary algebra A. We present various general constructions of Borel subalgebras, establish a triangular factorization of A, and relate the concept to graded Kazhdan–Lusztig theories in the sense of Cline et al. (Tôhoku Math. J. 45 (1993), 511–534). Various interesting types of Borel subalgebras arise naturally in different contexts. For example, `excellent" Borel subalgebras come about by abstracting the theory of Schubert varieties. Numerous examples from algebraic groups, q-Schur algebras, and quantum groups are considered in detail.     

代数的张量积的同调满同态的构造

利用已知的代数的同调满同态来构造其张量积代数的同调满同态.设A,B,C,D是域k上的有限维代数,如果环同态f:A→C和g:B→D是环的同调满同态,则fg:AB→CD也是环的同调满同态.     

Totally acyclic complexes over noetherian schemes

We define a notion of total acyclicity for complexes of flat quasi-coherent sheaves on a semi-separated noetherian scheme, and study these complexes using the pure derived category of flat quasi-coherent sheaves. We prove that a scheme is Gorenstein if and only if every acyclic complex of flat quasi-coherent sheaves is totally acyclic. Our formalism also removes the need for a dualising complex in several known results for rings, including Jørgensen's proof of the existence of Gorenstein projective precovers.     

Toric cycles in the complement to a complex curve in (C×)2

The amoeba of a complex curve in the 2‐dimensional complex torus is its image under the projection onto the real parts of the logarithmic coordinates. A toric cycle in the complement to a curve is a fiber of this projection over a point in the complement to the amoeba of the curve. We consider amoebas of complex algebraic curves defined by so‐called Harnack polynomials. We prove that toric cycles are homologically independent in the complement to a such curve.     

粘合、倾斜同调维数与高维Auslander-Reiten理论

本文主要研究阿贝尔范畴粘合$(\mathscr{A}, \mathscr{B}, \mathscr{C})$中$\mathscr{A}$, $\mathscr{B}$与$\mathscr{C}$之间的倾斜同调维数关系. 特别地,对遗传的阿贝尔范畴$\mathscr{B}$,给出了粘合$(\mathscr{A}, \mathscr{B}, \mathscr{C})$中的范畴之间的$n$-几乎可裂序列间的联系.     

图表追踪的一种机器实现方法

交换图表追踪法是同调代数中一种重要的证明方法.本文提出了一种基于前推的交换图表追踪的机器实现方法,它可以证明同调代数中一系列基于交换图的定理.该方法从定理的结论出发,通过构造辅助元素以及对辅助元素进行推导,能在有限步内给出定理的证明.本文提出的前推方法已经在Java开发平台上实现,同调代数中的五引理、九引理、蛇引理等的证明都可以由我们的程序自动产生.     

Precovering and Preenveloping Ideals

In [8 Salce , L. ( 1979 ). Cotorsion theories for abelian groups . Symposia Mathematica 23 : 11 – 23 . [Google Scholar]] L. Salce introduced the notion of a cotorsion pair (?, 𝒞) in the category of abelian groups. But his definitions and basic results carry over to more general abelian categories and have proven to be useful in a variety of settings. A significant result of cotorsion theory proven by Eklof and Trlifaj is that if a pair (?, 𝒞) of classes of R-modules is cogenerated by a set, then it is complete [1 Eklof , P. C. , Trlifaj , J. ( 2001 ). How to make Ext vanish . Bull. London Math. Soc. 33 : 31 – 41 .[Crossref], [Web of Science ®] , [Google Scholar]]. Recently Fu, Herzog, Guil, and Torrecillas developed the ideal approximation theory [6 Herzog , I. ( 2014 ). Phantom morphisms and Salce's Lemma . Contemp. Math. 607 : 57 – 83 .[Crossref] , [Google Scholar]], [4 Fu X. H. , Guil Asensio , P. A. , Herzog , I. , Torrecillas , B. ( 2013 ). Ideal Approximation Theory . Adv. in Math. 244 : 750 – 790 .[Crossref], [Web of Science ®] , [Google Scholar]]. In this article we look at a result motivated by the Eklof and Trlifaj argument for an ideal ? when it is generated by a set of homomorphisms.