Claude Cibils, Maria Julia Redondo, Manuel Saorin
Given a finite--dimensional monomial algebra $A$ we consider the trivial extension $TA$ and provide formulae, depending on the characteristic of the field, for the dimensions of the summands $HH_1(A)$ and $\Alt(DA)$ of the first Hochschild cohomology group $HH^1(TA)$. From these a formula for the dimension of $HH^1(TA)$ can be derived.
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Finite generation for Hochschild cohomology of Gorenstein monomial algebras
本文证明了有限维单项代数满足Snashall-Solberg的有限生成条件当且仅当为Gorenstein代数，给出了单项代数中Erdmann-Holloway-Snashall-Solberg-Taillefer定理的逆定理，并给出了有限生成的必要充分组合准则。
For a finite dimensional monomial algebra $Λ$ over a field $K$ we show that the Hochschild cohomology ring of $Λ$ modulo the ideal generated by homogeneous nilpotent elements is a commutative finitely generated $K$-algebra of Krull dimension at most one. This was conjectured to be true for any finite dimensional algebra over a field by Snashall-Solberg.
The Hochschild cohomology groups under gluing arrows
该研究延续此前对有限维单项式代数粘合幂等元时Hochschild上同调群的比较工作，进一步探究粘合两个箭线时该上同调群的对比情况。
In this paper we give necessary and sufficient conditions for the variety of a simple module over a (D,A)-stacked monomial algebra to be nontrivial. This class of algebras was introduced in [Green and Snashall, The Hochschild cohomology ring modulo nilpotence of a stacked monomial algebra, Colloq. Math. 105 (2006), 233-258] and generalizes Koszul and D-Koszul monomial algebras. As a consequence we show that if the variety of every simple module over such an algebra is nontrivial then the algebra is D-Koszul. We give examples of (D,A)-stacked monomial algebras which are not selfinjective but ne…
We consider associative algebras L over a field provided with a direct sum decomposition of a two-sided ideal M and a sub-algebra A - examples are provided by trivial extensions or triangular type matrix algebras. In this relative and split setting we describe a long exact sequence computing the Hochschild cohomology of L. We study the connecting homomorphism using the cup-product and we infer several results, in particular the first Hochschild cohomology group of a trivial extension never vanishes.
Using combinatorics of chains going back to works of Anick, Green, Happel and Zacharia, we give, for any monomial algebra $A$, an explicit description of its minimal model. This also provides us with formulas for a canonical $A_\infty$-structure on the Ext-algebra of the trivial $A$-module. We do this by exploiting the combinatorics of chains going back to works of Anick, Green, Happel and Zacharia, and the algebraic discrete Morse theory of Jöllenbeck, Welker and Sköldberg. We then show how this result can be used to obtain models for algebras with a chosen Gröbner basis, and briefly outline …
We investigate maximal tori in the Hochschild cohomology Lie algebra $HH^1(A)$ of a finite dimensional algebra $A$, and their connection with the fundamental groups associated to presentations of $A$. We prove that every maximal torus in $HH^1(A)$ arises as the dual of some fundamental group of $A$, extending work of Farkas, Green and Marcos; de la Peña and Saorín; and Le Meur. Combining this with known invariance results for Hochschild cohomology, we deduce that (in rough terms) the largest rank of a fundamental group of $A$ is a derived invariant quantity, and among self-injective algebras, …
We show that the graded commutative ring structure of the Hochschild cohomology HH*(A) is trivial in case A is a triangular quadratic string algebra. Moreover, in case A isgentle, the Lie algebra structure on HH*(A) is also trivial.
We obtain a decomposition for the Hochschild cochain complex of a split algebra and we study some properties of the cohomology of each term of this decomposition. Then, we consider the case of trivial extensions, specially of Frobenius algebras. In particular, we determine completely the cohomology of the trivial extension of a finite dimensional Hopf algebra. Finally, as an application, we obtain a result about the Hochschild cohomology of Frobenius algebras.
Algebraic versions of $\mathbb{T}^2$ and of $\mathbb{P}^1\times\mathbb{P}^1$ and Hochschild cohomology
本文研究了三角代数及其变形的Hochschild上同调，揭示了其杯积结构与维度变化的非直观特性。
On the first relative Hochschild cohomology
本文研究第一相对Hochschild上同调的Lie代数结构，证明在补图简单时其为可解Lie代数，并计算了根方零代数与对偶扩张代数的具体结构。
Let p be a prime and F a field containing a primitive pth root of unity. Let E/F be a cyclic extension of degree p and G_E < G_F the associated absolute Galois groups. We determine precise conditions for the cohomology group H^n(E)=H^n(G_E,Fp) to be free or trivial as an Fp[Gal(E/F)]-module. We examine when these properties for H^n(E) are inherited by H^k(E), k>n, and, by analogy with cohomological dimension, we introduce notions of cohomological freeness and cohomological triviality. We give examples of H^n(E) free or trivial for each n in N with prescribed cohomological dimension.
We enumerate the low dimensional cells in the Voronoi cell complexes attached to the modular groups $SL_N(Z)$ and $GL_N(Z)$ for $N=8,9,10,11$, using quotient sublattices techniques for $N=8,9$ and linear programming methods for higher dimensions. These enumerations allow us to compute some cohomology of these groups and prove that $K_8(Z) = 0$. We deduce from it new knowledge on the Kummer-Vandiver conjecture.
Batalin-Vilkovisky structure on Hochschild cohomology with coefficients in the dual algebra
本文研究以对偶代数 $A^*$ 为系数的 Hochschild 上同调，证明在一定条件下其具有 Batalin-Vilkovisky 代数结构，并覆盖对称代数、Frobenius 代数及单项代数情形。
We prove that the Drinfeld double of an arbitrary finite group scheme has finitely generated cohomology. That is to say, for G any finite group scheme, and D(G) the Drinfeld double of the group ring kG, we show that the self-extension algebra of the trivial representation for D(G) is a finitely generated algebra, and that for each D(G)-representation V the extensions from the trivial representation to V form a finitely generated module over the aforementioned algebra. As a corollary, we find that all categories rep(G)*_M dual to rep(G) are of also of finite type (i.e. have finitely generated c…
A main contribution of this paper is the explicit construction of comparison morphisms between the standard bar resolution and Bardzell's minimal resolution for truncated quiver algebras (TQA's). As a direct application we describe explicitely the Yoneda product and derive several results on the structure of the cohomology ring of TQA's. For instance, we show that the product of odd degree cohomology classes is always zero. We prove that TQA's associated with quivers with no cycles or with neither sinks nor sources have trivial cohomology rings. On the other side we exhibit a fundamental examp…
We generalize the cohomology theory for linear cycle sets introduced by Lebed and Vendramin. Our cohomology classifies extensions of linear cycle sets by trivial ideals, whereas the cohomology of Lebed and Vendramin only deals with central ideals $I$ (which are automatically trivial). Therefore our theory gives an analog to the theory of extensions of braces by trivial ideals constructed by Bachiller, but from a cohomological point of view. We also study the general notions of extensions of linear cycle sets and the equivalence of extensions.
Building on work of Livernet and Richter, we prove that E_n-homology and E_n-cohomology of a commutative algebra with coefficients in a symmetric bimodule can be interpreted as functor homology and cohomology. Furthermore we show that the associated Yoneda algebra is trivial.
We consider a non-standard totalisation functor to produce a cochain complex from a given double complex: instead of sums or products, totalisation is defined via truncated products of modules. We give an elementary proof of the fact that a double complex with exact rows (resp, columns) yields an acyclic cochain complex under totalisation using right (resp, left) truncated products. As an application we consider the algebraic mapping torus T(h) of a self map h of a cochain complex C. We show that if C consists of finitely presented modules then T(h) has trivial negative Novikov cohomology; if …
Motivic cohomology of cyclic coverings
本文研究了循环覆盖的motivic cohomology，证明在特定情况下，循环覆盖的chow群在张量有理数后或不张量时均变得平凡。
We study the relationship between trivial cocycles on the Torelli group and invariants of oriented integral homology 3-spheres. We give ncecessary and sufficient conditions for a function defined on the union of the Torelli groups to be an invariant of homology spheres. We apply this study to give a new purely algebraic construction of the Casson invariant and prove in this setting its surgery properties. As a by-product we get a new 2-torsion cohomology class in the second integral cohomology of the Torelli group.
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