Given a finite-dimensional algebra $Λ$ and $A \geqslant 1$, we construct a new algebra $\tildeΛ_A$, called the stretched algebra, and relate the homological properties of $Λ$ and $\tildeΛ_A$. We investigate Hochschild cohomology and the finiteness condition (Fg), and use stratifying ideals to show that $Λ$ has (Fg) if and only if $\tildeΛ_A$ has (Fg). We also consider projective resolutions and apply our results in the case where $Λ$ is a $d$-Koszul algebra for some $d \geqslant 2$.
We consider the socle deformations arising from formal deformations of a class of Koszul self-injective special biserial algebras which occur in the study of the Drinfeld double of the generalized Taft algebras. We show, for these deformations, that the Hochschild cohomology ring modulo nilpotence is a finitely generated commutative algebra of Krull dimension 2.
We introduce the Hochschild extensions of dg algebras, which are $A_\infty$-algebras. We show that all exact Hochschild extensions are symmetric Hochschild extensions, more precisely, every exact Hochschild extension of a finite dimensional complete typical dg algebra is a symmetric $A_\infty$-algebra. Moreover, we prove that the Koszul dual of trivial extension is Calabi-Yau completion and the Koszul dual of exact Hochschild extension is deformed Calabi-Yau completion, more precisely, the Koszul dual of the trivial extension of a finite dimensional complete dg algebra is the Calabi-Yau comple…
We formulate the (co)bar construction theory of dg $K$-(co)rings and the calculus theory of the Hochschild homology and cohomology of dg $K$-rings. As applications, we compare the Hochschild (co)homologies of a complete typical dg $K$-ring and its Koszul dual. Moreover, we show that the Koszul dual of a finite dimensional complete typical $d$-symmetric dg $K$-ring is a $d$-Calabi-Yau dg algebra whose Hochschild cohomology is a Batalin-Vilkovisky algebra. Furthermore, we prove that the Hochschild cohomologies of a finite dimensional complete typical $d$-symmetric dg $K$-ring and its Koszul dual…
We define Gerstenhaber's graded Lie bracket directly on complexes other than the bar complex, under some conditions. The Koszul complex of a Koszul algebra in particular satisfies our conditions. As examples we recover the Schouten-Nijenhuis bracket for a polynomial ring and the Gerstenhaber bracket for a group algebra of a cyclic group of prime order.
Let $Λ=kQ/I$ be a Koszul algebra over a field $k$, where $Q$ is a finite quiver. An algorithmic method for finding a minimal projective resolution $\mathbb{F}$ of the graded simple modules over $Λ$ is given in Green-Solberg. This resolution is shown to have a "comultiplicative" structure in Green-Hartman-Marcos-Solberg, and this is used to find a minimal projective resolution $\mathbb{P}$ of $Λ$ over the enveloping algebra $Λ^e$. Using these results we show that the multiplication in the Hochschild cohomology ring of $Ł$ relative to the resolution $\mathbb{P}$ is given as a cup product and als…
Let $B$ be a quantum algebra possessing a semiclassical limit $A$. We show that under certain hypotheses $B^e$ can be thought of as a deformation of the Poisson enveloping algebra of $A$, and we give a criterion for the Hochschild cohomology of $B$ to be a deformation of the Poisson cohomology of $A$ in the case that $B$ is Koszul. We verify that condition for the algebra of $2\times 2$ quantum matrices and calculate its Hochschild cohomology and the Poisson cohomology of its semiclassical limit.
We construct chain maps between the bar and Koszul resolutions for a quantum symmetric algebra (skew polynomial ring). This construction uses a recursive technique involving explicit formulae for contracting homotopies. We use these chain maps to compute the Gerstenhaber bracket, obtaining a quantum version of the Schouten-Nijenhuis bracket on a symmetric algebra (polynomial ring). We compute brackets also in some cases for skew group algebras arising as group extensions of quantum symmetric algebras.
For A a dg (or A-infinity) algebra and M a module over A, we study the image of the characteristic morphism $χ_M: HH^*(A, A) \to Ext_A(M, M)$ and its interaction with the higher structure on the Yoneda algebra $Ext_A(M, M)$. To this end, we introduce and study a notion of A-infinity centre for minimal A-infinity algebras, agreeing with the usual centre in the case that there is no higher structure. We show that the image of $χ_M$ lands in the A-infinity centre of $Ext_A(M, M)$. When A is augmented over k, we show (under mild connectedness assumptions) that the morphism $χ_k: HH^*(A, A) \to Ext…
One of our main results is a classification all the weakly symmetric radical cube zero finite dimensional algebras over an algebraically closed field having a theory of support via the Hochschild cohomology ring satisfying Dade's Lemma. Along the way we give a characterization of when a finite dimensional Koszul algebra has such a theory of support in terms of the graded centre of the Koszul dual.
The Hochschild homology and cohomology group can be described in terms of the homology and cohomology of the classifying space of the groupoid of the adjoint action of the group under the suitable assumption of the finiteness of the supports of cohomology groups. The difference between homology and cohomology leads to a correction of the results in the book by D. J. Benson (1995),(1991).
In this article we introduce the notion of multi-Koszul algebra for the case of a locally finite dimensional nonnegatively graded connected algebra, as a generalization of the notion of (generalized) Koszul algebras defined by R. Berger for homogeneous algebras, which were in turn an extension of Koszul algebras introduced by S. Priddy. It also extends and generalizes the definition recently introduced by the author and A. Rey. In order to simplify the exposition we consider the minimal graded projective resolution of the algebra A as a bimodule, which may be used to compute the corresponding …
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 use the theory of varieties for modules arising from Hochschild cohomology to give an alternative version of the wildness criterion of Bergh and Solberg: If a finite dimensional self-injective algebra has a module of complexity at least 3 and satisfies some finiteness assumptions on Hochschild cohomology, then the algebra is wild. We show directly how this is related to the analogous theory for Hopf algebras that we developed. We give applications to many different types of algebras: Hecke algebras, reduced universal enveloping algebras, small half-quantum groups, and Nichols (quantum symme…
Braverman and Gaitsgory gave necessary and sufficient conditions for a nonhomogeneous quadratic algebra to satisfy the Poincare-Birkhoff-Witt property when its homogeneous version is Koszul. We widen their viewpoint and consider a quotient of an algebra that is free over some (not necessarily semisimple) subalgebra. We show that their theorem holds under a weaker hypothesis: We require the homogeneous version of the nonhomogeneous quadratic algebra to be the skew group algebra (semidirect product algebra) of a finite group acting on a Koszul algebra, obtaining conditions for the Poincare-Birkh…
Let $(K,\mathcal O, k)$ be a $p$-modular system with $k$ algebraically closed and $\mathcal O$ unramified, and let $Λ$ be an $\mathcal O$-order in a separable $K$-algebra. We call a $Λ$-lattice $L$ rigid if ${\rm Ext}^1_Λ(L,L)=0$, in analogy with the definition of rigid modules over a finite-dimensional algebra. By partitioning the $Λ$-lattices of a given dimension into "varieties of lattices", we show that there are only finitely many rigid $Λ$-lattices $L$ of any given dimension. As a consequence we show that if the first Hochschild cohomology of $Λ$ vanishes, then the Picard group and the o…
The classical Hochschild--Kostant--Rosenberg (HKR) theorem computes the Hochschild homology and cohomology of smooth commutative algebras. In this paper, we generalise this result to other kinds of algebraic structures. Our main insight is that producing HKR isomorphisms for other types of algebras is directly related to computing quasi-free resolutions in the category of left modules over an operad; we establish that an HKR-type result follows as soon as this resolution is diagonally pure. As examples we obtain a permutative and a pre-Lie HKR theorem for smooth commutative and smooth brace al…
A new homological dimension is introduced to measure the quality of resolutions of `singular' finite dimensional algebras (of infinite global dimension) by `regular' ones (of finite global dimension). Upper bounds are established in terms of extensions and of Hochschild cohomology, and finiteness in general is derived from homological conjectures. 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 equiva…
We investigate deformations of a skew group algebra that arise from a finite group acting on a polynomial ring. When the characteristic of the underlying field divides the order of the group, a new type of deformation emerges that does not occur in characteristic zero. This analogue of Lusztig's graded affine Hecke algebra for positive characteristic can not be forged from the template of symplectic reflection and related algebras as originally crafted by Drinfeld. By contrast, we show that in characteristic zero, for arbitrary finite groups, a Lusztig-type deformation is always isomorphic to …
Potential algebras feature in the minimal model program and noncommutative resolution of singularities, and the important cases are when they are finite dimensional, or of linear growth. We develop techniques, involving Gröbner basis theory and generalized Golod-Shafarevich type theorems for potential algebras, to determine finiteness conditions in terms of the potential. We consider two-generated potential algebras. Using Gröbner bases techniques and arguing in terms of associated truncated algebra we prove that they cannot have dimension smaller than $8$. This answers a question of Wemyss \c…
