In this paper, we study the Koszul property of the homogeneous coordinate ring of a generic collection of lines in $\mathbb {P}^n$ and the homogeneous coordinate ring of a collection of lines in general linear position in $\mathbb {P}^n.$ We show that if $\mathcal {M}$ is a collection of m lines in general linear position in $\mathbb {P}^n$ with $2m \leq n+1$ and R is the coordinate ring of $\mathcal {M},$ then R is Koszul. Furthermore, if $\mathcal {M}$ is a generic collection of m lines in $\mathbb {P}^n$ and R is the coordinate ring of $\mathcal {M}$ with m even and $m +1\leq n$ or m is odd and $m +2\leq n,$ then R is Koszul. Lastly, we show that if $\mathcal {M}$ is a generic collection of m lines such that

$$ \begin{align*} m> \frac{1}{72}\left(3(n^2+10n+13)+\sqrt{3(n-1)^3(3n+5)}\right),\end{align*} $$

$n \leq 6$

$m \leq 6$

For a three-dimensional quantum polynomial algebra $A=\mathcal {A}(E,\sigma )$ , Artin, Tate, and Van den Bergh showed that A is finite over its center if and only if $|\sigma |<\infty $ . Moreover, Artin showed that if A is finite over its center and $E\neq \mathbb P^{2}$ , then A has a fat point module, which plays an important role in noncommutative algebraic geometry; however, the converse is not true in general. In this paper, we will show that if $E\neq \mathbb P^{2}$ , then A has a fat point module if and only if the quantum projective plane ${\sf Proj}_{\text {nc}} A$ is finite over its center in the sense of this paper if and only if $|\nu ^{*}\sigma ^{3}|<\infty $ where $\nu $ is the Nakayama automorphism of A. In particular, we will show that if the second Hessian of E is zero, then A has no fat point module.

In noncommutative algebraic geometry an Artin–Schelter regular (AS-regular) algebra is one of the main interests, and every three-dimensional quadratic AS-regular algebra is a geometric algebra, introduced by Mori, whose point scheme is either $\mathbb {P}^{2}$ or a cubic curve in $\mathbb {P}^{2}$ by Artin et al. [‘Some algebras associated to automorphisms of elliptic curves’, in: The Grothendieck Festschrift, Vol. 1, Progress in Mathematics, 86 (Birkhäuser, Basel, 1990), 33–85]. In the preceding paper by the authors Itaba and Matsuno [‘Defining relations of 3-dimensional quadratic AS-regular algebras’, Math. J. Okayama Univ. 63 (2021), 61–86], we determined all possible defining relations for these geometric algebras. However, we did not check their AS-regularity. In this paper, by using twisted superpotentials and twists of superpotentials in the Mori–Smith sense, we check the AS-regularity of geometric algebras whose point schemes are not elliptic curves. For geometric algebras whose point schemes are elliptic curves, we give a simple condition for three-dimensional quadratic AS-regular algebras. As an application, we show that every three-dimensional quadratic AS-regular algebra is graded Morita equivalent to a Calabi–Yau AS-regular algebra.

Classification of AS-regular algebras is one of the main interests in noncommutative algebraic geometry. We say that a $3$-dimensional quadratic AS-regular algebra is of Type EC if its point scheme is an elliptic curve in $\mathbb {P}^{2}$. In this paper, we give a complete list of geometric pairs and a complete list of twisted superpotentials corresponding to such algebras. As an application, we show that there are only two exceptions up to isomorphism among all $3$-dimensional quadratic AS-regular algebras that cannot be written as a twist of a Calabi–Yau AS-regular algebra by a graded algebra automorphism.

We prove a representation stability result for the codimension-one cohomology of the level-three congruence subgroup of $\mathbf{SL}_{n}(\mathbb{Z})$. This is a special case of a question of Church, Farb, and Putman which we make more precise. Our methods involve proving finiteness properties of the Steinberg module for the group $\mathbf{SL}_{n}(K)$ for $K$ a field. This also lets us give a new proof of Ash, Putman, and Sam’s homological vanishing theorem for the Steinberg module. We also prove an integral refinement of Church and Putman’s homological vanishing theorem for the Steinberg module for the group $\mathbf{SL}_{n}(\mathbb{Z})$.

The cup product in the cohomology of algebras over quadratic operads has been studied in the general setting of Koszul duality for operads. We study the cup product on the cohomology of n-ary totally associative algebras with an operation of even (homological) degree. This cup product endows the cohomology with the structure of an n-ary partially associative algebra with an operation of even or odd degree depending on the parity of n. In the cases n=3 and n=4, we provide an explicit definition of this cup product and prove its basic properties.

We introduce a notion of Koszul A∞-algebra that generalizes Priddy's notion of a Koszul algebra and we use it to construct small A∞-algebra models for Hochschild cochains. As an application, this yields new techniques for computing free loop space homology algebras of manifolds that are either formal or coformal (over a field or over the integers). We illustrate these techniques in two examples.

Let $k$ be a field and $R$ a standard graded $k$-algebra. We denote by $\operatorname{H}^{R}$ the homology algebra of the Koszul complex on a minimal set of generators of the irrelevant ideal of $R$. We discuss the relationship between the multiplicative structure of $\operatorname{H}^{R}$ and the property that $R$ is a Koszul algebra. More generally, we work in the setting of local rings and we show that certain conditions on the multiplicative structure of Koszul homology imply strong homological properties, such as existence of certain Golod homomorphisms, leading to explicit computations of Poincaré series. As an application, we show that the Poincaré series of all finitely generated modules over a stretched Cohen–Macaulay local ring are rational, sharing a common denominator.

We prove a conjecture of Rouquier relating the decomposition numbers in category ${\mathcal{O}}$ for a cyclotomic rational Cherednik algebra to Uglov’s canonical basis of a higher level Fock space. Independent proofs of this conjecture have also recently been given by Rouquier, Shan, Varagnolo and Vasserot and by Losev, using different methods. Our approach is to develop two diagrammatic models for this category ${\mathcal{O}}$; while inspired by geometry, these are purely diagrammatic algebras, which we believe are of some intrinsic interest. In particular, we can quite explicitly describe the representations of the Hecke algebra that are hit by projectives under the $\mathsf{KZ}$-functor from the Cherednik category ${\mathcal{O}}$ in this case, with an explicit basis. This algebra has a number of beautiful structures including categorifications of many aspects of Fock space. It can be understood quite explicitly using a homogeneous cellular basis which generalizes such a basis given by Hu and Mathas for cyclotomic KLR algebras. Thus, we can transfer results proven in this diagrammatic formalism to category ${\mathcal{O}}$ for a cyclotomic rational Cherednik algebra, including the connection of decomposition numbers to canonical bases mentioned above, and an action of the affine braid group by derived equivalences between different blocks.

We present a calculus that is well-adapted to homogeneous quadratic algebras. We define this calculus on Koszul cohomology – resp. homology – by cup products – resp. cap products. The Koszul homology and cohomology are interpreted in terms of derived categories. If the algebra is not Koszul, then Koszul (co)homology provides different information than Hochschild (co)homology. As an application of our calculus, the Koszul duality for Koszul cohomology algebras is proved for any quadratic algebra, and this duality is extended in some sense to Koszul homology. So, the true nature of the Koszul duality theorem is independent of any assumption on the quadratic algebra. We compute explicitly this calculus on a non-Koszul example.

Classification of AS-regular algebras is one of the major projects in non-commutative algebraic geometry. In this paper, we will study when given AS-regular algebras are graded Morita equivalent. In particular, for every geometric AS-regular algebra A, we define another graded algebra A, and show that if two geometric AS-regular algebras A and A' are graded Morita equivalent, then A and A' are isomorphic as graded algebras. We also show that the converse holds in many three-dimensional cases. As applications, we apply our results to Frobenius Koszul algebras and Beilinson algebras.

In this paper we construct, for F1 and F2 subbundles of a vector bundle E, a ‘Koszul duality’ equivalence between derived categories of 𝔾m-equivariant coherent(dg-)sheaves on the derived intersection , and the corresponding derived intersection . We also propose applications to Hecke algebras.

This article presents a study of an algebra spanned by the faces of a hyperplane arrangement. The quiver with relations of the algebra is computed and the algebra is shown to be a Koszul algebra. It is shown that the algebra depends only on the intersection lattice of the hyperplane arrangement. A complete system of primitive orthogonal idempotents for the algebra is constructed and other algebraic structure is determined including: a description of the projective indecomposable modules, the Cartan invariants, projective resolutions of the simple modules, the Hochschild homology and cohomology, and the Koszul dual algebra. A new cohomology construction on posets is introduced, and it is shown that the face semigroup algebra is isomorphic to the cohomology algebra when this construction is applied to the intersection lattice of the hyperplane arrangement.

Let $A$ be a graded algebra finitely generated in degree 1 over a field $k$. Point modules over $A$ introduced by Artin, Tate and Van den Bergh play an important role in studying $A$ in noncommutative algebraic geometry. In this paper, we define a dual notion of point module in terms of Koszul duality, which we call a co-point module. Using co-point modules, we will construct counter-examples to the following condition due to Auslander: for every finitely generated right module $\pi$ over a ring $R$, there is a natural number $n_M\in {\mathbb N}$ such that, for any finitely generated right module $N$ over $R$, ${\rm Ext}^i_R(M, N)=0$ for all $i\gg 0$ implies ${\rm Ext}^i_R(M, N)=0$ for all $i>n_M$.