Let $A$ be a noetherian Koszul Artin-Schelter regular algebra, and let $fin A_2$ be a central regular element of $A$. The quotient algebra $A/(f)$ is usually called a (noncommutative) quadric hypersurface. In this paper, we use the Clifford deformati
on to study the quadric hypersurfaces obtained from the tensor products. We introduce a notion of simple graded isolated singularity and proved that, if $B/(g)$ is a simple graded isolated singularity of 0-type, then there is an equivalence of triangulated categories $underline{text{mcm}},A/(f)congunderline{text{mcm}},(Aotimes B)/(f+g)$ of the stable categories of maximal Cohen-Macaulay modules. This result may be viewed as a generalization of Kn{o}rrers periodicity theorem. As an application, we study the double branch cover $(A/(f))^#=A[x]/(f+x^2)$ of a noncommutative conic $A/(f)$.
We introduce the notion of right pre-resolutions (quasi-resolutions) for noncommutative isolated singularities, which is a weaker version of quasi-resolutions introduced by Qin-Wang-Zhang. We prove that right quasi-resolutions for noetherian bounded
below and locally finite graded algebra with right injective dimension 2 are always Morita equivalent. When we restrict to noncommutative quadric hypersurfaces, we prove that a noncommutative quadric hypersurface, which is a noncommutative isolated singularity, always admits a right pre-resolution. Besides, we provide a method to verify whether a noncommutative quadric hypersurface is an isolated singularity. An example of noncommutative quadric hypersurfaces with detailed computations of indecomposable maximal Cohen-Macaulay modules and right pre-resolutions is included as well.
Let $E$ be a Koszul Frobenius algebra. A Clifford deformation of $E$ is a finite dimensional $mathbb Z_2$-graded algebra $E(theta)$, which corresponds to a noncommutative quadric hypersurface $E^!/(z)$, for some central regular element $zin E^!_2$. I
t turns out that the bounded derived category $D^b(text{gr}_{mathbb Z_2}E(theta))$ is equivalent to the stable category of the maximal Cohen-Macaulay modules over $E^!/(z)$ provided that $E^!$ is noetherian. As a consequence, $E^!/(z)$ is a noncommutative isolated singularity if and only if the corresponding Clifford deformation $E(theta)$ is a semisimple $mathbb Z_2$-graded algebra. The preceding equivalence of triangulated categories also indicates that Clifford deformations of trivial extensions of a Koszul Frobenius algebra are related to the Kn{o}rrer Periodicity Theorem for quadric hypersurfaces. As an application, we recover Kn{o}rrer Periodicity Theorem without using of matrix factorizations.
We compute the Nakayama automorphism of a PBW-deformation of a Koszul Artin-Schelter Gorenstein algebra of finite global dimension, and give a criterion for an augmented PBW-deformation of a Koszul Calabi-Yau algebra to be Calabi-Yau. The relations b
etween the Calabi-Yau property of augmented PBW-deformations and that of non-augmented cases are discussed. The Nakayama automorphisms of PBW-deformations of Koszul Artin-Schelter Gorenstein algebras of global dimensions 2 and 3 are given explicitly. We show that if a PBW-deformation of a graded Calabi-Yau algebra is still Calabi-Yau, then it is defined by a potential under some mild conditions. Some classical results are also recovered. Our main method used in this paper is elementary and based on linear algebra. The results obtained in this paper will be applied in a subsequent paper.
Let $H$ be a finite dimensional semisimple Hopf algebra, $A$ a differential graded (dg for short) $H$-module algebra. Then the smash product algebra $A#H$ is a dg algebra. For any dg $A#H$-module $M$, there is a quasi-isomorphism of dg algebras: $mat
hrm{RHom}_A(M,M)#Hlongrightarrow mathrm{RHom}_{A#H}(Mot H,Mot H)$. This result is applied to $d$-Koszul algebras, Calabi-Yau algebras and AS-Gorenstein dg algebras
Let $H$ be a Hopf algebra, $A/B$ be an $H$-Galois extension. Let $D(A)$ and $D(B)$ be the derived categories of right $A$-modules and of right $B$-modules respectively. An object $M^cdotin D(A)$ may be regarded as an object in $D(B)$ via the restrict
ion functor. We discuss the relations of the derived endomorphism rings $E_A(M^cdot)=op_{iinmathbb{Z}}Hom_{D(A)}(M^cdot,M^cdot[i])$ and $E_B(M^cdot)=op_{iinmathbb{Z}}Hom_{D(B)}(M^cdot,M^cdot[i])$. If $H$ is a finite dimensional semisimple Hopf algebra, then $E_A(M^cdot)$ is a graded subalgebra of $E_B(M^cdot)$. In particular, if $M$ is a usual $A$-module, a necessary and sufficient condition for $E_B(M)$ to be an $H^*$-Galois graded extension of $E_A(M)$ is obtained. As an application of the results, we show that the Koszul property is preserved under Hopf Galois graded extensions.
