Search

Theory of matrix factorizations is useful to study hypersurfaces in commutative algebra. To study noncommutative hypersurfaces, which are important objects of study in noncommutative algebraic geometry, we introduce a notion of noncommutative matrix factorization for an arbitrary nonzero non-unit element of a ring. First we show that the category of noncommutative graded matrix factorizations is invariant under the operation called twist (this result is a generalization of the result by Cassidy-Conner-Kirkman-Moore). Then we give two category equivalences involving noncommutative matrix factorizations and totally reflexive modules (this result is analogous to the famous result by Eisenbud for commutative hypersurfaces). As an application, we describe indecomposable noncommutative graded matrix factorizations over skew exterior algebras., 25 pages

We study multiplication modules. The rings are not assumed to be commutative. Several criteria with some applications are given for a direct sum of modules to be a multiplication module.

We develop a notion of degree for functions between two abelian groups that allows us to generalize the Chevalley-Warning Theorems from fields to noncommutative rings or abelian groups of prime power order.

We give an explicit formula showing how the double Poisson algebra introduced in [14] appears as a particular part of a pre-Calabi-Yau structure, i.e. cyclically invariant, with respect to the natural inner form, solution of the Maurer-Cartan equation on A ⊕ A ⁎ . Specific part of this solution is described, which is in one-to-one correspondence with the double Poisson algebra structures. The result holds for any associative algebra A and emphasises the special role of the fourth component of pre-Calabi-Yau structure in this respect. As a consequence we have that appropriate pre-Calabi-Yau structures induce a Poisson brackets on representation spaces ( Rep n A ) G l n for any associative algebra A.

Let $f(X_1,\dots, X_n)$ be a nonzero multilinear noncommutative polynomial. If $A$ is a unital algebra with a surjective inner derivation, then every element in $A$ can be written as $f(a_1,\dots,a_n)$ for some $a_i\in A$., Comment: 21 pages, 0 figures, submited to Journal of Algebra

Let k be an algebraically closed field and A a Z -graded finitely generated simple k -algebra which is a domain of Gelfand–Kirillov dimension 2. We show that the category of Z -graded right A-modules is equivalent to the category of quasicoherent sheaves on a certain quotient stack. The theory of these simple algebras is closely related to that of a class of generalized Weyl algebras (GWAs). We prove a translation principle for the noncommutative schemes of these GWAs, shedding new light on the classical translation principle for the infinite-dimensional primitive quotients of U ( sl 2 ) .

In this paper, we introduce the notion of a noncommutative Poisson bialgebra, and establish the equivalence between matched pairs, Manin triples and noncommutative Poisson bialgebras. Using quasi-representations and the corresponding cohomology theory of noncommutative Poisson algebras, we study coboundary noncommutative Poisson bialgebras which leads to the introduction of the Poisson Yang-Baxter equation. A skew-symmetric solution of the Poisson Yang-Baxter equation naturally gives a (coboundary) noncommutative Poisson bialgebra. Rota-Baxter operators, more generally O -operators on noncommutative Poisson algebras, and noncommutative pre-Poisson algebras are introduced, by which we construct skew-symmetric solutions of the Poisson Yang-Baxter equation in some special noncommutative Poisson algebras obtained from these structures.

We consider the algebra $\mathcal O(\mathsf M)$ of observables and the (formally) versal morphism $\eta: A \to \mathcal O(\mathsf M)$ defined by the noncommutative deformation functor $\mathsf{Def}_{\mathsf M}$ of a family $\mathsf M = \{ M_1, \dots, M_r \}$ of right modules over an associative $k$-algebra $A$. By the Generalized Burnside Theorem, due to Laudal, $\eta$ is an isomorphism when $A$ is finite dimensional, $\mathsf M$ is the family of simple $A$-modules, and $k$ is an algebraically closed field. The purpose of this paper is twofold: First, we prove a form of the Generalized Burnside Theorem that is more general, where there is no assumption on the field $k$. Secondly, we prove that the $\mathcal O$-construction is a closure operation when $A$ is any finitely generated $k$-algebra and $\mathsf M$ is any family of finite dimensional $A$-modules, in the sense that $\eta_B: B \to \mathcal O^B(\mathsf M)$ is an isomorphism when $B = \mathcal O(\mathsf M)$ and $\mathsf M$ is considered as a family of $B$-modules., Comment: 9 pages

We establish a version of Knorrer's Periodicity Theorem in the context of noncommutative invariant theory. Namely, let A be a left noetherian AS-regular algebra, let f be a normal and regular element of A of positive degree, and take B = A / ( f ) . Then there exists a bijection between the set of isomorphism classes of indecomposable non-free maximal Cohen-Macaulay modules over B and those over (a noncommutative analog of) its second double branched cover ( B # ) # . Our results use and extend the study of twisted matrix factorizations, which was introduced by the first three authors with Cassidy. These results are applied to the noncommutative Kleinian singularities studied by the second and fourth authors with Chan and Zhang.

We define twisted Hochschild homology for Green functors. This construction is the algebraic analogue of the relative topological Hochschild homology $THH_{C_n}(-)$, and it describes the $E_2$ term of the K\"unneth spectral sequence for relative $THH$. Applied to ordinary rings, we obtain new algebraic invariants. Extending Hesselholt's construction of the Witt vectors of noncommutative rings, we interpret our construction as providing Witt vectors for Green functors., Comment: Minor revisions. Published version

Let M and N be finitely generated modules over a d-dimensional commutative Noetherian ring R. We show that, if M maps onto N ⊕ ( t + d ) locally, then M maps onto N ⊕ t globally. In a similar manner, we demonstrate that, if M locally admits N ⊕ ( t + d ) as a direct summand, then M globally admits N ⊕ t as a direct summand. We also generalize splitting theorems due to Serre, De Stefani–Polstra–Yao, and Bass by exploiting the j-spectrum of R and by allowing M and N to be modules over a possibly noncommutative module-finite R-algebra.

The notion of a noncommutative quasi-resolution is introduced for a noncommutative noetherian algebra with singularities, even for a non-Cohen-Macaulay algebra. If A is a commutative normal Gorenstein domain, then a noncommutative quasi-resolution of A naturally produces a noncommutative crepant resolution (NCCR) of A in the sense of Van den Bergh, and vice versa. Under some mild hypotheses, we prove that (i) in dimension two, all noncommutative quasi-resolutions of a given noncommutative algebra are Morita equivalent, and (ii) in dimension three, all noncommutative quasi-resolutions of a given noncommutative algebra are derived equivalent. These assertions generalize important results of Van den Bergh, Iyama-Reiten and Iyama-Wemyss in the commutative and central-finite cases.

Standard table algebras that have an irreducible character of degree two, and with all other irreducible characters of multiplicity one, are classified to exact isomorphism.

For a finite group acting on a polynomial ring, the Chevalley–Shephard–Todd Theorem proves that the fixed subring is isomorphic to a polynomial ring if and only if the group is generated by pseudo-reflections. In recent years, progress was made in work of Kirkman, Kuzmanovich, Zhang, and others to extend this result to Artin–Schelter regular algebras by expanding pseudo-reflections to quasi-reflections. Naturally, the question arises if the theory generalizes further to non-connected noncommutative algebras. Our objects of study will be preprojective algebras which are certain factor algebras of path algebras corresponding to extended Dynkin diagrams of type A, D or E. This work answers the question what conditions need to be satisfied by the fixed ring in order to make a rich theory possible. On our way, we will point out additional difficulties in establishing quasi-reflections using the trace and reveal situations which do not occur for Artin–Schelter regular algebras.

Let A be a commutative ring, and let \a = \frak{a} be a finitely generated ideal in it. It is known that a necessary and sufficient condition for the derived \a-torsion and \a-adic completion functors to be nicely behaved is the weak proregularity of \a. In particular, the MGM Equivalence holds. Because weak proregularity is defined in terms of elements of the ring (specifically, it involves limits of Koszul complexes), it is not suitable for noncommutative ring theory. In this paper we introduce a new condition on a torsion class T in a module category: weak stability. Our first main theorem is that in the commutative case, the ideal \a is weakly proregular if and only if the corresponding torsion class T_{\a} is weakly stable. We then study weak stability of torsion classes in module categories over noncommutative rings. There are three main theorems in this context: - For a torsion class T that is weakly stable, quasi-compact and finite dimensional, the right derived torsion functor is isomorphic to a left derived tensor functor. - The Noncommutative MGM Equivalence, under the same assumptions on T. - A theorem about symmetric derived torsion for complexes of bimodules. This last theorem is a generalization of a result of Van den Bergh from 1997, and corrects an error in a paper of Yekutieli-Zhang from 2003., Comment: 49 pages. Final version, a few minor improvements, to appear in J. Algebra

In this paper we produce noncommutative algebras derived equivalent to deformations of schemes with tilting bundles. We do this in two settings, first proving that a tilting bundle on a scheme lifts to a tilting bundle on an infinitesimal deformations of that scheme and then expanding this result to $\mathbb{C}^*$-equivariant deformations over schemes with a good $\mathbb{C}^*$-action. In both these situations the endomorphism algebra of the lifted tilting bundle produces a deformation of the original endomorphism algebra, and this is a graded deformation in the $\mathbb{C}^*$-equivariant case. We apply our results to rational surface singularities, generalising the deformed preprojective algebras, and also to symplectic situations where the deformations produced are related to symplectic reflection algebras., Comment: 30 pages

Using the theory of signatures of hermitian forms over algebras with involution, developed by us in earlier work, we introduce a notion of positivity for symmetric elements and prove a noncommutative analogue of Artin's solution to Hilbert's 17th problem, characterizing totally positive elements in terms of weighted sums of hermitian squares. As a consequence we obtain an earlier result of Procesi and Schacher and give a complete answer to their question about representation of elements as sums of hermitian squares., Comment: 22 pages, some minor clarifications added to version 2

Drinfeld orbifold algebras are a type of deformation of skew group algebras generalizing graded Hecke algebras of interest in representation theory, algebraic combinatorics, and noncommutative geometry. In this article, we classify all Drinfeld orbifold algebras for symmetric groups acting by the natural permutation representation. This provides, for nonabelian groups, infinite families of examples of Drinfeld orbifold algebras that are not graded Hecke algebras. We include explicit descriptions of the maps recording commutator relations and show there is a one-parameter family of such maps supported only on the identity and a three-parameter family of maps supported only on 3-cycles and 5-cycles. Each commutator map must satisfy properties arising from a Poincare–Birkhoff–Witt condition on the algebra, and our analysis of the properties illustrates reduction techniques using orbits of group element factorizations and intersections of fixed point spaces.

We investigate the possibility of constructing Kochen–Specker uncolorable sets of idempotent matrices whose entries lie in various rings, including the rational numbers, the integers, and finite fields. Most notably, we show that there is no Kochen–Specker coloring of the n × n idempotent integer matrices for n ≥ 3 , thereby illustrating that Kochen–Specker contextuality is an inherent feature of pure matrix algebra. We apply this to generalize recent no-go results on noncommutative spectrum functors, showing that any contravariant functor from rings to sets (respectively, topological spaces or locales) that restricts to the Zariski prime spectrum functor for commutative rings must assign the empty set (respectively, empty space or locale) to the matrix ring M n ( R ) for any integer n ≥ 3 and any ring R . An appendix by Alexandru Chirvasitu shows that Kochen–Specker colorings of idempotents in partial subalgebras of M 3 ( F ) for a perfect field F can be extended to partial algebra morphisms into the algebraic closure of F .

To a finite ranked poset Γ we associate a finite-dimensional graded quadratic algebra R Γ . Assuming Γ satisfies a combinatorial condition known as uniform, R Γ is related to a well-known algebra, the splitting algebra A Γ . First introduced by Gelfand, Retakh, Serconek, and Wilson, splitting algebras originated from the problem of factoring non-commuting polynomials. Given a finite ranked poset Γ, we ask: Is R Γ Koszul? The Koszulity of R Γ is related to a combinatorial topology property of Γ called Cohen–Macaulay. Kloefkorn and Shelton proved that if Γ is a finite ranked cyclic poset, then Γ is Cohen–Macaulay if and only if Γ is uniform and R Γ is Koszul. We define a new generalization of Cohen–Macaulay, weakly Cohen–Macaulay. This new class includes non-uniform posets and posets with disconnected open subintervals. Using a spectral sequence associated to Γ and the notion of a noncommutative Koszul filtration for R Γ , we prove: if Γ is a finite ranked cyclic poset, then Γ is weakly Cohen–Macaulay if and only if R Γ is Koszul. In addition, we prove that Γ is Cohen–Macaulay if and only if Γ is uniform and weakly Cohen–Macaulay.

In this paper we propose a general method for computing a minimal free right resolution of a finitely presented graded right module over a finitely presented graded noncommutative algebra. In particular, if such module is the base field of the algebra then one obtains its graded homology. The approach is based on the possibility to obtain the resolution via the computation of syzygies for modules over commutative algebras. The method behaves algorithmically if one bounds the degree of the required elements in the resolution. Of course, this implies a complete computation when the resolution is a finite one. Finally, for a monomial right module over a monomial algebra we provide a bound for the degrees of the non-zero Betti numbers of any single homological degree in terms of the maximal degree of the monomial relations of the module and the algebra., Comment: 23 pages, to appear in Journal of Algebra

We prove first (Proposition 3) that, over any ring $R$, an acyclic complex of projective modules is totally acyclic if and only if the cycles of every acyclic complex of Gorenstein projective modules are Gorenstein projective. The dual result for injective and Gorenstein injective modules also holds over any ring $R$ (Proposition 4). And, when $R$ is a GF-closed ring, the analogue result for flat/Gorenstein flat modules is also true (Proposition 5). Then we show (Theorem 2) that over a left noetherian ring $R$, a third equivalent condition can be added to those in Proposition 4, more precisely, we prove that the following are equivalent: 1. Every acyclic complex of injective modules is totally acyclic. 2. The cycles of every acyclic complex of Gorenstein injective modules are Gorenstein injective. 3. Every complex of Gorenstein injective modules is dg-Gorenstein injective. Theorem 3 shows that the analogue result for complexes of flat and Gorenstein flat modules holds over any left coherent ring $R$. We prove (Corollary 1) that, over a commutative noetherian ring $R$, the equivalent statements in Theorem 3 hold if and only if the ring is Gorenstein. We also prove (Theorem 4) that when moreover $R$ is left coherent and right $n$-perfect (that is, every flat right $R$-module has finite projective dimension $\leq n$) then statements 1, 2, 3 in Theorem 2 are also equivalent to the following: 4. Every acyclic complex of projective right $R$-modules is totally acyclic. 5. Every acyclic complex of Gorenstein projective right $R$-modules is in $\widetilde{\mathcal{GP}}$. 6. Every complex of Gorenstein projective right $R$-modules is dg-Gorenstein projective. Corollary 2 shows that when $R$ is commutative noetherian of finite Krull dimension, the equivalent conditions (1)-(6) from Theorem 4 are also equivalent to those in Theorem 3 and hold if and only if $R$ is an Iwanaga-Gorenstein ring. Thus we improve slightly on a result of Iyengar's and Krause's; in [22] they proved that for a commutative noetherian ring $R$ with a dualizing complex, the class of acyclic complexes of injectives coincides with that of totally acyclic complexes of injectives if and only if $R$ is Gorenstein. We are able to remove the dualizing complex hypothesis and add more equivalent conditions. In the second part of the paper we focus on two sided noetherian rings that satisfy the Auslander condition. We prove (Theorem 7) that for such a ring $R$ that also has finite finitistic flat dimension, every complex of injective (left and respectively right) $R$-modules is totally acyclic if and only if $R$ is an Iwanaga-Gorenstein ring., Oberwolfach Preprints;2016,14

We study the non-uniqueness of factorizations of non zero-divisors into atoms (irreducibles) in noncommutative rings. To do so, we extend concepts from the commutative theory of non-unique factorizations to a noncommutative setting. Several notions of factorizations as well as distances between them are introduced. In addition, arithmetical invariants characterizing the non-uniqueness of factorizations such as the catenary degree, the $\omega$-invariant, and the tame degree, are extended from commutative to noncommutative settings. We introduce the concept of a cancellative semigroup being permutably factorial, and characterize this property by means of corresponding catenary and tame degrees. Also, we give necessary and sufficient conditions for there to be a weak transfer homomorphism from a cancellative semigroup to its reduced abelianization. Applying the abstract machinery we develop, we determine various catenary degrees for classical maximal orders in central simple algebras over global fields by using a natural transfer homomorphism to a monoid of zero-sum sequences over a ray class group. We also determine catenary degrees and the permutable tame degree for the semigroup of non zero-divisors of the ring of $n \times n$ upper triangular matrices over a commutative domain using a weak transfer homomorphism to a commutative semigroup., Comment: 45 pages

A quantum P 3 is a noncommutative analogue of a polynomial ring on four variables, and, herein, it is taken to be a regular algebra of global dimension four. It is well known that if a generic quadratic quantum P 3 exists, then it has a point scheme consisting of exactly twenty distinct points and a one-dimensional line scheme. In this article, we compute the line scheme of a family of algebras whose generic member is a candidate for a generic quadratic quantum P 3 . We find that, as a closed subscheme of P 5 , the line scheme of the generic member is the union of seven curves; namely, a nonplanar elliptic curve in a P 3 , four planar elliptic curves and two nonsingular conics.

We study various properties of the Frobenius map φ ( x ) = x p on a noncommutative algebra R over a field F of positive characteristic p. We call R perfect whenever φ is surjective. More generally, we say that R is additively (respectively, multiplicatively) p-power-closed if, for every x , y ∈ R , there exists a positive integer n such that x p n + y p n (respectively, x p n y p n ) is a p-power in R. If, for example, φ n were a ring homomorphism (as when R is commutative), then R would be additively and multiplicatively p-power-closed. We show that a finite-dimensional unital algebra R is Lie nilpotent if and only if R is additively (respectively, multiplicatively) p-power-closed and each of its simple homomorphic images has Schur index 1. Since not all finite-dimensional perfect division algebras are Lie nilpotent, the Schur index condition cannot be omitted. We deduce that similar results hold for the class of all finitely generated PI-algebras. Moreover, for this same class, we give a positive solution to the following problem reminiscent of the problems of Kurosh and Levitzki: is every finitely generated perfect algebra finite-dimensional?

This paper develops the tools of formal algebraic geometry in the setting of noncommutative manifolds, roughly ringed spaces locally modeled on the free associative algebra. We define a notion of noncommutative coordinate system, which is a principal bundle for an appropriate group of local coordinate changes. These bundles are shown to carry a natural flat connection with properties analogous to the classical Gelfand-Kazhdan structure. Every noncommutative manifold has an underlying smooth variety given by abelianization. A basic question is existence and uniqueness of noncommutative thickenings of a smooth variety, i.e., finding noncommutative manifolds abelianizing to a given smooth variety. We obtain new results in this direction by showing that noncommutative coordinate systems always arise as reductions of structure group of the commutative bundle of coordinate systems on the underlying smooth variety; this also explains a relationship between $\mathcal{D}$-modules on the commutative variety and sheaves of modules for the noncommutative structure sheaf., Comment: Preliminary version, comments welcome! Several corrections from first version

The Weil restriction functor, introduced in the late fifties, was recently extended by Karpenko to the category of Chow motives with integer coefficients. In this article we introduce the noncommutative (=NC) analogue of the Weil restriction functor, where schemes are replaced by dg algebras, and extend it to Kontsevich's categories of NC Chow motives and NC numerical motives. Instead of integer coefficients, we work more generally with coefficients in a binomial ring. Along the way, we extend Karpenko's functor to the classical category of numerical motives, and compare this extension with its NC analogue. As an application, we compute the (NC) Chow motive of the Weil restriction of every smooth projective scheme whose category of perfect complexes admits a full exceptional collection. Finally, in the case of central simple algebras, we describe explicitly the NC analogue of the Weil restriction functor using solely the degree of the field extension. This leads to a “categorification” of the classical corestriction homomorphism between Brauer groups.

Let $A$ be an associative unital algebra, $B_k$ its successive quotients of lower central series and $N_k$ the successive quotients of ideals generated by lower central series. The geometric and algebraic aspects of $B_k$ and $N_k$ have been of great interest since the pioneering work of \cite{feigin2007}. In this paper, we will concentrate on the case where $A$ is a noncommutative polynomial algebra over $\mathbb{Z}$ modulo a single homogeneous relation. Both the torsion part and the free part of $B_k$ and $N_k$ are explored. Many examples are demonstrated in detail, and several general theorems are proved. Finally we end up with an appendix about the torsion subgroups of $N_k(A_n(\mathbb{Z}))$ and some open problems., Comment: 12 pages. Updates in v2: minor typos corrected, new references added

Results of Vancliff, Van Rompay and Willaert in 1998 [8] prove that point modules over a regular graded Clifford algebra (GCA) are determined by (commutative) quadrics of rank at most two that belong to the quadric system associated to the GCA. In 2010, in [4] , Cassidy and Vancliff generalized the notion of a GCA to that of a graded skew Clifford algebra (GSCA). The results in this article show that the results of [8] may be extended, with suitable modification, to GSCAs. In particular, using the notion of μ -rank introduced recently by the authors in [9] , the point modules over a regular GSCA are determined by (noncommutative) quadrics of μ -rank at most two that belong to the noncommutative quadric system associated to the GSCA.

In this paper we find noncommutative analogues of the coordinate rings of the unipotent radicals of the standard Borel subgroups in SL n + 1 . Two subalgebras of the quantized coordinate ring of the standard Borel in SL n + 1 are defined, both of which can be considered quantizations of the unipotent radical. Presentations are given for these algebras and they are proven to be isomorphic. It is the shown that these algebras also arise as coinvariants of a natural comodule algebra action using the Hopf Algebra structure of O q ( SL n + 1 ) . Finally, using a dual paring, it is shown that these algebras are isomorphic U q ± ( sl n + 1 ) .

Let k be a perfect field and let K/k be a finite extension of fields. An arithmetic noncommutative projective line is a noncommutative space equal to the projectivization of the noncommutative symmetric algebra of a k-central two -sided vector space of rank two over K. We study the geometry of these spaces. More precisely, we prove they are integral, we classify vector bundles over them, we classify them up to isomorphism, and we classify isomorphisms between them. Using the classification of isomorphisms, we compute the automorphism group of an arithmetic noncommutative projective line., 41 pages, to appear in J. Algebra

We define Jordan quadruple systems by the polynomial identities of degrees 4 and 7 satisfied by the Jordan tetrad { a , b , c , d } = a b c d + d c b a as a quadrilinear operation on associative algebras. We find further identities in degree 10 which are not consequences of the defining identities. We introduce four infinite families of finite dimensional Jordan quadruple systems, and construct the universal associative envelope for a small system in each family. We obtain analogous results for the anti-tetrad [ a , b , c , d ] = a b c d − d c b a . Our methods rely on computer algebra, especially linear algebra on large matrices, the LLL algorithm for lattice basis reduction, representation theory of the symmetric group, noncommutative Grobner bases, and Wedderburn decompositions of associative algebras.

We study the interplay between the classes of right quasi-Euclidean rings and right K-Hermite rings, and relate them to projective-free rings and Cohn's GE2-rings using the method of noncommutative Euclidean divisions and matrix factorizations into idempotents. Right quasi-Euclidean rings are closed under matrix extensions, and a left quasi-Euclidean ring is right quasi-Euclidean if and only if it is right Bezout. Singular matrices over left and right quasi-Euclidean domains are shown to be products of idempotent matrices, generalizing an earlier result of Laffey for singular matrices over commutative Euclidean domains.

In this paper, we define a notion of noncommutative graded isolated singularity, and study AS-Gorenstein isolated singularities and the categories of graded maximal Cohen–Macaulay modules over them. In particular, for an AS-Gorenstein algebra A of dimension d ⩾ 2 , we show that A is a graded isolated singularity if and only if the stable category of graded maximal Cohen–Macaulay modules over A has the Serre functor. Using this result, we also show the existence of cluster tilting objects in the categories of graded maximal Cohen–Macaulay modules over Veronese subalgebras of certain AS-regular algebras.

We present an algorithm that finds all toric noncommutative crepant resolutions of a given toric 3-dimensional Gorenstein singularity. The algorithm embeds the quivers of these algebras inside a real 3-dimensional torus such that the relations are homotopy relations. One can project these embedded quivers down to a 2-dimensional torus to obtain the corresponding dimer models. We discuss some examples and use the algorithm to show that all toric noncommutative crepant resolutions of a finite quotient of the conifold singularity can be obtained by mutating one basic dimer model. We also discuss how this algorithm might be extended to higher dimensional singularities.

In this paper, we construct a vast collection of maximal numerically Calabi–Yau orders utilising a noncommutative analogue of the well-known commutative cyclic covering trick. Such orders play an integral role in the Mori program for orders on projective surfaces and although we know a substantial amount about them, there are relatively few known examples.

The graded modules over noncommutative algebras often have minimal free resolutions of infinite length, resulting in infinite Castelnuovo–Mumford regularity . In Kang et al. (2010) [6] , we introduced a generalized notion of Castelnuovo–Mumford regularity to overcome this difficulty. In this paper, we compute the generalized Castelnuovo–Mumford regularity for integrable highest weight representations of all affine Kac–Moody algebras. It is shown that the generalized regularity depends only on the type and rank of algebras and the level of representations.

In this paper we define the notion of “graded algebra with symmetries”. This notion is a generalization of the extended differential forms. We prove that for a graded algebra with symmetries T, we associate a subalgebra Ω ⁎ which generalizes the noncommutative differential forms. Using this algebra Ω ⁎ , we can define the Hochschild and cyclic homologies, cup i-products and the Steenrod squares.