If $k$ is a field and $R$ is a commutative $k$-algebra, we explore the question of when the ring $D_{R|k}$ of $k$-linear differential operators on $R$ is isomorphic to its opposite ring. Under mild hypotheses, we prove this is the case whenever $R$ Gorenstein local or when $R$ is a ring of invariants. As a key step in the proof we show that in many cases of interest canonical modules admit right $D$-module structures.
After this work was completed we realized that some of our results were already proved in higher generality by Yekutieli, albeit using more sophisticated methods.
In Picard-Vessiot theory, the Galois theory for linear differential equations, the Picard-Vessiot ring plays an important role, since it is the Picard-Vessiot ring which is a torsor (principal homogeneous space) for the Galois group (scheme). Like fields are simple rings having only (0) and (1) as ideals, the Picard-Vessiot ring is a differentially simple ring, i.e. a differential ring having only (0) and (1) as differential ideals. Having in mind that the classical Galois theory is a theory of extensions of fields, i.e. of simple rings, it is quite natural to ask whether one can also set up a…
In the article "Picard-Vessiot theory of differentially simple rings" we established a Picard-Vessiot theory over differentially simple rings which may not be fields. Differential modules over such rings were proven to be locally free but do not have to be free as modules. In this article, we give a family of examples of non-free differential modules, and compute Picard-Vessiot rings as well as Galois groups for them.
Levasseur and Stafford described the rings of differential operators on various classical invariant rings of characteristic zero; in each of the cases that they considered, the differential operators form a simple ring. Towards an attack on the simplicity of rings of differential operators on invariant rings of linearly reductive groups over the complex numbers, Smith and Van den Bergh asked if differential operators on the corresponding rings of positive prime characteristic lift to characteristic zero differential operators. We prove that, in general, this is not the case for determinantal h…
Let $δ$ be a derivation of a locally nilpotent ring $R$. Then the differential polynomial ring $R[X; δ]$ cannot be mapped onto a ring with a non-zero idempotent. This answers a recent question by Greenfeld, Smoktunowicz and Ziembowski.
We study iterated differential polynomial rings over a locally nilpotent ring and show that a large class of such rings are Behrens radical. This extends results of Chebotar and Chen et al.
The differential Brauer monoid of a differential commutative ring R s defined. Its elements are the isomorphism classes of differential Azumaya R algebras with operation from tensor product subject to the relation that two such algebras are equivalent if matrix algebras over them are differentially isomorphic. The Bauer monoid, which is a group, is the same thing without the differential requirement. The differential Brauer monoid is then determined from the Brauer monoids of R and its ring of constants and the submonoid whose underlying Azumaya algebras are matrix rings.
We consider the Noetherian properties of the ring of differential operators of an affine semigroup algebra. First we show that it is always right Noetherian. Next we give a condition, based on the data of the difference between the semigroup and its scored closure, for the ring of differential operators being anti-isomorphic to another ring of differential operators. Using this, we prove that the ring of differential operators is left Noetherian if the condition is satisfied. Moreover we give some other conditions for the ring of differential operators being left Noetherian. Finally we conject…
In this article we give a characterization of left (right) quasi-duo differential polynomial rings. In particular, we show that a differential polynomial ring is left quasi-duo if and only if it is right quasi-duo. This yields a partial answer to a question posed by Lam and Dugas in 2005. We provide non-trivial examples of such rings and give a complete description of the maximal ideals of an arbitrary quasi-duo differential polynomial ring. Moreover, we show that there is no left (right) quasi-duo differential polynomial ring in several indeterminates.
We answer a question by Shestakov on the Jacobson radical in differential polynomial rings. We show that if R is a locally nilpotent ring with a derivation D then R[X;D] need not be Jacobson radical. We also show that J(R[X;D])\cap R is a nil ideal of R in the case where D is a locally nilpotent derivation and R is an algebra over an uncountable field.
In this paper, we will consider derived equivalences for differential graded endomorphism algebras by Keller's approaches. First we construct derived equivalences of differential graded algebras which are endomorphism algebras of the objects from a triangle in the homotopy category of differential graded algebras. We also obtain derived equivalences of differential graded endomorphism algebras from a standard derived equivalence of finite dimensional algebras. Moreover, under some conditions, the cohomology rings of these differential graded endomorphism algebras are also derived equivalent. T…
Whereas Holm proved that the ring of differential operators on a generic hyperplane arrangement is finitely generated as an algebra, the problem of its Noetherian properties is still open. In this article, after proving that the ring of differential operators on a central arrangement is right Noetherian if and only if it is left Noetherian, we prove that the ring of differential operators on a central 2-arrangement is Noetherian. In addition, we prove that its graded ring associated to the order filtration is not Noetherian when the number of the consistuent hyperplanes is greater than 1.
It is shown that, under some natural assumptions, the tensor product of differentially smooth algebras and the skew-polynomial rings over differentially smooth algebras are differentially smooth.
Let R be a ring with a derivation δ. In this paper, we prove that an analogue of Amitsur's property holds for left T-nilpotent radideals of pseudo-differential operator rings R((x^{-1}; δ)), where R is a delta-compatible ring. As a direct consequence of this fact, we obtain an alternative characterization of the prime radical of R((x^{-1}; δ)).
The Gelfand--Kirillov dimension has gained importance since its introduction as an tool in the study of non-commutative infinite dimensional algebras and their modules. In this paper we show a dichotomy for the Gelfand--Kirillov dimension of simple modules over certain simple rings of differential operators.
In this paper we discuss under which conditions cyclic essential extensions of simple modules over a differential operator ring R[z;d] are Artinian. In particular, we study the case when R is either d-simple or d-primitive. Furthermore, we obtain important results when R is an affine algebra of Kull dimension 2. As an application we characterize the differential operator rings C[x,y][z;d] for which cyclic essential extensions of simple modules are Artinian.
Using the language of operated algebras, we construct and investigate a class of operator rings and enriched modules induced by a derivation or Rota-Baxter operator. In applying the general framework to univariate polynomials, one is led to the integro-differential analogs of the classical Weyl algebra. These are analyzed in terms of skew polynomial rings and noncommutative Groebner bases.
Let $R$ be a ring satisfying a polynomial identity and let $δ$ be a derivation of $R$. We show that if $N$ is the nil radical of $R$ then $δ(N)\subseteq N$ and the Jacobson radical of $R[x;δ]$ is equal to $N[x;δ]$. As a consequence, we have that if $R$ is locally nilpotent then $R[x;δ]$ is locally nilpotent. This affirmatively answers a question of Smoktunowicz and Ziembowski.
In this article, the ring of flows of autonomous differential equations of order one on integral domains is constructed. First, we build the autonomous ring $\Opa(\hurw_{R}[[x]])$ and then its structure is studied. Next, we build the ring of formal exponential generating series of the ring $\Opa(\hurw_ {R} [[x]]) $, where it is possible to find solutions of differential equations of order one when the vector field of the system can be decomposed in sum or product of functions.
For ring of differential operators on smooth affine algebraic variety over perfect field of prime characteristic a set of algebra generators and a set of defining relations are found explicitly.
We show that given two smooth affine varieties over $\mathbb{C}$ such that their rings of differential operators are Morita equivalent, then corresponding cotangent bundles are isomorphic as symplectic varieties.
Let $k$ be an algebraically closed field of characteristic zero, let $X$ and $Y$ be smooth irreducible algebraic curves over $k$, and let $D(X)$ and $D(Y)$ denote respectively the quotient division rings of the ring of differential operators of $X$ and $Y$. We show that if there is a $k$-algebra embedding of $D(X)$ into $D(Y)$ then the genus of $X$ must be less than or equal to the genus of $Y$, answering a question of the first-named author and Smoktunowicz.
We study injective hulls of simple modules over differential operator rings $R[θ; d]$, providing necessary conditions under which these modules are locally Artinian. As a consequence we characterize Ore extensions of $S=K[x][θ;σ, d]$ for $d$ a $K$-linear derivation and $σ$ a $K$-linear automorphism of $K[x]$ such that injective hulls of simple $S$-modules are locally Artinian.
Let W be a finite dimensional representation of a linearly reductive group G over a field k. Motivated by their work on classical rings of invariants, Levasseur and Stafford asked whether the ring of invariants under G of the symmetric algebra of W has a simple ring of differential operators. In this paper, we show that this is true in prime characteristic. Indeed, if R is a graded subring of a polynomial ring over a perfect field of characteristic p>0 and if the inclusionof R into S splits, then D_k(R) is a simple ring. In the last section of the paper, we discuss how one might try to deduce …
Differential graded algebra techniques have played a crucial role in the development of homological algebra, especially in the study of homological properties of commutative rings carried out by Serre, Tate, Gulliksen, Avramov, and others. In this article, we extend the construction of the Koszul complex and acyclic closure to a more general setting. As an application of our constructions, we shine some light on the structure of the Ext algebra of quotients of skew polynomial rings by ideals generated by normal elements. As a consequence, we give a presentation of the Ext algebra when the elem…
Let $K$ be a field of characteristic $p>0$. It is proved that each automorphism $\s \in \Aut_K(\CDPn)$ of the ring $\CDPn$ of differential operators on a polynomial algebra $P_n= K[x_1, ..., x_n]$ is {\em uniquely} determined by the elements $\s (x_1), ... ,\s (x_n)$, and the set $\Frob (\CDPn)$ of all the extensions of the Frobenius from certain maximal commutative polynomial subalgebras of $\CDPn$, like $P_n$, is equal to $\Aut_K(\CDPn) \cdot \CF$ where $\CF$ is the set of all the extensions of the Frobenius from $P_n$ to $\CDPn$ that leave invariant the subalgebra of scalar differential ope…
Two-dimensional integrable differential calculi for classes of Ore extensions of the polynomial ring and the Laurent polynomial ring in one variable are constructed. Thus it is concluded that all affine pointed Hopf domains of Gelfand-Kirillov dimension two which are not polynomial identity rings are differentially smooth.
We consider the following question: When are rings of differential operators coherent? If $A$ is a finitely generated smooth domain over a field $k$ of characteristic $0$, then the ring $D$ of differential operators on $A$ is a Noetherian ring and a finitely generated $k$-algebra. However, when $k$ has characteristic $p > 0$ or when $A$ is singular, this is no longer true. In fact, Bernstein, Gelfand and Gelfand showed that for the cubic cone $A = k[x,y,z]/(x^3 + y^3 + z^3)$, the ring $D$ is neither Noetherian nor finitely generated if $k$ has characteristic $0$, and the same is true for the p…
Following the definition of quantum differential operators given by Lunts and Rosenberg in (Sel. math., New ser. 3 (1997) 335--359), we show that the ring of quantum differential operators on the affine line is the ring generated by x and \del, the familiar differential operators on the line, along with two additional operators which we call \del^β^1 and \del^β^-1. We describe this ring both as a subring of the ring of graded endomorphisms and as a ring given by generators and relations. From this starting point, we are able to describe the ring of quantum differential operators on affine n-sp…
We define when a noetherian ring R is called a right ( or a left) weakly krull symmetric ring . We then prove that if R is a right ( or a left ) krull homogenous ring then R is a right ( or a left ) weakly krull symmetric ring . This result modifies the main result of [2] . The key terms introduced in this paper are of independent interest .
Let $R$ be a ring and $σ$ an endomorphism of $R$. In this note, we study the transfert of the symmetry ($σ$-symmetry) and reversibility ($σ$-reversibility) from $R$ to its skew power series ring $R[[x;σ]]$. Moreover, we study on the relationship between the Baerness, quasi-Baerness and p.p.-property of a ring $R$ and these of the skew power series ring $R[[x;σ]]$ in case $R$ is right $σ$-reversible. As a consequence we obtain a generalization of \cite{hong/2000}.
Let $k$ be an algebraically closed field of characteristic 0 and let $A$ be a finitely generated $k$-algebra that is a domain whose Gelfand-Kirillov dimension is in $[2,3)$. We show that if $A$ has a nonzero locally nilpotent derivation then $A$ has quadratic growth. In addition to this, we show that $A$ either satisfies a polynomial identity or $A$ is isomorphic to a subalgebra of $\mathcal{D}(X)$, the ring of differential operators on an irreducible smooth affine curve $X$, and $A$ is birationally isomorphic to $\mathcal{D}(X)$.
Let $A$ be a $d\times n$ integer matrix whose column vectors generate the lattice $\Z^d$, and let $D(R_A)$ be the ring of differential operators on the affine toric variety defined by $A$. We show that the classification of $A$-hypergeometric systems and that of $\Z^d$-graded simple $D(R_A)$-modules (up to shift) are the same. We then show that the set of $\Z^d$-homogeneous primitive ideals of $D(R_A)$ is finite. Furthermore, we give conditions for the algebra $D(R_A)$ being simple.
We describe the center of the ring $\Diff(n)$ of $\h$-deformed differential operators of type A. We establish an isomorphism between certain localizations of $\Diff(n)$ and the Weyl algebra $\text{W}_n$ extended by $n$ indeterminates.
Given a non-associative unital ring $R$, a monoid $G$ and a set $π$ of additive maps $R \rightarrow R$, we introduce the Ore monoid ring $R[π; G]$, and, in a special case, the differential monoid ring. We show that these structures generalize, in a natural way, not only the classical Ore extensions and differential polynomial rings, but also the constructions, introduced by Cojuhari, defined by so-called $D$-structures $π$. Moreover, for commutative monoids, we give necessary and sufficient conditions for differential monoid rings to be simple. We use this in a special case to obtain new and s…
This paper presents the right-left symmetry of the CS and max-min CS conditions on nonsingular rings, and generalization to nonsingular modules. We prove that a ring is right nonsingular right CS and left Utumi if and only if it is left nonsingular left CS and right Utumi. A nonsingular Utumi ring is right max (resp. right min, right max-min) CS if and only if it is left min (resp. left max, left max-min) CS. In addition, a semiprime nonsingular ring is right max-min CS with finite right uniform dimension if and only if it is left max-min CS with finite left uniform dimension.
The universal enveloping algebra U(g) of a Lie algebra g acts on its representation ring R through D(R), the ring of differential operators on R. A quantised universal enveloping algebra (or "quantum group") is a deformation of a universal enveloping algebra and acts not through the differential operators of its representation ring but through the quantised differential operators of its representation ring. We present this situation for the quantum group of sl_2.
We classify subalgebras of a ring of differential operators which are big in the sense that the extension of associated graded rings is finite. We show that these subalgebras correspond, up to automorphisms, to uniformly ramified finite morphisms. This generalizes a theorem of Levasseur-Stafford on the generators of the invariants of a Weyl algebra under a finite group.
In this note we show that if a noetherian ring R is left and right Krull-homogenous and if: Λ={P\textexclamdown ε spec.R/ |R/P\textexclamdown|_r =|R|_r} and v ={Qj ε spec.R| |R/Qj|l=|R|l} and P =\cap P\textexclamdownε ΛP\textexclamdown and Q = \cap QjεVQj then the following hold true; (1) Pn =0, for some integer n\geq1 (2) Λ=v
Let $K$ be a field of characteristic $p>0$. It is proved that the group $\Aut_{ord}(\CD (L_n))$ of order preserving automorphisms of the ring $\CD (L_n)$ of differential operators on a Laurent polynomial algebra $L_n:= K[x_1^{\pm 1}, ..., x_n^{\pm 1}]$ is isomorphic to a skew direct product of groups $\Zp^n \rtimes \Aut_K(L_n)$ where $\Zp$ is the ring of $p$-adic integers. Moreover, the group $\Aut_{ord}(\CD (L_n))$ is found explicitly. Similarly, $\Aut_{ord}(\CDPn)\simeq \Aut_K(P_n)$ where $P_n: =K[x_1, ..., x_n]$ is a polynomial algebra.
Let $K$ be a field of arbitrary characteristic, $\CA$ be a commutative $K$-algebra which is a domain of essentially finite type (eg, the algebra of functions on an irreducible affine algebraic variety), $\ga_r$ be its {\em Jacobian ideal}, $\CD (\CA )$ be the algebra of differential operators on the algebra $\CA$. The aim of the paper is to give a simplicity criterion for the algebra $\CD (\CA )$: {\em The algebra $\CD (\CA )$ is simple iff $\CD (\CA ) \ga_r^i\CD (\CA )= \CD (\CA )$ for all $i\geq 1$ provided the field $K$ is a perfect field.} Furthermore, a simplicity criterion is given for t…
Let $R$ be an integral domain of characteristic zero. We prove that a function $D\colon R\to R$ is a derivation of order $n$ if and only if $D$ belongs to the closure of the set of differential operators of degree $n$ in the product topology of $R^R$, where the image space is endowed with the discrete topology. In other words, $f$ is a derivation of order $n$ if and only if, for every finite set $F\subset R$, there is a differential operator $D$ of degree $n$ such that $f=D$ on $F$. We also prove that if $d_1, \dots, d_n$ are nonzero derivations on $R$, then $d_1 \circ \ldots \circ d_n$ is a d…
We construct a cofibrantly generated model structure on the category of differential non-negatively graded quasi-coherent commutative $D_X$-algebras, where $D_X$ is the sheaf of differential operators of a smooth afine algebraic variety X. The paper contains an extensive appendix on D-modules, sheaves versus global sections, some more technical model categorical issues, as well as on relative Sullivan algebras. This article is the first of a series of works -located at the interface of homotopical algebra, algebraic geometry, and mathematical physics - on a derived D-geometric approach to the …
We study the existence of a Batalin-Vilkovisky differential on the complete cohomology ring of a Frobenius algebra.We construct a Batalin-Vilkovisky differential on the complete cohomology ring in the case of Frobenius algebras with diagonalizable Nakayama automorphisms.
For the algebra $\mI_1= K<x, \frac{d}{dx}, \int>$ of polynomial integro-differential operators over a field $K$ of characteristic zero, a classification of simple modules is given. It is proved that $\mI_1$ is a left and right coherent algebra. The {\em Strong Compact-Fredholm Alternative} is proved for $\mI_1$. The endomorphism algebra of each simple $\mI_1$-module is a {\em finite dimensional} skew field. In contrast to the first Weyl algebra, the centralizer of a non-scalar integro-differential operator can be a noncommutative, non-Noetherian, non-finitely generated algebra which is not a d…
A torsion theory is called differential (higher differential) if a derivation (higher derivation) can be extended from any module to the module of quotients corresponding to the torsion theory. We study conditions equivalent to higher differentiability of a torsion theory. It is known that the Lambek, Goldie and any perfect torsion theories are differential. We show that these classes of torsion theories are higher differential as well. Then, we study conditions under which a higher derivation extended to a right module of quotients extends also to a right module of quotients with respect to a…
