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We introduce differential primary decompositions for ideals in a commutative ring. Ideal membership is characterized by differential conditions. The minimal number of conditions needed is the arithmetic multiplicity. Minimal differential primary decompositions are unique up to change of bases. Our results generalize the construction of Noetherian operators for primary ideals in the analytic theory of Ehrenpreis-Palamodov, and they offer a concise method for representing affine schemes. The case of modules is also addressed. We implemented an algorithm in Macaulay2 that computes the minimal decomposition for an ideal in a polynomial ring.
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An ideal in a polynomial ring encodes a system of linear partial differential equations with constant coefficients. Primary decomposition organizes the solutions to the PDE. This paper develops a novel structure theory for primary ideals in a polynomial ring. We characterize primary ideals in terms of PDE, punctual Hilbert schemes, relative Weyl algebras, and the join construction. Solving the PDE described by a primary ideal amounts to computing Noetherian operators in the sense of Ehrenpreis and Palamodov. We develop new algorithms for this task, and we present efficient implementations.
In their work on differential operators in positive characteristic, Smith and Van den Bergh define and study the derived functors of differential operators; they arise naturally as obstructions to differential operators reducing to positive characteristic. In this note, we provide formulas for the ring of differential operators as well as these derived functors of differential operators. We apply these descriptions to show that differential operators behave well under reduction to positive characteristic under certain hypotheses. We show that these functors also detect a number of interesting properties of singularities.
An explicit lattice point realization is provided for the primary components of an arbitrary binomial ideal in characteristic zero. This decomposition is derived from a characteristic-free combinatorial description of certain primary components of binomial ideals in affine semigroup rings, namely those that are associated to faces of the semigroup. These results are intimately connected to hypergeometric differential equations in several variables.
We study the symmetric subquotient decomposition of the associated graded algebras $A^*$ of a non-homogeneous commutative Artinian Gorenstein (AG) algebra $A$. This decomposition arises from the stratification of $A^*$ by a sequence of ideals $A^*=C_A(0)supset C_A(1)supsetcdots$ whose successive quotients $Q(a)=C(a)/C(a+1)$ are reflexive $A^*$ modules. These were introduced by the first author, and have been used more recently by several groups, especially those interested in short Gorenstein algebras, and in the scheme length (cactus rank) of forms. For us a Gorenstein sequence is an integer sequence $H$ occurring as the Hilbert function for an AG algebra $A$, that is not necessarily homogeneous. Such a Hilbert function $H(A)$ is the sum of symmetric non-negative sequences $H_A(a)=H(Q_A(a))$, each having center of symmetry $(j-a)/2$ where $j$ is the socle degree of $A$: we call these the symmetry conditions, and the decomposition $mathcal{D}(A)=(H_A(0),H_A(1),ldots)$ the symmetric decomposition of $H(A)$. We here study which sequences may occur as the summands $H_A(a)$: in particular we construct in a systematic way examples of AG algebras $A$ for which $H_A(a)$ can have interior zeroes, as $H_A(a)=(0,s,0,ldots,0,s,0)$. We also study the symmetric decomposition sets $mathcal{D}(A)$, and in particular determine which sequences $H_A(a)$ can be non-zero when the dual generator is linear in a subset of the variables. Several groups have studied exotic summands of the Macaulay dual generator $F$. Studying these, we recall a normal form for the Macaulay dual generator of an AG algebra that has no exotic summands. We apply this to Gorenstein algebras that are connected sums. We give throughout many examples and counterexamples, and conclude with some open questions about symmetric decomposition.
We introduce a new formalism of differential operators for a general associative algebra A. It replaces Grothendiecks notion of differential operator on a commutative algebra in such a way that derivations of the commutative algebra are replaced by DDer(A), the bimodule of double derivations. Our differential operators act not on the algebra A itself but rather on F(A), a certain `Fock space associated to any noncommutative algebra A in a functorial way. The corresponding algebra D(F(A)), of differential operators, is filtered and gr D(F(A)), the associated graded algebra, is commutative in some `twisted sense. The resulting double Poisson structure on gr D(F(A)) is closely related to the one introduced by Van den Bergh. Specifically, we prove that gr D(F(A))=F(T_A(DDer(A)), provided A is smooth. It is crucial for our construction that the Fock space F(A) carries an extra-structure of a wheelgebra, a new notion closely related to the notion of a wheeled PROP. There are also notions of Lie wheelgebras, and so on. In that language, D(F(A)) becomes the universal enveloping wheelgebra of a Lie wheelgebroid of double derivations. In the second part of the paper we show, extending a classical construction of Koszul to the noncommutative setting, that any Ricci-flat, torsion-free bimodule connection on DDer(A) gives rise to a second order (wheeled) differential operator, a noncommutative analogue of the BV-operator.
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