Mathematics

Levasseur and Stafford described the rings of differential operators on various classical invariant rings of characteristic zero; in each of the cases 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 hypersurfaces, as well as for Pfaffian and symmetric determinantal hypersurfaces. We also prove that, with few exceptions, these hypersurfaces do not admit a mod p 2 lift of the Frobenius endomorphism.

Foxby defined the (Krull) dimension of a complex of modules over a commutative Noetherian ring in terms of the dimension of its homology modules. In this note it is proved that the dimension of a bounded complex of free modules of finite rank can be computed directly from the matrices representing the differentials of the complex.

Let M be a free module of rank m over a commutative unital ring R and let N be its free submodule. We consider the problem of when a given element of the exterior product ? p M is divisible, in a sense, over elements of the exterior product ? r N , where r?�p . Precisely, we give conditions under which a given η??? p M can be expressed as a finite sum of elements of ? r N multiplied (via the exterior product) by elements of ? p?�r M . Necessary and sufficient conditions for such divisibility take a simple form, provided that the submodule is embedded in M with singularities having the depth larger then p?�r+1 . In the special case where r=rankN the divisibility property means that η=Ω?��?where Ω is the product ? 1 ?�⋯??? r of elements of a basis of N and γ is an element of ? p?�r M . More detailed statements of these results are then used to state criteria for existence and uniqueness of algebraic residua when the "divisor" is defined by elements f 1 ,?? f k ?�R . Special cases are multidimensional logarithmic residua in complex analysis.

Divisors whose Jacobian ideal is of linear type have received a lot of attention recently because of its connections with the theory of D-modules. In this work we are interested on divisors of expected Jacobian type, that is, divisors whose gradient ideal is of linear type and the relation type of its Jacobian ideal coincides with the reduction number with respect to the gradient ideal plus one. We provide conditions in order to be able to describe precisely the equations of the Rees algebra of the Jacobian ideal. We also relate the relation type of the Jacobian ideal to some D-module theoretic invariant given by the degree of the Kashiwara operator.

Let D be an integral domain, S(D)=I(D) ( I t (D)) the set of proper nonzero ideals (proper t -ideals) of D, Max(D) (t - Max(D) the set of maximal ( t -) ideals of D, and let P be a predicate on S(D) with nonempty truth set Π S(D) ⊆S(D) , where P can be: "---is invertible" or "---is divisorial" etc. . We say S(D) meets P (S(D)⊲P) if ∀s∈S(D)∃π∈ Π S(D) (P) (s⊆ π) . Clearly S(D)⊲P⇔Max(D) ( t - Max(D))⊆ Π S(D) (P) . We show that if S(D) ⊲P, we have no control over dimD . We also show that I(D) ⊲P does not imply I(R) while I t (D) ⊲P implies I t (R) ⊲P, for most choices of P, when R=D[X] and have examples to show that generally S(D)⊲P does not extend to rings of fractions. We study restrictions that may control the dimension of when S(D)⊲P. We also say S(D)⊲P with a twist (S(D) ⊲ t P) if ∀s∈S(D) ∃π∈ Π D (P)( s n ⊆π for some n∈N) and study along the same lines as S(D)⊲P and provide examples.

Let M be a module over a commutative ring R . The annihilating-submodule graph of M , denoted by AG(M) , is a simple graph in which a non-zero submodule N of M is a vertex if and only if there exists a non-zero proper submodule K of M such that NK=(0) , where NK , the product of N and K , is denoted by (N:M)(K:M)M and two distinct vertices N and K are adjacent if and only if NK=(0) . This graph is a submodule version of the annihilating-ideal graph and under some conditions, is isomorphic with an induced subgraph of the Zariski topology-graph G( τ T ) which was introduced in (The Zariski topology-graph of modules over commutative rings, Comm. Algebra., 42 (2014), 3283--3296). In this paper, we study the domination number of AG(M) and some connections between the graph-theoretic properties of AG(M) and algebraic properties of module M .

We study homological properties of a locally complete intersection ring by importing facts from homological algebra over exterior algebras. One application is showing that the thick subcategories of the bounded derived category of a locally complete intersection ring are self-dual under Grothendieck duality. This was proved by Stevenson when the ring is a quotient of a regular ring modulo a regular sequence; we offer two independent proofs in the more general setting. Second, we use these techniques to supply new proofs that complete intersections possess symmetry of complexity.

We describe all the trees with the property that the corresponding edge ideal of the square of the tree has a linear resolution. As a consequence, we give a complete characterization of those trees T for which the square is co-chordal, that is the complement of the square, ( T 2 ) c , is a chordal graph. For particular classes of trees such as paths and double brooms we determine the Krull dimension and the projective dimension.

A graded ideal I in K[ x 1 ,…, x n ] , where K is a field, is said to have almost maximal finite index if its minimal free resolution is linear up to the homological degree pd(I)−2 , while it is not linear at the homological degree pd(I)−1 , where pd(I) denotes the projective dimension of I . In this paper we classify the graphs whose edge ideals have this property. This in particular shows that for edge ideals the property of having almost maximal finite index does not depend on the characteristic of K . We also compute the non-linear Betti numbers of these ideals. Finally, we show that for the edge ideal I of a graph G with almost maximal finite index, the ideal I s has a linear resolution for s≥2 if and only if the complementary graph G ¯ does not contain induced cycles of length 4 .

In this paper, we propose a new method for localization of polynomial ideal, which we call "Local Primary Algorithm". For an ideal I and a prime ideal P , our method computes a P -primary component of I after checking if P is associated with I by using "double ideal quotient" (I:(I:P)) and its variants which give us a lot of information about localization of I .