In this paper we investigate principal prime ideals in commutative rings. Among other things we characterize the principal prime ideals that are both minimal and maximal and characterize the maximal ideals of a polynomial ring that are principal. Our main result is that if (p) is a principal prime ideal of an atomic ring R, then ht(p)≤1.

The holonomic rank of the A-hypergeometric system MA(β) is the degree of the toric ideal IA for generic parameters; in general, this is only a lower bound. To the semigroup ring of A we attach the ranking arrangement and use this algebraic invariant and the exceptional arrangement of non-generic parameters to construct a combinatorial formula for the rank jump of MA(β). As consequences, we obtain a refinement of the stratification of the exceptional arrangement by the rank of MA(β) and show that the Zariski closure of each of its strata is a union of translates of linear subspaces of the parameter space. These results hold for generalized A-hypergeometric systems as well, where the semigroup ring of A is replaced by a non-trivial weakly toric module M⊆ℂ[ℤA] . We also provide a direct proof of the main result in [M. Saito, Isomorphism classes of A-hypergeometric systems, Compositio Math. 128 (2001), 323–338] regarding the isomorphism classes of MA (β) .

Two fundamental questions in the theory of Gröbner bases are decision (‘Is a basis G of a polynomial ideal a Gröbner basis?’) and transformation (‘If it is not, how do we transform it into a Gröbner basis?’) This paper considers the first question. It is well known that G is a Gröbner basis if and only if a certain set of polynomials (the S-polynomials) satisfy a certain property. In general there are m(m−1)/2 of these, where m is the number of polynomials in G, but criteria due to Buchberger and others often allow one to consider a smaller number. This paper presents two original results. The first is a new characterization theorem for Gröbner bases that makes use of a new criterion that extends Buchberger’s criteria. The second is the identification of a class of polynomial systems G for which the new criterion has dramatic impact, reducing the worst-case scenario from m(m−1)/2 S-polynomials to m−1.

For a split semisimple Chevalley group scheme G with Lie algebra over an arbitrary base scheme S, we consider the quotient of by the adjoint action of G. We study in detail the structure of over S. Given a maximal torus T with Lie algebra and associated Weyl group W, we show that the Chevalley morphism π : /W → /G is an isomorphism except for the group Sp2n over a base with 2-torsion. In this case this morphism is only dominant and we compute it explicitly. We compute the adjoint quotient in some other classical cases, yielding examples where the formation of the quotient → /G commutes, or does not commute, with base change on S.

We prove the semisimplicity conjecture for A-motives over finitely generated fields K. This conjecture states that the rational Tate modules V𝔭(M) of a semisimple A-motive M are semisimple as representations of the absolute Galois group of K. This theorem is in analogy with known results for abelian varieties and Drinfeld modules, and has been sketched previously by Tamagawa. We deduce two consequences of the theorem for the algebraic monodromy groups G𝔭(M) associated to an A-motive M by Tannakian duality. The first requires no semisimplicity condition on M and states that G𝔭(M) may be identified naturally with the Zariski closure of the image of the absolute Galois group of K in the automorphism group of V𝔭(M). The second states that the connected component of G𝔭(M) is reductive if M is semisimple and has a separable endomorphism algebra.

Let f be a probability density function on (a, b) ⊂ (0, ∞), and consider the class C f of all probability density functions of the form Pf, where P is a polynomial. Assume that if X has its density in C f then the equilibrium probability density x ↦ P(X > x) / E(X) also belongs to C f : this happens, for instance, when f(x) = Ce−λx or f(x) = C(b − x)λ−1. We show in the present paper that these two cases are the only possibilities. This surprising result is achieved with an unusual tool in renewal theory, by using ideals of polynomials.

Let X be a smooth projective variety of dimension n in Pr, and let π:X→Pn+c be a general linear projection, with c>0. In this paper we bound the scheme-theoretic complexity of the fibers of π. In his famous work on stable mappings, Mather extended the classical results by showing that the number of distinct points in the fiber is bounded by B:=n/c+1, and that, when n is not too large, the degree of the fiber (taking the scheme structure into account) is also bounded by B. A result of Lazarsfeld shows that this fails dramatically for n≫0. We describe a new invariant of the scheme-theoretic fiber that agrees with the degree in many cases and is always bounded by B. We deduce, for example, that if we write a fiber as the disjoint union of schemes Y′ and Y′′ such that Y′ is the union of the locally complete intersection components of Y, then deg Y′+deg Y′′red≤B. Our method also gives a sharp bound on the subvariety of Pr swept out by the l-secant lines of X for any positive integer l, and we discuss a corresponding bound for highly secant linear spaces of higher dimension. These results extend Ran’s ‘dimension +2 secant lemma’.

Let I be an ideal of a commutative Noetherian local ring R, and M and N two finitely generated modules. Let t be a positive integer. We mainly prove that (i) if HIi(M,N) is Artinian for all i<t, then HIi(M,N) is I-cofinite for all i<t and Hom(R/I,HIt(M,N)) is finitely generated; (ii) if d=pd(M)<∞ and dim N=n<∞, then HId+n(M,N) is I-cofinite. We also prove that if M is a nonzero cyclic R-module, then HIi(N) is finitely generated for all i<t if and only if HIi(M,N) is finitely generated for all i<t.

The face ring of a homology manifold (without boundary) modulo a generic system of parameters is studied. Its socle is computed and it is verified that a particular quotient of this ring is Gorenstein. This fact is used to prove that the algebraic g-conjecture for spheres implies all enumerative consequences of its far-reaching generalization (due to Kalai) to manifolds. A special case of Kalai’s conjecture is established for homology manifolds that have a codimension-two face whose link contains many vertices.

We define and study virtual representation spaces for vectors having both positive and negative dimensions at the vertices of a quiver without oriented cycles. We consider the natural semi-invariants on these spaces which we call virtual semi-invariants and prove that they satisfy the three basic theorems: the first fundamental theorem, the saturation theorem and the canonical decomposition theorem. In the special case of Dynkin quivers with n vertices, this gives the fundamental interrelationship between supports of the semi-invariants and the tilting triangulation of the (n−1)-sphere.

We show that the multiplicity of a prime p as a factor of the resultant of two polynomials with integer coefficients is at least the degree of their greatest common divisor modulo p. This answers an open question by Konyagin and Shparlinski.

Natural sufficient conditions for a polynomial to have a local minimum at a point are considered. These conditions tend to hold with probability 1. It is shown that polynomials satisfying these conditions at each minimum point have nice presentations in terms of sums of squares. Applications are given to optimization on a compact set and also to global optimization. In many cases, there are degree bounds for such presentations. These bounds are of theoretical interest, but they appear to be too large to be of much practical use at present. In the final section, other more concrete degree bounds are obtained which ensure at least that the feasible set of solutions is not empty.

In this paper we prove that a generalized version of the Minimal Resolution Conjecture given by Mustaţă holds for certain general sets of points on a smooth cubic surface $X\,\subset \,{{\mathbb{P}}^{3}}$ . The main tool used is Gorenstein liaison theory and, more precisely, the relationship between the free resolutions of two linked schemes.

Let (R,𝔪) be a commutative Noetherian local ring, let I be an ideal of R and let M and N be finitely generated R-modules. Assume that , . First, we give the formula for the attached primes of the top generalized local cohomology module HId+n(M,N); later, we prove that if Att(HId+n(M,N))=Att(HJd+n(M,N)), then HId+n(M,N)=HJd+n(M,N).

A notion of Hochschild cohomology HH*(𝒜) of an abelian category 𝒜 was defined by Lowen and Van den Bergh (Adv. Math. 198 (2005), 172–221). They also showed the existence of a characteristic morphism χ from the Hochschild cohomology of 𝒜 into the graded centre ℨ*(Db(𝒜)) of the bounded derived category of 𝒜. An element c∈HH2(𝒜) corresponds to a first-order deformation 𝒜c of 𝒜 (Lowen and Van den Bergh, Trans. Amer. Math. Soc. 358 (2006), 5441–5483). The problem of deforming an object M∈Db(𝒜) to Db(𝒜c) was treated by Lowen (Comm. Algebra 33 (2005), 3195–3223). In this paper we show that the element χ(c)M∈Ext𝒜2(M,M) is precisely the obstruction to deforming M to Db(𝒜c). Hence, this paper provides a missing link between the above works. Finally we discuss some implications of these facts in the direction of a ‘derived deformation theory’.

We give the first examples over finite fields of rings of invariants that are not finitely generated. (The examples work over arbitrary fields, for example the rational numbers.) The group involved can be as small as three copies of the additive group. The failure of finite generation comes from certain elliptic fibrations or abelian surface fibrations having positive Mordell–Weil rank. Our work suggests a generalization of the Morrison–Kawamata cone conjecture on Calabi–Yau fiber spaces to klt Calabi–Yau pairs. We prove the conjecture in dimension two under the assumption that the anticanonical bundle is semi-ample.

Let k be an algebraically closed field of positive characteristic p. We consider which finite groups G have the property that every faithful action of G on a connected smooth projective curve over k lifts to characteristic zero. Oort conjectured that cyclic groups have this property. We show that if a cyclic-by-p group G has this property, then G must be either cyclic or dihedral, with the exception of A4 in characteristic two. This proves one direction of a strong form of the Oort conjecture.

We describe an algorithm for computing parameter-test-ideals in certain local Cohen–Macaulay rings. The algorithm is based on the study of a Frobenius map on the injective hull of the residue field of the ring and on the application of Sharp’s notion of ‘special ideals’. Our techniques also provide an algorithm for computing indices of nilpotency of Frobenius actions on top local cohomology modules of the ring and on the injective hull of its residue field. The study of nilpotent elements on injective hulls of residue fields also yields a great simplification of the proof of the celebrated result in the article Generators of D-modules in positive characteristic (J. Alvarez-Montaner, M. Blickle and G. Lyubeznik, Math. Res. Lett. 12 (2005), 459–473).

Let R and S be commutative rings, not necessarily with identity. We investigate the ideals, prime ideals, radical ideals, primary ideals, and maximal ideals of R×S. Unlike the case where R and S have an identity, an ideal (or primary ideal, or maximal ideal) of R×S need not be a ‘subproduct’ I×J of ideals. We show that for a ring R, for each commutative ring S every ideal (or primary ideal, or maximal ideal) is a subproduct if and only if R is an e-ring (that is, for r∈R, there exists er∈R with err=r) (or u-ring (that is, for each proper ideal A of R, )), the Abelian group (R/R2 ,+)has no maximal subgroups).

In this paper we prove that most ropes of arbitrary multiplicity supported on smooth curves can be smoothed. By a rope being smoothable we mean that the rope is the flat limit of a family of smooth, irreducible curves. To construct a smoothing, we connect, on the one hand, deformations of a finite morphism to projective space and, on the other hand, morphisms from a rope to projective space. We also prove a general result of independent interest, namely that finite covers onto smooth irreducible curves embedded in projective space can be deformed to a family of 1:1 maps. We apply our general theory to prove the smoothing of ropes of multiplicity 3 on P1. Even though this paper focuses on ropes of dimension 1, our method yields a general approach to deal with the smoothing of ropes of higher dimension.