Mathematics

Let X=SpecA denote a regular affine scheme, over a field k , with 1/2∈k and dimX=d . Let P denote a projective A -module of rank n≥2 . Let π 0 (LO(P)) denote the (Nori) Homotopy Obstruction set, and CH ˜ n (X, Λ n P) denote the Chow Witt group. In this article, we define a natural (set theoretic) map} Θ P : π 0 (LO(P))⟶ CH ˜ n (X, Λ n P)

Continuing a well established tradition of associating convex bodies to monomial ideals, we initiate a program to construct asymptotic Newton polyhedra from decompositions of monomial ideals. This is achieved by forming a graded family of ideals based on a given decomposition. We term these graded families powers since they generalize the notions of ordinary and symbolic powers. We introduce a novel family of irreducible powers. Irreducible powers and symbolic powers of monomial ideals are studied by means of the corresponding irreducible polyhedron and symbolic polyhedron respectively. Asymptotic invariants for these graded families are expressed as solutions to linear optimization problems on the respective convex bodies. This allows to establish a lower bound on the Waldschmidt constant of a monomial ideal, an asymptotic invariant which can be defined using the symbolic polyhedron, by means of an analogous invariant stemming from the irreducible polyhedron, which we introduce under the name of naive Waldschmidt constant.

One deals with degenerations by coordinate sections of the square generic Hankel matrix over a field k of characteristic zero, along with its main related structures, such as the determinant of the matrix, the ideal generated by its partial derivatives, the polar map defined by these derivatives, the Hessian matrix and the ideal of the submaximal minors of the matrix. It is proved that the polar map is dominant for any such degenerations, and not homaloidal in the generic case. The problem of whether the determinant f of the matrix is a factor of the Hessian with the (Segre) expected multiplicity is considered, for which the expected lower bound of the dual variety of V(f) is established.

We are interested in characterising the commutative rings for which a 1 -tilting cotorsion pair (A,T) provides for covers, that is when the class A is a covering class. We use Hrbek's bijective correspondence between the 1 -tilting cotorsion pairs over a commutative ring R and the faithful finitely generated Gabriel topologies on R . Moreover, we use results of Bazzoni-Positselski, in particular a generalisation of Matlis equivalence and their characterisation of covering classes for 1 -tilting cotorsion pairs arising from flat injective ring epimorphisms. Explicitly, if G is the Gabriel topology associated to the 1 -tilting cotorsion pair (A,T) , and R G is the ring of quotients with respect to G , we show that if A is covering then G is a perfect localisation (in Stenström's sense) and the localisation R G has projective dimension at most one. Moreover, we show that A is covering if and only if both the localisation R G and the quotient rings R/J are perfect rings for every J∈G . Rings satisfying the latter two conditions are called G -almost perfect.

In this paper, we study cubic surfaces in characteristic two from the perspective of positive characteristic commutative algebra and completely classify those which are Frobenius split. In particular, we explicitly describe the finitely many non- F -pure cubics (up to projective change of coordinates in P 3 ), exactly one of which is smooth. We also describe the configurations of lines on these cubic surfaces; a cubic surface in characteristic two is Frobenius split unless every pair of intersecting lines meets in an Eckardt point, which, in the smooth case, means no three lines form a "triangle".

We study the cyclotomic exponent sequence of a numerical semigroup S, and we compute its values at the gaps of S, the elements of S with unique representations in terms of minimal generators, and the Betti elements b?�S for which the set {a?�Betti(S):a ??S b} is totally ordered with respect to ??S (we write a ??S b whenever a?�b?�S, with a,b?�S ). This allows us to characterize certain semigroup families, such as Betti-sorted or Betti-divisible numerical semigroups, as well as numerical semigroups with a unique Betti element, in terms of their cyclotomic exponent sequences. Our results also apply to cyclotomic numerical semigroups, which are numerical semigroups with a finitely supported cyclotomic exponent sequence. We show that cyclotomic numerical semigroups with certain cyclotomic exponent sequences are complete intersections, thereby making progress towards proving the conjecture of Ciolan, García-Sánchez and Moree (2016) stating that S is cyclotomic if and only if it is a complete intersection.

A numerical semigroup S is cyclotomic if its semigroup polynomial P S is a product of cyclotomic polynomials. The number of irreducible factors of P S (with multiplicity) is the polynomial length ??S) of S. We show that a cyclotomic numerical semigroup is complete intersection if ??S)?? . This establishes a particular case of a conjecture of Ciolan, García-Sánchez and Moree (2016) claiming that every cyclotomic numerical semigroup is complete intersection. In addition, we investigate the relation between ??S) and the embedding dimension of S.

In this paper, we consider the iterated trimming complex associated to data yielding a complex of length 3 . We compute an explicit algebra structure in this complex in terms of the algebra structures of the associated input data. Moreover, it is shown that many of these products become trivial after descending to homology. We apply these results to the problem of realizability for Tor-algebras of grade 3 perfect ideals, and show that under mild hypotheses, the process of "trimming" an ideal preserves Tor-algebra class. In particular, we construct new classes of ideals in arbitrary regular local rings defining rings realizing Tor-algebra classes G(r) and H(p,q) for a prescribed set of homological data.

The big from small construction was introduced by Kustin and Miller and can be used to construct resolutions of tightly double linked Gorenstein ideals. In this paper, we expand on the DG-algebra techniques introduced Kustin for building matrix factorizations and construct a DG-algebra structure on the length 4 big from small construction. The techniques employed involve the construction of a morphism from a Tate-like complex to an acyclic DG-algebra exhibiting Poincaré duality. This induces homomorphisms which, after suitable modifications, satisfy a list of identities that end up perfectly encapsulating the required associativity and DG axioms of the desired product structure for the big from small construction.

We introduce the notion of E-depth of graded modules over polynomial rings to measure the depth of certain Ext modules. First, we characterize graded modules over polynomial rings with (sufficiently) large E-depth as those modules whose (sufficiently) partial general initial submodules preserve the Hilbert function of local cohomology modules supported at the irrelevant maximal ideal, extending a result of Herzog and Sbarra on sequentially Cohen-Macaulay modules. Second, we describe the cone of local cohomology tables of modules with sufficiently high E-depth, building on previous work of the second author and Smirnov. Finally, we obtain a non-Artinian version of a socle-lemma proved by Kustin and Ulrich.