Note on bounds for multiplicities

Let S=K[x₁,…,x_n] be a polynomial ring and R=S/I be a graded K-algebra where I⊂S is a graded ideal. Herzog, Huneke and Srinivasan have conjectured that the multiplicity of R is bounded above by a function of the maximal shifts in the minimal graded free resolution of R over S. We prove the conjecture in the case that codim(R)=2 which generalizes results in (J. Pure Appl. Algebra 182 (2003) 201; Trans. Amer. Math. Soc. 350 (1998) 2879). We also give a proof for the bound in the case in which I is componentwise linear. For example, stable and squarefree stable ideals belong to this class of ideals.     

Borel-plus-powers monomial ideals

Let S=K[x₁,…,x_n] be a standard graded polynomial ring over a field K. In this paper, we show that the lex-plus-powers ideal has the largest graded Betti numbers among all Borel-plus-powers monomial ideals with the same Hilbert function. In addition in the case of characteristic 0, by using this result, we prove the lex-plus-powers conjecture for graded ideals containing , where p is a prime number.     

Henriksen?s contributions to residue class rings of analytic and entire functions

The author surveys, summarizes and generalizes results of Golasiński and Henriksen, and of others, concerning certain residue class rings.Let A(R) denote the ring of analytic functions over reals R and E(K) the ring of entire functions over R or complex numbers C. It is shown that if m is a maximal ideal of A(R), then A(R)/m is isomorphic either to the reals or a real-closed field that is η₁-set, while if m is a maximal ideal of E(K), then E(K)/m is isomorphic to one of these latter two fields or to complex numbers.     

Stillman's question for exterior algebras and Herzog's conjecture on Betti numbers of syzygy modules

Let K be a field of characteristic 0 and consider exterior algebras of finite dimensional K-vector spaces. In this short paper we exhibit principal quadric ideals in a family whose Castelnuovo–Mumford regularity is unbounded. This negatively answers the analogue of Stillman's Question for exterior algebras posed by I. Peeva. We show that, via the Bernstein–Gel'fand–Gel'fand correspondence, these examples also yields counterexamples to a conjecture of J. Herzog on the Betti numbers in the linear strand of syzygy modules over polynomial rings.     

The minimum distance of sets of points and the minimum socle degree

Let K be a field of characteristic 0. Let be a reduced finite set of points, not all contained in a hyperplane. Let be the maximum number of points of Γ contained in any hyperplane, and let . If I⊂R=K[x₀,…,x_n] is the ideal of Γ, then in Tohaˇneanu (2009) [12] it is shown that for n=2,3, d(Γ) has a lower bound expressed in terms of some shift in the graded minimal free resolution of R/I. In these notes we show that this behavior holds true in general, for any n≥2: d(Γ)≥A_n, where A_n=min{a_i−n} and ⊕_iR(−a_i) is the last module in the graded minimal free resolution of R/I. In the end we also prove that this bound is sharp for a whole class of examples due to Juan Migliore (2010) [10].     

Strongly irreducible ideals of a commutative ring

An ideal I of a ring R is said to be strongly irreducible if for ideals J and K of R, the inclusion J∩K⊆I implies that either J⊆I or K⊆I. The relationship among the families of irreducible ideals, strongly irreducible ideals, and prime ideals of a commutative ring R is considered, and a characterization is given of the Noetherian rings which contain a non-prime strongly irreducible ideal.     

Strongly Jónsson and strongly HS modules

Let R be a commutative ring with identity and let M be an infinite unitary R-module. Then M is a Jónsson module provided every proper R-submodule of M has smaller cardinality than M. In this note, we strengthen this condition and call an R-module M (which may be finite) strongly Jónsson provided distinct R-submodules of M have distinct cardinalities. We present a classification of these modules, and then we study a sort of dual notion. Specifically, we consider modules M   for which M/N$M/N$ and M/K$M/K$ have distinct cardinalities for distinct R-submodules N and K of M; we call such modules strongly HS (see the introduction for etymology). We conclude the paper with a classification of the strongly HS modules over an arbitrary commutative ring.     

Convex polarities over ordered fields

We use tools and methods from real algebraic geometry (spaces of ultrafilters, elimination of quantifiers) to formulate a theory of convexity in K^N over an arbitrary ordered field. By defining certain ideal points (which can be viewed as generalizations of recession cones) we obtain a generalized notion of polar set. These satisfy a form of polar duality that applies to general convex sets and does not reduce to classical duality if K is the field of real numbers. As an application we give a partial classification of total orderings of Artinian local rings and two applications to ordinary convex geometry over the real numbers.     

2-Cosemisimplicial objects in a 2-category, permutohedra and deformations of pseudofunctors

In this paper we take up again the deformation theory for K-linear pseudofunctors initiated in Elgueta (Adv. Math. 182 (2004) 204-277). We start by introducing a notion of 2-cosemisimplicial object in an arbitrary 2-category and analyzing the corresponding coherence question, where the permutohedra make their appearance. We then describe a general method to obtain usual cochain complexes of K-modules from (enhanced) 2-cosemisimplicial objects in the 2-category of small K-linear categories and prove that the deformation complex introduced in Elgueta (to appear) can be obtained by this method from a 2-cosemisimplicial object that can be associated to . Finally, using this 2-cosemisimplicial object of and a generalization to the context of K-linear categories of the deviation calculus introduced by Markl and Stasheff for K-modules (J. Algebra 170 (1994) 122), it is shown that the obstructions to the integrability of an nth-order deformation of indeed correspond to cocycles in the third cohomology group , a question which remained open in Elgueta (Adv. Math. 182 (2004) 204-277).     

Cohomology and deformation theory of monoidal 2-categories I

In this paper we define a cohomology theory for an arbitrary K-linear semistrict semigroupal 2-category (called for short a Gray semigroup) and show that its first-order (unitary) deformations, up to the suitable notion of equivalence, are in bijection with the elements of the second cohomology group. Fundamental to the construction is a double complex, similar to the Gerstenhaber-Schack double complex for bialgebras, the role of the multiplication and the comultiplication being now played by the composition and the tensor product of 1-morphisms. We also identify the cohomologies describing separately the deformations of the tensor product, the associator and the pentagonator. To obtain the above results, a cohomology theory for an arbitrary K-linear (unitary) pseudofunctor is introduced describing its purely pseudofunctorial deformations, and generalizing Yetter's cohomology for semigroupal functors (in: M. Kapranov, E. Getzler (Eds.), Higher Category Theory, AMS Contemporary Mathematics, Vol. 230, Amer. Math. Soc., Providence, RI, 1998, pp. 117-134). The corresponding higher order obstructions will be considered in detail in a future paper.     

Explicit linear minimal free resolutions over a natural class of Rees algebras

Homological properties of the Rees algebra R of a Koszul K-algebra A over a field K, with respect to the maximal homogeneous ideal, are studied. In particular, for a finitely generated graded A-module N with linear minimal free R-resolution over A, the minimal free resolution of is explicitly constructed. This resolution is again linear.Received: 23 October 2000     

Gorenstein dimension of modules over homomorphisms

Given a homomorphism of commutative noetherian rings R→S and an S-module N, it is proved that the Gorenstein flat dimension of N over R, when finite, may be computed locally over S. When, in addition, the homomorphism is local and N is finitely generated over S, the Gorenstein flat dimension equals , where E is the injective hull of the residue field of R. This result is analogous to a theorem of André on flat dimension.     

Minimal free resolution of the associated graded ring of monomial curves of generalized arithmetic sequences

Let A(C) be the coordinate ring of a monomial curve C⊆Aⁿ corresponding to the numerical semigroup S minimally generated by a sequence a₀,…,a_n. In the literature, little is known about the Betti numbers of the corresponding associated graded ring gr_m(A) with respect to the maximal ideal m of A=A(C). In this paper we characterize the numerical invariants of a minimal free resolution of gr_m(A) in the case a₀,…,a_n is a generalized arithmetic sequence.     

The Lichtenbaum-Hartshorne theorem for modules which are finite over a ring homomorphism

Let φ:(R,m)→S be a flat ring homomorphism such that mS≠S. Assume that M is a finitely generated S-module with dim_R(M)=d. If the set of support of M has a special property, then it is shown that if and only if for each prime ideal satisfying , we have . This gives a generalization of the Lichtenbaum-Hartshorne vanishing theorem for modules which are finite over a ring homomorphism. Furthermore, we provide two extensions of Grothendieck’s non-vanishing theorem. Applications to connectedness properties of the support are given.     

Sharp bounds for the number of roots of univariate fewnomials

Let K be a field and t?0. Denote by Bm(t,K) the supremum of the number of roots in K^?, counted with multiplicities, that can have a non-zero polynomial in K[x] with at most t+1 monomial terms. We prove, using an unified approach based on Vandermonde determinants, that Bm(t,L)?t²Bm(t,K) for any local field L with a non-archimedean valuation v:L→R∪{∞} such that vZ≠0_|≡0 and residue field K, and that Bm(t,K)?(t²−t+1)(pf−1) for any finite extension K/Qp with residual class degree f and ramification index e, assuming that p>t+e. For any finite extension K/Qp, for p odd, we also show the lower bound Bm(t,K)?(2t−1)(pf−1), which gives the sharp estimation Bm(2,K)=3(pf−1) for trinomials when p>2+e.     

Generalized local cohomology and the canonical element conjecture

We study a generalization of the Canonical Element Conjecture. In particular we show that given a nonregular local ring (A,m) and an i>0, there exist finitely generated A-modules M such that the canonical map from to is nonzero. Moreover, we show that even when M has an infinite projective dimension and i>dim(A), studying these maps sheds light on the Canonical Element Conjecture.     

Gröbner bases for spaces of quadrics of codimension 3

Let R=⊕_i≥0R_i be an Artinian standard graded K-algebra defined by quadrics. Assume that dimR₂≤3 and that K is algebraically closed, of characteristic ≠2. We show that R is defined by a Gröbner basis of quadrics with, essentially, one exception. The exception is given by K[x,y,z]/I where I is a complete intersection of three quadrics not containing a square of a linear form.     

Nonvanishing cohomology and classes of Gorenstein rings

We give counterexamples to the following conjecture of Auslander: given a finitely generated module M over an Artin algebra Λ, there exists a positive integer n_M such that for all finitely generated Λ-modules N, if Ext_Λⁱ(M,N)=0 for all i?0, then Ext_Λⁱ(M,N)=0 for all i?n_M. Some of our examples moreover yield homologically defined classes of commutative local rings strictly between the class of local complete intersections and the class of local Gorenstein rings.