Rosanna Utano

Extremal Betti numbers of graded ideals

The possible extremal Betti numbers of graded ideals in the polynomial ring K[x1,…,xn] in n variables with coefficients in a field K are studied, completing our results in [7]. In case char(K) = 0 we determine, given any integers r < n, the conditions under which there exists a graded ideal I ⊂ K[x1,…, xn] with extremal Betti numbers % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A!

Upper bounds for the betti numbers of graded ideals of a given length in the exterior algebra

\beta_{k_{i}k_{i}+\ell_{i}}\ {\rm for}\ i=1,\cdots,r

Connected Graphs Arising from Products of Veronese Varieties

. We also treat a similar problem for squarefree lexsegment ideals.

Classes of Graded Ideals with Given Data in the Exterior Algebra

Let E be the exterior algebra of a finite dimensional vector space over a field K. Hilbert functions and Betti numbers of graded ideals i ⊂ E are studied. It is shown that there exists a unique graded ideal J ⊂ E such that for all graded ideals I ⊂ E generated in degree ≥ 2 such that one has for all i. Here is the i-th Betti number of a graded E-module M. The unique ideal J with the above property has “maximal” Hiibert function. Our result is the analogue of a theorem of Valla [10] which he proved for graded ideals in the polynomial ring. We further prove a similar inequality for the Bass numbers of E/I, and give a combinatorial application.

On the Symmetric Algebra of the First Syzygy of a Graded Maximal Ideal

Given a Segre squarefree Veronese configuration, following Bernd Sturmfels, we improve the study of the graphs associated to the configuration. We determine two special families of toric ideals and a finite set of moves for each of them, which still guarantee simultaneously the connection of all graphs arising from each family of moves.

A note on the Zariski Lemma for hopf algebras

Let ℱ be the family of graded ideals J in the exterior algebra E of a n-dimensional vector space over a field K such that e(E/J) = dim K (E/J) = e, indeg(E/J) = i and H E/J (i) = dim K (E/J) i are fixed integers. It is shown that there exists a unique lexsegment graded ideal J(n, e, i) ∈ ℱ whose Betti numbers give an upper bound for the Betti numbers of the ideals of ℱ. The authors continue the computation of upper bounds for the Betti numbers of graded ideals with given data started in Crupi and Utano (1999).

Extremal Betti Numbers Of Lexsegment Ideals

Let Syz1(𝔪) be the first syzygy of the graded maximal ideal 𝔪 of a polynomial ring K[x1,…, xn] over a field K. Using the theory of s-sequences, the dimension and depth of the symmetric algebra Sym(Syz1(𝔪)) are calculated. As a conclusion, Sym(Syz1(𝔪)) is not Cohen–Macaulay for any n ≥ 4.

Minimal Resolutions of Some Monomial Submodules

The formulation of the lemma of Zariski is given for coactions of a class of Hopf algebras of the additive group on algebras. Known formulations and consequences of the lemma of Zariski for derivations and differentiations are revised.