scholarly journals PROPERTIES OF $T$-SPREAD PRINCIPAL BOREL IDEALS GENERATED IN DEGREE TWO

2020 ◽  
Vol 35 (1) ◽  
pp. 131
Author(s):  
Bahareh Lajmiri ◽  
Farhad Rahmati
Keyword(s):  
Complete Intersection ◽  
Borel Ideal ◽  
Lyubeznik Numbers ◽  
The Stability ◽  
Torsion Free ◽  
Borel Ideals

In this paper, we have studied the stability of $t$-spread principal Borel ideals in degree two. We have proved that $\Ass^\infty(I) =\Min(I)\cup \{\mathfrak{m}\}$ , where $I=B_t(u)\subset S$ is a $t$-spread Borel ideal generated in degree $2$ with $u=x_ix_n, t+1\leq i\leq n-t.$ Indeed, $I$ has the property that $\Ass(I^m)=\Ass(I)$ for all $m\geq 1$ and $i\leq t,$ in other words, $I$ is normally torsion free. Moreover, we have shown that $I$ is a set theoretic complete intersection if and only if $u=x_{n-t}x_n$. Also, we have derived some results on the vanishing of Lyubeznik numbers of these ideals.  

scholarly journals ON THE STABILITY OF SYZYGY BUNDLES

2011 ◽  
Vol 22 (04) ◽  
pp. 515-534 ◽  
Author(s):  
IUSTIN COANDĂ
Keyword(s):  
Projective Space ◽  
Complete Intersection ◽  
Vector Bundles ◽  
Elementary Proof ◽  
Base Point ◽  
Vector Spaces ◽  
Similar Argument ◽  
Vanishing Theorem ◽  
The Stability ◽  
Koszul Cohomology

We are concerned with the problem of the stability of the syzygy bundles associated to base-point-free vector spaces of forms of the same degree d on the projective space of dimension n. We deduce directly, from M. Green's vanishing theorem for Koszul cohomology, that any such bundle is stable if its rank is sufficiently high. With a similar argument, we prove the semistability of a certain syzygy bundle on a general complete intersection of hypersurfaces of degree d in the projective space. This answers a question of H. Flenner [Comment. Math. Helv.59 (1984) 635–650]. We then give an elementary proof of H. Brenner's criterion of stability for monomial syzygy bundles, avoiding the use of Klyachko's results on toric vector bundles. We finally prove the existence of stable syzygy bundles defined by monomials of the same degree d, of any possible rank, for n at least 3. This extends the similar result proved, for n = 2, by L. Costa, P. Macias Marques and R. M. Miro-Roig [J. Pure Appl. Algebra214 (2010) 1241–1262]. The extension to the case n at least 3 has been also, independently, obtained by P. Macias Marques in his thesis [arXiv:0909.4646/math.AG (2009)].

scholarly journals Torsion in tensor powers of modules

2015 ◽  
Vol 219 ◽  
pp. 113-125
Author(s):  
Olgur Celikbas ◽  
Srikanth B. Iyengar ◽  
Greg Piepmeyer ◽  
Roger Wiegand
Keyword(s):  
Local Ring ◽  
Complete Intersection ◽  
Tensor Products ◽  
Central Theme ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Free Modules ◽  
Tensor Powers ◽  
Torsion Free ◽  
Nonzero Torsion

AbstractTensor products usually have nonzero torsion. This is a central theme of Auslander's 1961 paper; the theme continues in the work of Huneke and Wiegand in the 1990s. The main focus in this article is on tensor powers of a finitely generated module over a local ring. Also, we study torsion-free modulesNwith the property thatM ⊗RNhas nonzero torsion unlessMis very special. An important example of such a moduleNis the Frobenius powerpeRover a complete intersection domainRof characteristicp> 0.

scholarly journals Some properties of complete intersections in “good” projective varieties

1976 ◽  
Vol 61 ◽  
pp. 103-111 ◽  
Author(s):  
Lorenzo Robbiano
Keyword(s):  
Complete Intersection ◽  
Singular Surface ◽  
Complete Intersections ◽  
Closed Field ◽  
Irreducible Curve ◽  
Projective Varieties ◽  
Algebraically Closed Field ◽  
Torsion Free

In [10] it was proved that, if X denotes a non singular surface which is a complete intersection in (k an algebraically closed field of characteristic 0) and C an irreducible curve on X, which is a set-theoretic complete intersection in X, then C is actually a complete intersection in X; the key point was to show that Pic (X) modulo the subgroup generated by the class of is torsion-free.

scholarly journals Borel degenerations of arithmetically Cohen–Macaulay curves in ℙ3

2014 ◽  
Vol 24 (05) ◽  
pp. 715-739 ◽  
Author(s):  
Gunnar Fløystad ◽  
Margherita Roggero
Keyword(s):  
Hilbert Scheme ◽  
Codimension Two ◽  
Borel Ideal ◽  
Borel Ideals

We investigate Borel ideals on the Hilbert scheme components of arithmetically Cohen–Macaulay (ACM) codimension two schemes in ℙn. We give a basic necessary criterion for a Borel ideal to be on such a component. Then considering ACM curves in ℙ3 on a quadric we compute in several examples all the Borel ideals on their Hilbert scheme component. Based on this we conjecture which Borel ideals are on such a component, and for a range of Borel ideals we prove that they are on the component.

scholarly journals Picard groups of punctured spectra of dimension three local hypersurfaces are torsion-free

2011 ◽  
Vol 148 (1) ◽  
pp. 145-152 ◽  
Author(s):  
Hailong Dao
Keyword(s):  
Abelian Group ◽  
Local Ring ◽  
Complete Intersection ◽  
Homological Algebra ◽  
Local Rings ◽  
Noetherian Local Ring ◽  
Picard Groups ◽  
Crepant Resolutions ◽  
Torsion Free

AbstractLet (R,m) be a Noetherian local ring and UR=Spec(R)−{m} be the punctured spectrum of R. Gabber conjectured that if R is a complete intersection of dimension three, then the abelian group Pic(UR) is torsion-free. In this note we prove Gabber’s statement for the hypersurface case. We also point out certain connections between Gabber’s conjecture, Van den Bergh’s notion of non-commutative crepant resolutions and some well-studied questions in homological algebra over local rings.

scholarly journals The Elman-Lam-Krüskemper Theorem

ISRN Algebra ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-8 ◽  
Author(s):  
Karim Johannes Becher ◽  
David B. Leep
Keyword(s):  
Elementary Proof ◽  
Stability Index ◽  
Real Field ◽  
Witt Ring ◽  
Fundamental Ideal ◽  
The Stability ◽  
Torsion Part ◽  
Torsion Free

For a (formally) real field K, the vanishing of a certain power of the fundamental ideal in the Witt ring of K(-1) implies that the same power of the fundamental ideal in the Witt ring of K is torsion free. The proof of this statement involves a fact on the structure of the torsion part of powers of the fundamental ideal in the Witt ring of K. This fact is very difficult to prove in general, but has an elementary proof under an assumption on the stability index of K. We present an exposition of these results.

LINEAR QUOTIENTS AND MULTIGRADED SHIFTS OF BOREL IDEALS

2019 ◽  
Vol 100 (1) ◽  
pp. 48-57 ◽  
Author(s):  
SHAMILA BAYATI ◽  
IMAN JAHANI ◽  
NADIYA TAGHIPOUR
Keyword(s):  
Borel Ideal ◽  
Linear Quotients ◽  
Borel Ideals ◽  
The Ideal

We investigate whether the property of having linear quotients is inherited by ideals generated by multigraded shifts of a Borel ideal and a squarefree Borel ideal. We show that the ideal generated by the first multigraded shifts of a Borel ideal has linear quotients, as do the ideal generated by the $k$th multigraded shifts of a principal Borel ideal and an equigenerated squarefree Borel ideal for each $k$. Furthermore, we show that equigenerated squarefree Borel ideals share the property of being squarefree Borel with the ideals generated by multigraded shifts.

scholarly journals Torsion in tensor powers of modules

2015 ◽  
Vol 219 ◽  
pp. 113-125 ◽  
Author(s):  
Olgur Celikbas ◽  
Srikanth B. Iyengar ◽  
Greg Piepmeyer ◽  
Roger Wiegand
Keyword(s):  
Local Ring ◽  
Complete Intersection ◽  
Tensor Products ◽  
Central Theme ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Free Modules ◽  
Tensor Powers ◽  
Torsion Free ◽  
Nonzero Torsion

AbstractTensor products usually have nonzero torsion. This is a central theme of Auslander's 1961 paper; the theme continues in the work of Huneke and Wiegand in the 1990s. The main focus in this article is on tensor powers of a finitely generated module over a local ring. Also, we study torsion-free modules N with the property that M ⊗R N has nonzero torsion unless M is very special. An important example of such a module N is the Frobenius power peR over a complete intersection domain R of characteristic p > 0.

Open discussion

1982 ◽  
Vol 99 ◽  
pp. 605-613
Author(s):  
P. S. Conti
Keyword(s):  
Buenos Aires ◽  
Large Mass ◽  
List Type ◽  
Common Phenomenon ◽  
Open Discussion ◽  
The Stability ◽  
Ring Nebulae

Conti: One of the main conclusions of the Wolf-Rayet symposium in Buenos Aires was that Wolf-Rayet stars are evolutionary products of massive objects. Some questions:–Do hot helium-rich stars, that are not Wolf-Rayet stars, exist?–What about the stability of helium rich stars of large mass? We know a helium rich star of ∼40 MO. Has the stability something to do with the wind?–Ring nebulae and bubbles : this seems to be a much more common phenomenon than we thought of some years age.–What is the origin of the subtypes? This is important to find a possible matching of scenarios to subtypes.

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