scholarly journals Koszul cohomology and singular curves

2010 ◽  
Vol 59 (1) ◽  
pp. 121-125
Author(s):  
Edoardo Ballico ◽  
Claudio Fontanari ◽  
Luca Tasin
Keyword(s):  
Koszul Cohomology

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)].

A Lefschetz type result for Koszul cohomology

2004 ◽  
Vol 114 (4) ◽  
Author(s):  
Marian Aprodu ◽  
Jan Nagel
Keyword(s):  
Type Result ◽  
Lefschetz Type ◽  
Koszul Cohomology

scholarly journals Koszul Cohomology and Algebraic Geometry

2009 ◽  
Author(s):  
Marian Aprodu ◽  
Jan Nagel
Keyword(s):  
Algebraic Geometry ◽  
Koszul Cohomology

scholarly journals de Rham cohomology of local cohomology modules: The graded case

2015 ◽  
Vol 217 ◽  
pp. 1-21
Author(s):  
Tony J. Puthenpurakal
Keyword(s):  
Characteristic Zero ◽  
Weyl Algebra ◽  
Local Cohomology ◽  
Isolated Singularity ◽  
De Rham Cohomology ◽  
Local Cohomology Modules ◽  
De Rham ◽  
Positive Integers ◽  
Cohomology Modules ◽  
Koszul Cohomology

AbstractLetKbe a field of characteristic zero, and letR = K[X1,… ,Xn]. Let An(K) = K⟨X1,… ,Xn,∂1,… ,∂n⟩ be thenth Weyl algebra overK. We consider the case whenRandAn(K) are graded by giving degXi= ωiand deg∂i=–ωifori= 1,…,n(hereωiare positive integers). Set. LetIbe a graded ideal inR. By a result due to Lyubeznik the local cohomology modulesare holonomic (An(K))-modules for eachi≥0. In this article we prove that the de Rham cohomology modulesare concentrated in degree —ω; that is,forj ≠ –ω. As an application whenA = R/(f) is an isolated singularity, we relatetoHn-1(∂(f);A), the (n –1)th Koszul cohomology ofAwith respect to∂1(f),…,∂n(f).

scholarly journals de Rham cohomology of local cohomology modules: The graded case

2015 ◽  
Vol 217 ◽  
pp. 1-21 ◽  
Author(s):  
Tony J. Puthenpurakal
Keyword(s):  
Characteristic Zero ◽  
Weyl Algebra ◽  
Local Cohomology ◽  
Isolated Singularity ◽  
De Rham Cohomology ◽  
Local Cohomology Modules ◽  
De Rham ◽  
Positive Integers ◽  
Cohomology Modules ◽  
Koszul Cohomology

AbstractLet K be a field of characteristic zero, and let R = K[X1,… ,Xn]. Let An(K) = K⟨X1,… ,Xn,∂1,… ,∂n⟩ be the nth Weyl algebra over K. We consider the case when R and An(K) are graded by giving deg Xi = ωi and deg ∂i = –ωi for i = 1,…,n (here ωi are positive integers). Set . Let I be a graded ideal in R. By a result due to Lyubeznik the local cohomology modules are holonomic (An(K))-modules for each i≥0. In this article we prove that the de Rham cohomology modules are concentrated in degree —ω; that is, for j ≠ –ω. As an application when A = R/(f) is an isolated singularity, we relate to Hn-1(∂(f);A), the (n – 1)th Koszul cohomology of A with respect to ∂1(f),…, ∂n(f).

scholarly journals Local cohomology of Du Bois singularities and applications to families

2017 ◽  
Vol 153 (10) ◽  
pp. 2147-2170 ◽  
Author(s):  
Linquan Ma ◽  
Karl Schwede ◽  
Kazuma Shimomoto
Keyword(s):  
Commutative Algebra ◽  
Local Cohomology ◽  
American Mathematical Society ◽  
Vanishing Theorem ◽  
Du Bois ◽  
Local Cohomology Modules ◽  
Du Bois Singularities ◽  
Cohomology Modules ◽  
Kodaira Vanishing ◽  
Koszul Cohomology

In this paper we study the local cohomology modules of Du Bois singularities. Let $(R,\mathfrak{m})$ be a local ring; we prove that if $R_{\text{red}}$ is Du Bois, then $H_{\mathfrak{m}}^{i}(R)\rightarrow H_{\mathfrak{m}}^{i}(R_{\text{red}})$ is surjective for every $i$. We find many applications of this result. For example, we answer a question of Kovács and Schwede [Inversion of adjunction for rational and Du Bois pairs, Algebra Number Theory 10 (2016), 969–1000; MR 3531359] on the Cohen–Macaulay property of Du Bois singularities. We obtain results on the injectivity of $\operatorname{Ext}$ that provide substantial partial answers to questions in Eisenbud et al. [Cohomology on toric varieties and local cohomology with monomial supports, J. Symbolic Comput. 29 (2000), 583–600] in characteristic $0$. These results can also be viewed as generalizations of the Kodaira vanishing theorem for Cohen–Macaulay Du Bois varieties. We prove results on the set-theoretic Cohen–Macaulayness of the defining ideal of Du Bois singularities, which are characteristic-$0$ analogs and generalizations of results of Singh–Walther and answer some of their questions in Singh and Walther [On the arithmetic rank of certain Segre products, in Commutative algebra and algebraic geometry, Contemporary Mathematics, vol. 390 (American Mathematical Society, Providence, RI, 2005), 147–155]. We extend results on the relation between Koszul cohomology and local cohomology for $F$-injective and Du Bois singularities first shown in Hochster and Roberts [The purity of the Frobenius and local cohomology, Adv. Math. 21 (1976), 117–172; MR 0417172 (54 #5230)]. We also prove that singularities of dense $F$-injective type deform.

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