A finiteness theorem for the Hilbert functions of complete intersection local rings

1997 ◽  
Vol 225 (4) ◽  
pp. 543-558 ◽  
Author(s):  
V. Srinivas ◽  
Vijaylaxmi Trivedi
Keyword(s):  
Complete Intersection ◽  
Hilbert Functions ◽  
Local Rings ◽  
Finiteness Theorem

scholarly journals Proof of de Smit’s conjecture: a freeness criterion

2017 ◽  
Vol 153 (11) ◽  
pp. 2310-2317
Author(s):  
Sylvain Brochard
Keyword(s):  
Complete Intersection ◽  
Local Rings ◽  
Embedding Dimension ◽  
Fermat's Last Theorem ◽  
Multiplicity One ◽  
Fermat’S Last Theorem ◽  
Invent Math

Let $A\rightarrow B$ be a morphism of Artin local rings with the same embedding dimension. We prove that any $A$-flat $B$-module is $B$-flat. This freeness criterion was conjectured by de Smit in 1997 and improves Diamond’s criterion [The Taylor–Wiles construction and multiplicity one, Invent. Math. 128 (1997), 379–391, Theorem 2.1]. We also prove that if there is a nonzero $A$-flat $B$-module, then $A\rightarrow B$ is flat and is a relative complete intersection. Then we explain how this result allows one to simplify Wiles’s proof of Fermat’s last theorem: we do not need the so-called ‘Taylor–Wiles systems’ any more.

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 Hilbert polynomials of j-transforms

2016 ◽  
Vol 161 (2) ◽  
pp. 305-337
Author(s):  
SHIRO GOTO ◽  
JOOYOUN HONG ◽  
WOLMER V. VASCONCELOS
Keyword(s):  
Hilbert Functions ◽  
Local Rings ◽  
Hilbert Polynomials ◽  
Graded Module ◽  
Systems Of Parameters ◽  
Partial Systems

AbstractWe study transformations of finite modules over Noetherian local rings that attach to a module M a graded module H(x)(M) defined via partial systems of parameters x of M. Despite the generality of the process, which are called j-transforms, in numerous cases they have interesting cohomological properties. We focus on deriving the Hilbert functions of j-transforms and studying the significance of the vanishing of some of its coefficients.

scholarly journals Multigraded Hilbert functions and toric complete intersection codes

2016 ◽  
Vol 459 ◽  
pp. 446-467 ◽  
Author(s):  
Mesut Şahin ◽  
Ivan Soprunov
Keyword(s):  
Complete Intersection ◽  
Hilbert Functions

scholarly journals A FREENESS CRITERION WITHOUT PATCHING FOR MODULES OVER LOCAL RINGS

2021 ◽  
pp. 1-13
Author(s):  
Sylvain Brochard ◽  
Srikanth B. Iyengar ◽  
Chandrashekhar B. Khare
Keyword(s):  
Complete Intersection ◽  
Local Rings ◽  
Finitely Generated ◽  
Flat Dimension ◽  
Special Case ◽  
Local Homomorphism

Abstract It is proved that if $\varphi \colon A\to B$ is a local homomorphism of commutative noetherian local rings, a nonzero finitely generated B-module N whose flat dimension over A is at most $\operatorname {edim} A - \operatorname {edim} B$ is free over B and $\varphi $ is a special type of complete intersection. This result is motivated by a ‘patching method’ developed by Taylor and Wiles and a conjecture of de Smit, proved by the first author, dealing with the special case when N is flat over A.

NORMAL HILBERT FUNCTIONS OF ONE-DIMENSIONAL LOCAL RINGS

2001 ◽  
Vol 29 (1) ◽  
pp. 333-341
Author(s):  
Valentina Barucci ◽  
Marco D'Anna ◽  
Ralf Fröberg
Keyword(s):  
Hilbert Functions ◽  
Local Rings ◽  
One Dimensional
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