scholarly journals A DESCENT THEOREM FOR FORMAL SMOOTHNESS

2016 ◽  
Vol 229 ◽  
pp. 113-140
Author(s):  
JAVIER MAJADAS
Keyword(s):  
Finite Type ◽  
Complete Intersection ◽  
Intersection Property ◽  
Local Rings ◽  
Similar Question ◽  
Regular Local Rings ◽  
Formal Smoothness

We give a descent result for formal smoothness having interesting applications: we deduce that quasiexcellence descends along flat local homomorphisms of finite type, we greatly improve Kunz’s characterization of regular local rings by means of the Frobenius homomorphisms as well as André and Radu relativization of this result, etc. In the second part of the paper, we study a similar question for the complete intersection property instead of formal smoothness, giving also some applications.

scholarly journals Characterizations of regular local rings via syzygy modules of the residue field

2018 ◽  
Vol 10 (3) ◽  
pp. 327-337
Author(s):  
Dipankar Ghosh ◽  
Anjan Gupta ◽  
Tony J. Puthenpurakal
Keyword(s):  
Residue Field ◽  
Local Rings ◽  
Syzygy Modules ◽  
Regular Local Rings

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 Complete Intersection Monomial Curves and the Cohen—Macaulayness of Their Tangent Cones

2019 ◽  
Vol 26 (04) ◽  
pp. 629-642
Author(s):  
Anargyros Katsabekis
Keyword(s):  
Complete Intersection ◽  
Tangent Cone ◽  
Affine Space ◽  
Intersection Property ◽  
Tangent Cones ◽  
Monomial Curves ◽  
Monomial Curve

Let C(n) be a complete intersection monomial curve in the 4-dimensional affine space. In this paper we study the complete intersection property of the monomial curve C(n + wv), where w > 0 is an integer and v ∈ ℕ4. In addition, we investigate the Cohen–Macaulayness of the tangent cone of C(n + wv).

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