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Uwe Storch's Mathematical Work Factorial domains Differential modules Further results The Books Bibliography Uwe Storch's Mathematical Work Hubert Flenner December 8, 2010 Hubert Flenner Uwe Storch's Mathematical Work Factorial domains Differential modules Further results The Books Bibliography Factorial domains The PhD thesis: Q-factorial domains The work with Bingener Brieskorn singularities Differential modules Differential modules of (semi-)analytic algebras Differential dependence of ideals Further results Number of equations The Oberseminar in Osnabr¨uck (Quasi-)Frobenius Algebras The Books Bibliography Hubert Flenner Uwe Storch's Mathematical Work Factorial domains Differential modules Further results The Books Bibliography Biographical data I *12.07.1940 in Leopoldshall I 1960/66 student University M¨unsterand Heidelberg I 1966 PhD M¨unster I 1966/67 Assistent M¨unster I 1967 Assistent University Bochum I 1972 Habilitation at Bochum I 1974 Full Professor Osnabr¨uck I 1981 Full Professor Bochum I Retired since 2006 Hubert Flenner Uwe Storch's Mathematical Work Factorial domains Differential modules Further results The Books Bibliography M¨unster:Behnke's Oberseminar 1965 Storch Spallek Behnke Holman Vetter Reiffen Hubert Flenner Uwe Storch's Mathematical Work 2. If A Q-factorial ) A[X ] is Q-factorial −1 3. If A Q-factorial ) S A is Q-factorial Factorial domains The PhD thesis: -factorial domains Differential modules Q The Oberseminar in Bochum Further results The work with Bingener The Books Brieskorn singularities Bibliography Q-factorial domains Adviser G. Scheja Definition A Krull domain A is called Q-factorial (or almost factorial) if Cl(A) ⊗ Q = 0 or, equivalently, if every element in Cl(A) is torsion. Standard properties: 1. factorial) Q-factorial Hubert Flenner Uwe Storch's Mathematical Work −1 3. If A Q-factorial ) S A is Q-factorial Factorial domains The PhD thesis: -factorial domains Differential modules Q The Oberseminar in Bochum Further results The work with Bingener The Books Brieskorn singularities Bibliography Q-factorial domains Adviser G. Scheja Definition A Krull domain A is called Q-factorial (or almost factorial) if Cl(A) ⊗ Q = 0 or, equivalently, if every element in Cl(A) is torsion. Standard properties: 1. factorial) Q-factorial 2. If A Q-factorial ) A[X ] is Q-factorial Hubert Flenner Uwe Storch's Mathematical Work Factorial domains The PhD thesis: -factorial domains Differential modules Q The Oberseminar in Bochum Further results The work with Bingener The Books Brieskorn singularities Bibliography Q-factorial domains Adviser G. Scheja Definition A Krull domain A is called Q-factorial (or almost factorial) if Cl(A) ⊗ Q = 0 or, equivalently, if every element in Cl(A) is torsion. Standard properties: 1. factorial) Q-factorial 2. If A Q-factorial ) A[X ] is Q-factorial −1 3. If A Q-factorial ) S A is Q-factorial Hubert Flenner Uwe Storch's Mathematical Work 2. In particular: if a finite group G acts on R0 and R = R0G then R is Q-factorial. 3. If Am[[T ]] Q-factorial 8m ) A[[T ]] is Q-factorial 4. If A complete Q-factorial, depthA ≥ 3 ) A[[T ]] is Q-factorial 5. If dim A = 2 then A is Q-factorial iff it has a rational singularity. r s t 6. Numerous examples, e.g. A = Cfx; y; zg=(x + y + z ) (r; s; t pairwise coprime) is Q-factorial iff (r; s; t) = (2; 3; 5). Factorial domains The PhD thesis: -factorial domains Differential modules Q The Oberseminar in Bochum Further results The work with Bingener The Books Brieskorn singularities Bibliography Q-factorial domains Some results: 1. If R ! R0 is an integral extension of Krull rings and R0 is Q-factorial then so is A. Hubert Flenner Uwe Storch's Mathematical Work 3. If Am[[T ]] Q-factorial 8m ) A[[T ]] is Q-factorial 4. If A complete Q-factorial, depthA ≥ 3 ) A[[T ]] is Q-factorial 5. If dim A = 2 then A is Q-factorial iff it has a rational singularity. r s t 6. Numerous examples, e.g. A = Cfx; y; zg=(x + y + z ) (r; s; t pairwise coprime) is Q-factorial iff (r; s; t) = (2; 3; 5). Factorial domains The PhD thesis: -factorial domains Differential modules Q The Oberseminar in Bochum Further results The work with Bingener The Books Brieskorn singularities Bibliography Q-factorial domains Some results: 1. If R ! R0 is an integral extension of Krull rings and R0 is Q-factorial then so is A. 2. In particular: if a finite group G acts on R0 and R = R0G then R is Q-factorial. Hubert Flenner Uwe Storch's Mathematical Work 4. If A complete Q-factorial, depthA ≥ 3 ) A[[T ]] is Q-factorial 5. If dim A = 2 then A is Q-factorial iff it has a rational singularity. r s t 6. Numerous examples, e.g. A = Cfx; y; zg=(x + y + z ) (r; s; t pairwise coprime) is Q-factorial iff (r; s; t) = (2; 3; 5). Factorial domains The PhD thesis: -factorial domains Differential modules Q The Oberseminar in Bochum Further results The work with Bingener The Books Brieskorn singularities Bibliography Q-factorial domains Some results: 1. If R ! R0 is an integral extension of Krull rings and R0 is Q-factorial then so is A. 2. In particular: if a finite group G acts on R0 and R = R0G then R is Q-factorial. 3. If Am[[T ]] Q-factorial 8m ) A[[T ]] is Q-factorial Hubert Flenner Uwe Storch's Mathematical Work 5. If dim A = 2 then A is Q-factorial iff it has a rational singularity. r s t 6. Numerous examples, e.g. A = Cfx; y; zg=(x + y + z ) (r; s; t pairwise coprime) is Q-factorial iff (r; s; t) = (2; 3; 5). Factorial domains The PhD thesis: -factorial domains Differential modules Q The Oberseminar in Bochum Further results The work with Bingener The Books Brieskorn singularities Bibliography Q-factorial domains Some results: 1. If R ! R0 is an integral extension of Krull rings and R0 is Q-factorial then so is A. 2. In particular: if a finite group G acts on R0 and R = R0G then R is Q-factorial. 3. If Am[[T ]] Q-factorial 8m ) A[[T ]] is Q-factorial 4. If A complete Q-factorial, depthA ≥ 3 ) A[[T ]] is Q-factorial Hubert Flenner Uwe Storch's Mathematical Work r s t 6. Numerous examples, e.g. A = Cfx; y; zg=(x + y + z ) (r; s; t pairwise coprime) is Q-factorial iff (r; s; t) = (2; 3; 5). Factorial domains The PhD thesis: -factorial domains Differential modules Q The Oberseminar in Bochum Further results The work with Bingener The Books Brieskorn singularities Bibliography Q-factorial domains Some results: 1. If R ! R0 is an integral extension of Krull rings and R0 is Q-factorial then so is A. 2. In particular: if a finite group G acts on R0 and R = R0G then R is Q-factorial. 3. If Am[[T ]] Q-factorial 8m ) A[[T ]] is Q-factorial 4. If A complete Q-factorial, depthA ≥ 3 ) A[[T ]] is Q-factorial 5. If dim A = 2 then A is Q-factorial iff it has a rational singularity. Hubert Flenner Uwe Storch's Mathematical Work Factorial domains The PhD thesis: -factorial domains Differential modules Q The Oberseminar in Bochum Further results The work with Bingener The Books Brieskorn singularities Bibliography Q-factorial domains Some results: 1. If R ! R0 is an integral extension of Krull rings and R0 is Q-factorial then so is A. 2. In particular: if a finite group G acts on R0 and R = R0G then R is Q-factorial. 3. If Am[[T ]] Q-factorial 8m ) A[[T ]] is Q-factorial 4. If A complete Q-factorial, depthA ≥ 3 ) A[[T ]] is Q-factorial 5. If dim A = 2 then A is Q-factorial iff it has a rational singularity. r s t 6. Numerous examples, e.g. A = Cfx; y; zg=(x + y + z ) (r; s; t pairwise coprime) is Q-factorial iff (r; s; t) = (2; 3; 5). Hubert Flenner Uwe Storch's Mathematical Work Factorial domains The PhD thesis: -factorial domains Differential modules Q The Oberseminar in Bochum Further results The work with Bingener The Books Brieskorn singularities Bibliography Hubert Flenner Uwe Storch's Mathematical Work Question of Samuel: is Cl(A) ! Cl(A[[T ]]) always an isomorphism? First counterexample by P. Salmon 1966. Scheja: if A is complete and factorial with depthA ≥ 3 then A[[X ]] is factorial. Theorem Storch Math. Ann. 183(1969): If A is analytic C-algebra, Cl(A) is finitely generated then Cl(A) ! Cl(AfT g) is an isomorphism. Moreover Cl(A) is finitely generated iff A is 1-rational. Definition (Danilov 70): A is said to have discrete divisor class groups if Cl(A) ! Cl(A[[T ]]) an isomorphism. Factorial domains The PhD thesis: -factorial domains Differential modules Q The Oberseminar in Bochum Further results The work with Bingener The Books Brieskorn singularities Bibliography Background Mumford (Pub. Math. 61), Brieskorn (Invent. 67) and Lipman (Pub.Math. 69) Hubert Flenner Uwe Storch's Mathematical Work Scheja: if A is complete and factorial with depthA ≥ 3 then A[[X ]] is factorial. Theorem Storch Math. Ann. 183(1969): If A is analytic C-algebra, Cl(A) is finitely generated then Cl(A) ! Cl(AfT g) is an isomorphism. Moreover Cl(A) is finitely generated iff A is 1-rational. Definition (Danilov 70): A is said to have discrete divisor class groups if Cl(A) ! Cl(A[[T ]]) an isomorphism. Factorial domains The PhD thesis: -factorial domains Differential modules Q The Oberseminar in Bochum Further results The work with Bingener The Books Brieskorn singularities Bibliography Background Mumford (Pub.
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