On the mystery of generic models
– an invitation –
Seminar at the
Dipartimento di Matematica (Tullio Levi-Civita) Università degli Studi di Padova
The generic ring
“
LetRbe a ring.
” – Which ring does this phrase refer to?
Z F2 Q[X] R OX
A
Thm.For any?propertyP of rings, the following are equivalent:
1 Thegeneric ringAhas propertyP.
2 Every?ring has propertyP.
3 The ring axioms entail propertyP.
Example A.For anyx,y,z ∈A,x+ (y+z) = (x+y) +z. Example B.
It isnot the casethat 1+16=0 inA. Example C (Anders Kock).The generic ring is afield:
∀x ∈A. (x=0⇒1 =0)⇒(∃y ∈A.xy=1) . Hence: When verifying a coherent sequent for all rings, can without loss of generality assume the field condition.
The generic ring
“LetRbe a ring.” – Which ring does this phrase refer to?
Z F2 Q[X] R OX
A
Thm.For any?propertyP of rings, the following are equivalent:
1 Thegeneric ringAhas propertyP.
2 Every?ring has propertyP.
3 The ring axioms entail propertyP.
Example A.For anyx,y,z ∈A,x+ (y+z) = (x+y) +z. Example B.
It isnot the casethat 1+16=0 inA. Example C (Anders Kock).The generic ring is afield:
∀x ∈A. (x=0⇒1 =0)⇒(∃y ∈A.xy=1) . Hence: When verifying a coherent sequent for all rings, can without loss of generality assume the field condition.
The generic ring
“LetRbe a ring.” – Which ring does this phrase refer to?
Z F2 Q[X] R OX A
Thm.For any?propertyP of rings, the following are equivalent:
1 Thegeneric ringAhas propertyP.
2 Every?ring has propertyP.
3 The ring axioms entail propertyP.
Example A.For anyx,y,z ∈A,x+ (y+z) = (x+y) +z. Example B.
It isnot the casethat 1+16=0 inA. Example C (Anders Kock).The generic ring is afield:
∀x ∈A. (x=0⇒1 =0)⇒(∃y ∈A.xy=1) . Hence: When verifying a coherent sequent for all rings, can without loss of generality assume the field condition.
The generic ring
“LetRbe a ring.” – Which ring does this phrase refer to?
Z F2 Q[X] R OX A
Thm.For any?propertyP of rings, the following are equivalent:
1 Thegeneric ringAhas propertyP.
2 Every?ring has propertyP.
3 The ring axioms entail propertyP.
Example A.For anyx,y,z ∈A,x+ (y+z) = (x+y) +z. Example B.
It isnot the casethat 1+16=0 inA. Example C (Anders Kock).The generic ring is afield:
∀x ∈A. (x=0⇒1 =0)⇒(∃y ∈A.xy=1) . Hence: When verifying a coherent sequent for all rings, can without loss of generality assume the field condition.
The generic ring
“LetRbe a ring.” – Which ring does this phrase refer to?
Z F2 Q[X] R OX A
Thm.For any?propertyP of rings, the following are equivalent:
1 Thegeneric ringAhas propertyP.
2 Every?ring has propertyP.
3 The ring axioms entail propertyP.
Example A.For anyx,y,z ∈A,x+ (y+z) = (x+y) +z.
Example B.
It isnot the casethat 1+16=0 inA. Example C (Anders Kock).The generic ring is afield:
∀x ∈A. (x=0⇒1 =0)⇒(∃y ∈A.xy=1) . Hence: When verifying a coherent sequent for all rings, can without loss of generality assume the field condition.
The generic ring
“LetRbe a ring.” – Which ring does this phrase refer to?
Z F2 Q[X] R OX A
Thm.For any?propertyP of rings, the following are equivalent:
1 Thegeneric ringAhas propertyP.
2 Every?ring has propertyP.
3 The ring axioms entail propertyP.
Example A.For anyx,y,z ∈A,x+ (y+z) = (x+y) +z.
Example B.Is 1+1=0 inA?
It isnot the casethat 1+16=0 inA. Example C (Anders Kock).The generic ring is afield:
∀x ∈A. (x=0⇒1 =0)⇒(∃y ∈A.xy=1) . Hence: When verifying a coherent sequent for all rings, can without loss of generality assume the field condition.
The generic ring
“LetRbe a ring.” – Which ring does this phrase refer to?
Z F2 Q[X] R OX A
Thm.For any?propertyP of rings, the following are equivalent:
1 Thegeneric ringAhas propertyP.
2 Every?ring has propertyP.
3 The ring axioms entail propertyP.
Example A.For anyx,y,z ∈A,x+ (y+z) = (x+y) +z.
Example B.It isnot the casethat 1+1=0 inA.
It isnot the casethat 1+16=0 inA. Example C (Anders Kock).The generic ring is afield:
∀x ∈A. (x=0⇒1 =0)⇒(∃y ∈A.xy=1) . Hence: When verifying a coherent sequent for all rings, can without loss of generality assume the field condition.
The generic ring
“LetRbe a ring.” – Which ring does this phrase refer to?
Z F2 Q[X] R OX A
Thm.For any?propertyP of rings, the following are equivalent:
1 Thegeneric ringAhas propertyP.
2 Every?ring has propertyP.
3 The ring axioms entail propertyP.
Example A.For anyx,y,z ∈A,x+ (y+z) = (x+y) +z.
Example B.It isnot the casethat 1+1=0 inA. But also:
It isnot the casethat 1+16=0 inA.
Example C (Anders Kock).The generic ring is afield:
∀x ∈A. (x=0⇒1 =0)⇒(∃y ∈A.xy=1) . Hence: When verifying a coherent sequent for all rings, can without loss of generality assume the field condition.
The generic ring
“LetRbe a ring.” – Which ring does this phrase refer to?
Z F2 Q[X] R OX A
Thm.For any?propertyP of rings, the following are equivalent:
1 Thegeneric ringAhas propertyP.
2 Every?ring has propertyP.
3 The ring axioms entail propertyP.
Example A.For anyx,y,z ∈A,x+ (y+z) = (x+y) +z.
Example B.It isnot the casethat 1+1=0 inA. But also:
It isnot the casethat 1+16=0 inA.
Hence: When verifying a coherent sequent for all rings, can without loss of generality assume the field condition.
The generic ring
“LetRbe a ring.” – Which ring does this phrase refer to?
Z F2 Q[X] R OX A
Thm.For any?propertyP of rings, the following are equivalent:
1 Thegeneric ringAhas propertyP.
2 Every?ring has propertyP.
3 The ring axioms entail propertyP.
Example A.For anyx,y,z ∈A,x+ (y+z) = (x+y) +z.
Example B.It isnot the casethat 1+1=0 inA. But also:
It isnot the casethat 1+16=0 inA. Example C (Anders Kock).The generic ring is afield:
A selection of noncoherent sequents
Thegeneric objectMvalidates:
1 ∀x,y ∈M.¬¬(x =y).
2 ∀x1, . . . ,xn ∈M.¬∀y ∈M.y =x1∨ · · · ∨y=xn. Thegeneric ringAvalidates:
1 ∀x∈A.(x=0⇒1 =0)⇒(∃y ∈A.xy=1).
2 ∀x∈A.¬¬(x=0).
Thegeneric local ringA0validates:
∀x∈ 0.(x =0 ⇒1 =0)⇒(∃y ∈ 0.xy=1).
An application in commutative algebra
LetAbe a reduced ring (xn =0⇒x=0). Letpbe thegeneric prime ideal? ofA. ThenAp :=A[p−1]validates:
Ap is afield: ∀x∈Ap.(¬(∃y ∈Ap.xy=1)⇒x =0).
Ap has¬¬-stable equality: ∀x,y ∈Ap.¬¬(x =y)⇒x =y.
Ap isanonymously Noetherian.
This observation unlocks a short and conceptual proof of Grothen- dieck’sgeneric freeness lemmain algebraic geometry.
Thm. (baby freeness) LetM be anA-module. Then 1 implies 3.
1 Mis finitely generated (⇐⇒Mpis finitely generated)
A systematic source
Gavin Wraith.Some recent developments in topos theory.
In: Proc. of the ICM (Helsinki, 1978).
Thm. (Nullstellensatz):The genericT-modelUT validates:
For any coherent sequentσ,
Arithmetic universes
Places where we can do mathematics (among others):
1 Set (sets)
2 Eff (data types)
3 sSet (simplicial sets)
4 Sh(X)(sheaves overX)
These are examples forarithmetic universes.
Definition.Anarithmetic universeis a category with finite lim- its (“×”), stable finite disjoint coproducts (“q”), stable effective quotients (“X/∼”) and parametrized list objects (“N”, “List(X)”). Thm. Any statement which is provable inpredicative con- structive mathematics(no powersets, noϕ∨¬ϕ, no¬¬ϕ⇒ϕ, no axiom of choice) is true in any arithmetic universe.
Further examples:
5 theinitialarithmetic universe
6 theclassifyingarithmetic universe for the theory of rings
Arithmetic universes
Places where we can do mathematics (among others):
1 Set (sets)
2 Eff (data types)
3 sSet (simplicial sets)
4 Sh(X)(sheaves overX) These are examples forarithmetic universes.
Definition.Anarithmetic universeis a category with finite lim- its (“×”), stable finite disjoint coproducts (“q”), stable effective quotients (“X/∼”) and parametrized list objects (“N”, “List(X)”). Thm. Any statement which is provable inpredicative con- structive mathematics(no powersets, noϕ∨¬ϕ, no¬¬ϕ⇒ϕ, no axiom of choice) is true in any arithmetic universe.
Further examples:
5 theinitialarithmetic universe
6 theclassifyingarithmetic universe for the theory of rings
Arithmetic universes
Places where we can do mathematics (among others):
1 Set (sets)
2 Eff (data types)
3 sSet (simplicial sets)
4 Sh(X)(sheaves overX) These are examples forarithmetic universes.
Definition.Anarithmetic universeis a category with finite lim- its (“×”), stable finite disjoint coproducts (“q”), stable effective quotients (“X/∼”) and parametrized list objects (“N”, “List(X)”).
Thm. Any statement which is provable inpredicative con- structive mathematics(no powersets, noϕ∨¬ϕ, no¬¬ϕ⇒ϕ, no axiom of choice) is true in any arithmetic universe.
Further examples:
5 theinitialarithmetic universe
6 theclassifyingarithmetic universe for the theory of rings
Arithmetic universes
Places where we can do mathematics (among others):
1 Set (sets)
2 Eff (data types)
3 sSet (simplicial sets)
4 Sh(X)(sheaves overX) These are examples forarithmetic universes.
Definition.Anarithmetic universeis a category with finite lim- its (“×”), stable finite disjoint coproducts (“q”), stable effective quotients (“X/∼”) and parametrized list objects (“N”, “List(X)”).
Thm. Any statement which is provable inpredicative con- structive mathematics(no powersets, noϕ∨¬ϕ, no¬¬ϕ⇒ϕ, no axiom of choice) is true in any arithmetic universe.
