Nullstellen and Subdirect Representation Walter Tholen York University, Toronto, Canada Union College 19-20 October 2013 Walter Tholen (York University) Nullstellen & Subdirect Rep 19-20 October 2013 1 / 21
HNB Theorem — Version 1 System of r polynomial equations in n variables with coefficients in a field k : f 1 ( x 1 , . . . , x n ) = 0 . . . ( f i ∈ k [ x 1 , . . . , x n ]) ( ⋆ ) f r ( x 1 , . . . , x n ) = 0 Examples x 1 − x 2 x 2 x 2 x 2 2 = 0 1 − 1 = 0 1 + 1 = 0 1 − 1 = 0 x 2 1 + 1 = 0 Walter Tholen (York University) Nullstellen & Subdirect Rep 19-20 October 2013 2 / 21
HNB Theorem — Version 1 D. Hilbert (1893) Every system ( ⋆ ) has a solution a = ( a 1 , . . . , a n ) in F n for any algebraically-closed extension field F of k , unless there are polynomials g 1 , . . . , g r ∈ k [ x ] with g 1 f 1 + . . . + g r f r = 1 Restriction is essential: 0 = g 1 ( a ) f 1 ( a ) + . . . + g r ( a ) f r ( a ) = 1 Walter Tholen (York University) Nullstellen & Subdirect Rep 19-20 October 2013 3 / 21
A Galois Correspondence Systems of equations Solution sets S ( P ) = { a ∈ k n | ∀ f ∈ P : f ( a ) = 0 } P ⊆ k [ x 1 , . . . , x n ] �− → X ⊆ F n J ( X ) = { f ∈ k [ x ] | ∀ a ∈ X : f ( a ) = 0 } ← − � P ⊆ J ( X ) ⇐ ⇒ X ⊆ S ( P ) P = J ( S ( P )) ← → X = S ( J ( X )) √ P = { f ∈ k [ x ] | ∃ m ≥ 1 : f m ∈ P } Note: P ⊆ Walter Tholen (York University) Nullstellen & Subdirect Rep 19-20 October 2013 4 / 21
HNB Theorem — Version 2 P � k [ x 1 , . . . , x n ] proper ideal √ ⇒ = J ( S ( P )) = P F algebraically-closed That is : ⇒ ∃ m ≥ 1 : f m ∈ P ) If f ∈ k [ x ], then ( ∀ a ∈ S ( P ) : f ( a ) = 0 ⇐ Walter Tholen (York University) Nullstellen & Subdirect Rep 19-20 October 2013 5 / 21
Version 2 implies Version 1 ( f 1 , . . . , f r ) = P √ P = J ( S ( P )) ← → S ( P ) ← → ∅ k [ x 1 , . . . , x n ] Note: Any P � k [ x 1 , . . . , x n ] is finitely generated: k Noetherian ring = ⇒ k [ x ] Noetherian Walter Tholen (York University) Nullstellen & Subdirect Rep 19-20 October 2013 6 / 21
Trying to prove Version 2 √ Need to show J ( S ( P )) ⊆ P : √ If f / ∈ P , must find a ∈ S ( P ) with f ( a ) � = 0. √ Consider A = k [ x 1 , . . . , x n ] / P Wish: Find π ϕ ✲ A ✲ F k [ x ] ✲ π ( f ) = u ✲ ϕ ( u ) � = 0 f Consider a = ( ϕ ( π ( x 1 )) , . . . , ϕ ( π ( x n ))). Then f ( a ) � = 0, but ∀ p ∈ P : p ( a ) = 0. Walter Tholen (York University) Nullstellen & Subdirect Rep 19-20 October 2013 7 / 21
HNB Theorem — Version 3 and 4 Version 3 A fin. gen. comm. k -algebra ∃ ϕ : A → F k ⊆ F alg. closed = ⇒ u �→ ϕ ( u ) � = 0 u ∈ A non-nilpotent Version 4 ∃ χ : A → K , χ ( u ) � = 0, � A fin. gen. comm. k -algebra ⇒ k ⊆ K field, = u ∈ A non-nilpotent K fin. gen. (as unital k -algebra) Walter Tholen (York University) Nullstellen & Subdirect Rep 19-20 October 2013 8 / 21
Version 4 implies Version 3 ✲ F k algebraically closed ✻ ❄ χ ✲ K A algebraic Walter Tholen (York University) Nullstellen & Subdirect Rep 19-20 October 2013 9 / 21
HNB Theorem — Version 5, Version 5 implies Version 4 A comm. ring = ⇒ ∃ Q � A prime , u / ∈ Q u non-nilpotent 5 ⇒ 4 L ✲ field of fractions ✻ integral domain σ ✲ A / Q ✲ ✲ K = R / M R = ( A / Q )[ σ ( u ) − 1 ] A M maximal ideal Walter Tholen (York University) Nullstellen & Subdirect Rep 19-20 October 2013 10 / 21
Proof of Version 5 S := { u n | n ≥ 1 } �∋ 0 Zorn: ∃ Q � A maximal w.r.t. Q ∩ S = ∅ Q is prime: a , b / ∈ Q = ⇒ ( a ) + Q , ( b ) + Q meet S ⇒ ∃ l , m : u l ∈ ( a ) + Q , u m ∈ b + Q = ⇒ u l + m ∈ ( ab ) + Q = = ⇒ ab / ∈ Q QED Walter Tholen (York University) Nullstellen & Subdirect Rep 19-20 October 2013 11 / 21
Decomposition Theorems � A comm., reduced = ⇒ { Q � A | Q prime } = (0) ✲ ✲ � A A / Q Q Lasker (1904), Noether (1920) R comm. Noetherian � = ⇒ P = Q 1 ∩ . . . ∩ Q n with Q i � R irreducible P � R ideal Alternative formulation of Version 5 � R comm., P � R radical ⇐ ⇒ P = { Q � R | P ⊆ Q , Q prime } Walter Tholen (York University) Nullstellen & Subdirect Rep 19-20 October 2013 12 / 21
HNB Theorem — Version 6 Birkhoff’s Subdirect Representation Theorem (1944) ✲ ✲ � A S i S i sdi i ∈ I A general finitary algebra = ⇒ ✲ ✲ ❄ S i ✲ ✲ � A S i i ∈ I A subdirectly irreducible (sdi) : ⇐ ⇒ If , then A ∼ = S i 0 ✲ ✲ ❄ S i This notion is categorical! Walter Tholen (York University) Nullstellen & Subdirect Rep 19-20 October 2013 13 / 21
HNB-categories A has (strong epi, mono)-factorizations A is weakly cowellpowered A has generator consisting of finitary objects (no existence requirements for limits or colimits) examples quasi-varieties of finitary algebras locally finitely presentable categories presheaf categories the category of topological spaces Walter Tholen (York University) Nullstellen & Subdirect Rep 19-20 October 2013 14 / 21
HNB Theorem — Version 7 A object of an HNB-category = ⇒ A has subdirect representation ✲ ✲ � A S i i ∈ I S sdi ⇐ ⇒ ∃ a � = b : P → S ∀ f : A → B ( fa � = fb ⇒ f monic) ⇐ ⇒ Con S \ { ∆ S } has a least element Set: 2 k -Vec: k CRng: Z p , Z p 2 Walter Tholen (York University) Nullstellen & Subdirect Rep 19-20 October 2013 15 / 21
HNB Theorem — Version 8 f : A → B morphism of an HNB-category with finite products S i ✲ ✲ s i p i (( p i ) i monic) each s i sdi ✲ ✲ A B f Walter Tholen (York University) Nullstellen & Subdirect Rep 19-20 October 2013 16 / 21
Set/B ∼ = Set B f : A → B ( A b ) b ∈ B A b = f − 1 b � A = A b , f | Ab = const b b ∈ B f sdi ⇐ ⇒ ∃ ! b 0 ∈ B : | A b 0 | = 2 and | A b | ≤ 1 ( b � = b 0 ) Walter Tholen (York University) Nullstellen & Subdirect Rep 19-20 October 2013 17 / 21
Constructive? Functorial? Generally: No! Zorn’s Lemma everywhere! Set: constructive, but not functorial: ✲ 2 0 = 1 ∅ ⊂ � ∩ � � � � � � � � ❄ � = 2 0 = 1 1 = = = = = = = = = = = = = = = = = = = x ❄ ❄ ✲ ✲ 2 I X Walter Tholen (York University) Nullstellen & Subdirect Rep 19-20 October 2013 18 / 21
Residually small HNB-categories { A ∈ A | A sdi } / ∼ A residually small ⇐ ⇒ small = ⇐ ⇒ A has cogenerator (of sdi objects) (if A is HNB, wellpowered) Set, AbGrp, Mod R : yes Grp, CompAbGrp : no Residually small finitary varieties: Taylor (1972) Walter Tholen (York University) Nullstellen & Subdirect Rep 19-20 October 2013 19 / 21
Equivalent Completeness properties for residually small HNB-categories ( A → [ A op , Set] has left adjoint) (i) A totally cocomplete (ii) A hypercomplete (iii) A small-complete with large intersections of monos (i) op A totally complete (ii) op A hypercocomplete (iii) op A small-cocomplete with large cointersections of epis Walter Tholen (York University) Nullstellen & Subdirect Rep 19-20 October 2013 20 / 21
Applications of HNB to algebraic geometry Recent work by M. Menni: An Exercise with Sufficient Cohesion, 2011 Walter Tholen (York University) Nullstellen & Subdirect Rep 19-20 October 2013 21 / 21
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