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Constructive Homological Algebra Constructive Homological Algebra Nis, June 24, 2013 Constructive Homological Algebra Constructive Mathematics Constructive mathematics may be viewed as mathematics done using intuitionistic logic Bridges, Lombardi, Richman What is remarkable is that this characterization does not mention the notion of algorithm, but is purely logical We are going to illustrate this characterization in constructive algebra and more specially constructive homological algebra 1 Constructive Homological Algebra Homological algebra Homological algebra: originates from Hilbert On the theory of algebraic forms Math. Annalen, vol. 36, 473-534, 1890 Important for the history of constructive mathematics: this is the paper where Hilbert proves the basis Theorem in a non constructive way In this paper it is only a lemma to prove the Syzygy Theorem (and finite generation of invariant polynomials) But also from works of Cayley (Koszul complex) on elimination On the theory of elimination, Cambridge and Dublin Mathematical Journal, 1848 2 Constructive Homological Algebra Homological algebra Homological algebra can be described as linear algebra over a ring From a logical point of view, one would expect most of results in algebra to be directly expressed in first-order logic This is not the case: most text books use Noetherian hypotheses 3 Constructive Homological Algebra The regular element property Theorem: If an ideal is regular it contains a regular element True if the ring is Noetherian, not in general \Among the most useful theorem" in algebra according to Kaplansky! cf. Richman The regular element property 1998 Unexpected solution to the regular element property due to Hochster presented in Northcott's book on Finite Free Resolutions 4 Constructive Homological Algebra Constructive homological algebra In Northcott's book, the statements are first-order schemas Some proofs however use existence of prime ideals and minimal prime ideals According to the Skolem-G¨odelcompleteness Theorem, there should be direct first-order proofs What are they? The elementary proofs are (surprisingly) always shorter than the proofs in Northcott and sometimes more informative 5 Constructive Homological Algebra First-order versus higher-order Two examples -Bezout domain versus Principal ideal domain -gcd domain versus Unique Factorization domain A Bezout domain is such that any finitely generated ideal is principal This can be expressed by a first-order formula 6 Constructive Homological Algebra First-order versus higher-order A principal ideal domain is such that any ideal is principal This is a higher-order condition (quantification over all subsets) Classically, principal is equivalent to Bezout and Noetherian Noetherian (any ideal is finitely generated) is higher-order 7 Constructive Homological Algebra First-order versus higher-order A GCD domain is such that any two elements have a greatest common divisor This is a first-order condition Classically UFD is equivalent to GCD and Noetherian Theorem: If R is a GCD domain then so is R[X] Corollary: k[X1;:::;Xn] is a GCD domain if k is a discrete field This can be proved intuitionistically (cf. Mines-Richman-Ruitenburg) NonNoetherian version that R[X] is UFD if R is UFD 8 Constructive Homological Algebra Intuitionistic versus classical Discrete field 0 6= 1 and x = 0 or x is invertible This is intuitionistically distinct from the notion of Heyting field where we have an apartness relation x#y and the condition x#0 implies x invertible For instance R is a Heyting field and not a discrete field 9 Constructive Homological Algebra Intuitionistic versus classical Domain? One definition may be that the equality is decidable (discrete set) and a = 0 _ b = 0 if ab = 0 (one may want to add 0 6= 1) An element a in a ring R is regular iff ax = 0 implies x = 0 Another possible definition of a domain is to have the property x = 0 or x is regular Classically the two definitions are equivalent Intuitionistically they are not 10 Constructive Homological Algebra Coherent rings A fundamental notion in constructive homological algebra is the notion of coherent ring This notion is also fundamental in the computer algebra HOMALG (M. Barakat et al.) A ring is coherent iff any finitely generated ideal is finitely presented Theorem: The ring k[X1;:::;Xn] is coherent Can be proved using Gr¨obnerbasis 11 Constructive Homological Algebra Coherent rings R is coherent iff for any matrix A there exists a matrix B such that AX = 0 iff 9Y:X = BY There is not (yet?) a good intuition why this notion is so important: it is logically complex (not geometric) and does not have good closure property (it is not the case that R[X] is coherent if R is coherent) 12 Constructive Homological Algebra Geometric formulae A first-order formula is positive iff it can be built from _; ^; 9 and (positive) atomic formulae (including ?) It is geometric if it is an universal quantification of an implication between two positive formulae The Bezout condition is geometric To be a GCD domain is not geometric (but is first-order) This class of formulae satisfies the Glivenko condition: if they are provable in classical first-order logic then they are intuitionistically provable 13 Constructive Homological Algebra Prime and minimal prime ideals Classically any nontrivial ring has a prime ideal Intuitionistically it is not possible in general to prove the existence of a prime ideal A possible solution, when the prime (resp. minimal prime) ideal is used in the proof in a generic way, is to use ideas from formal (or pointfree) topology We describe the prime ideals as ideal(!) objects being described by finite approximations 14 Constructive Homological Algebra Prime ideals A prime ideal is described by a predicate D(a) (meaning a is not in the prime ideal) with the following conditions D(0) = 0 D(1) = 1 D(ab) = D(a)^D(b) D(a+b) 6 D(a)_D(b) Intuitionitically we can fully describe the lattice generated by these elements and conditions We interpret D(a1; : : : ; an) = D(a1) _···_ D(an) as the radical of the ideal ha1; : : : ; ani Corollary: D(a) = 0 if, and only if, a is nilpotent Corollary: D(a1; : : : ; an) = 1 if, and only if, ha1; : : : ; ani = 1 15 Constructive Homological Algebra Prime ideals n Example: we say that a polynomial a0 + a1X + ··· + anX is primitive iff 1 = ha0; a1; : : : ; ani Proposition: A product of primitive polynomials is primitive Classically a polynomial is primitive iff it is not zero modulo any prime ideals and use the remark that R[X] is an integral domain if R is an integral domain Expressed intuitionistically this remark becomes Proposition: (Gauss-Joyal) D(a0; : : : ; an)^D(b0; : : : ; bm) = D(c0; : : : ; cm+n) n m m+n whenever (a0 + : : : anX )(b0 + ··· + bmX ) = c0 + ··· + cm+nX 16 Constructive Homological Algebra Minimal Prime ideals A similar analysis can be done for minimal prime ideals We add the (infinitary) condition W 1 = D(a) _ ab2N D(b) where N is the ideal of nilpotent element 17 Constructive Homological Algebra Constructive Finite Free Resolution Hilbert-Burch Theorem Theorem: If we have an exact sequence n A n+1 0 ! R −−!R ! ha0; : : : ; ani ! 0 then the elements a0; : : : ; an have a GCD, which is regular For a fixed size, this is a first-order statement 18 Constructive Homological Algebra Hilbert-Burch Theorem 0 1 u0 v0 For n = 2 with A = @ u1 v1 A u2 v2 Hypotheses: a0u0 + a1u1 + a2u2 = a0v0 + a1v1 + a2v2 = 0 If a0x0 + a1x1 + a2x2 = 0 then x0; x1; x2 is a linear combination of u0; u1; u2 and v0; v1; v2 Conclusion: 9g: gja0 ^ gja1 ^ gja2 ^ (8x: xja0 ^ xja1 ^ xja2^ ! xjg) Question: can we/how do we compute the gcd of a0; a1; a2 from the given data? Notice that the statement is not a Glivenko statement, so that we cannot be sure for the direct first-order proof to be intuitionistic 19 Constructive Homological Algebra Homological Algebra Linear algebra over a ring If A is a matrix we let ∆n(A) be the ideal generated by all n × n minors of A Over a field if A is a m × n matrix with m > n and we consider the system AX = C for ∆n(A) 6= 0. It has a solution if ∆n+1(B) = 0 where B is the matrix AC This is not the case over Z 2x = 3 4x = 6 20 Constructive Homological Algebra Homological Algebra R is coherent iff for any matrix A there exists a matrix B such that AX = 0 iff 9Y:X = BY For a finitely generated ideal I we have a map Rm −−A !I −−!0 and we can build a sequence, if R is coherent A A A A ::: −−!Rm3 −−3!Rm2 −−2!Rm1 −−1!Rm −−!I −−!0 21 Constructive Homological Algebra Free Resolution In particular if we have a finitely generated ideal I we have a map Rm −−A !I −−!0 and, if the ring is coherent, we can build a sequence A A A A ::: −−!Rm3 −−3!Rm2 −−2!Rm1 −−1!Rm −−!I −−!0 This is called a free resolution of the ideal This measures the \complexity" of the ideal: relations between generators, then relations between relations, and so on. 22 Constructive Homological Algebra Free Resolution If we have mk = 0 for k > N we say that I has a finite free resolution A A A A 0 −−!RmN −−N!::: −−2!Rm1 −−1!Rm −−!I −−!0 Counter-example: for R = k[X; Y ]=hXY i the ideal hXi has an infinite free resolution Theorem: (Hilbert Syzygy Theorem) In the ring k[X1;:::;Xn] any finitely generated ideal has a finite free resolution hX; Y; Zi has a free resolution with N = 2 23 Constructive Homological Algebra Regular rings A ring is regular iff any finitely generated ideal has a finite free resolution k[X1;:::;Xn] is regular This notion was introduced by Serre to capture the properties of a local ring at a smooth (non singular) point of an algebraic variety (to show that this notion is stable under localisation) Theorem:
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