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 over a

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 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 versus 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 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

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 ∃Y.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 ∨, ∧, ∃ 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 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 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) ∨ ab∈N 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

  u0 v0 For n = 2 with A =  u1 v1  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: ∃g. g|a0 ∧ g|a1 ∧ g|a2 ∧ (∀x. x|a0 ∧ x|a1 ∧ x|a2∧ → x|g)

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 ∃Y.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: If R is Noetherian, local and regular, then R is an UFD

24 Constructive Homological Algebra Noetherianity

Most presentation of homological algebra assumes the ring R to be Noetherian

A remarquable exception is the book by Northcott Finite Free Resolution

In this context most results are first-order schema, and we can hope to have direct elementary proofs

An example is the following result

Theorem: If R is regular, then R is a GCD domain

Domain means here that we have x = 0 or x is regular for all x in R

The implication R regular → R domain is geometric

25 Constructive Homological Algebra Regular element and ideal

We say that a is regular iff ax = 0 → x = 0

We say that a1, . . . , an is regular iff a1x = 0 ∧ · · · ∧ anx = 0 → x = 0 We say that I is regular iff xI = 0 → x = 0

Lemma: If a1, . . . , an, a and a1, . . . , an, b are regular then so is a1, . . . , an, ab Simple logical form

k k Corollary: If a1, . . . , an is regular then so is a1, . . . , an

26 Constructive Homological Algebra Vasconcellos Theorem

If we have

A A A A 0 −−→Rmk −−k→... −−2→Rm1 −−1→Rm −−→I −−→0

We define c(I) = m − m1 + m2 − ... to be the Euler characteristic of I One can show that it depends only on I and not on the choice of the resolution

Theorem: (Vasconcellos 1971) If c(I) = 0 then I = 0. If c(I) = 1 then I is regular. In all the other cases 1 = 0 in R

27 Constructive Homological Algebra Vasconcellos Theorem

The proof of Vasconcellos Theorem in Northcott’s book relies on the existence of minimal prime ideals, which is proved using Zorn’s Lemma

If we fix the size of the resolution, for instance

2 A 3 0 → R −−→R → ha0, a1, a2i → 0 the statement becomes first-order

It is even a geometric formula, hence we know a priori that it should have a simple logical proof

28 Constructive Homological Algebra Minimal Prime ideals

In Northcott one uses localization at a minimal prime ideal

To prove: from c(a0, a1, a2) 6= 0 and ha0, a1, a2i not regular we get a contradiction

Take x 6= 0 in (0 : I) and a minimal prime M over (0 : x)

Lemma: If hb1, . . . , bmi is regular then some of the bi is not in M

We then look at a minimal resolution in the localization RM

29 Constructive Homological Algebra Vasconcellos Theorem

Using such ideas, we extracted from the classical argument the following elementary glueing principle

Lemma: If u1, . . . , un is regular and b = 0 in R[1/u1],...,R[1/un] then b = 0 in R

k k Since u1, . . . , un is regular and b = 0 in R[1/u] iff there exists k such that ukb = 0

Local-global principle, compare with 1 = hu1, . . . , uni

Corollary: If u1, . . . , un is regular and b is regular in R[1/u1],...,R[1/un] then b is regular in R

30 Constructive Homological Algebra Vasconcellos Theorem

  u0 v0 We write A =  u1 v1  u2 v2

Since A represents an injective map both hu0, u1, u2i and hv0, v1, v2i are regular

31 Constructive Homological Algebra Vasconcellos Theorem

I = ha0, a1, a2i is regular in R[1/u0],R[1/u1],R[1/u2]

In R[1/u0], we can by change of basis, consider the sequence

0 0 → R2 −A−→R3 → I → 0

 10  0 with A =  0 v1 − u1v0/u0  0 v2 − u2v0/u0

32 Constructive Homological Algebra Vasconcellos Theorem

We can then simplify the sequence to

0 → R −−→R2 → I → 0

Reasoning in a similar way, we reduce the problem to

0 → R → I → 0 and it is clear that I is regular in this case

33 Constructive Homological Algebra A Regular Ring is a Domain

Assume that any finitely generated ideal has a finite free resolution

In particular hai has a finite free resolution

Hence we have a = 0 or a is regular

Classically, this means that R is an integral domain

34 Constructive Homological Algebra Injective maps

The glueing property for regular elements has replaced the use of minimal prime ideals

The same method can be used to give simple proofs of other standard result

Lemma: If A is a n × m matrix with n 6 m and ∆n(A) is regular then Rn −−A →Rm is injective

35 Constructive Homological Algebra Injective maps

The converse holds

n A m Lemma: If A is a n × m matrix with n 6 m and R −−→R is injective then and ∆n(A) is regular The same method proves the converse, by induction on n and considering the first column which is regular

36 Constructive Homological Algebra Regular Element Theorem

The following result holds only for Noetherian rings

Theorem: If a finitely generated ideal is regular then it contains a regular element

It is one reason why most treatment considers only Noetherian rings

Northcott presents a way to avoid this Noetherianity condition (due to Hochster).

37 Constructive Homological Algebra Regular Element Theorem

n Theorem: (McCoy) If a0, . . . , an is regular in R then a0 + a1X + ··· + anX is regular in R[X]

Thus, in general, we have a regular element but in R[X]

The solution exists in an enlarged universe

This result can be used instead of the Regular Element Theorem

38 Constructive Homological Algebra McCoy’s Theorem

n We write P = a0 +···+anX and we show by induction on m that if PQ = 0 m with Q = b0 + b1X + ··· + bmX , then Q = 0

We have anbm = 0 and P (anQ) = 0. Hence by induction, anQ = 0

Similarly, we get an−1Q = ··· = a0Q = 0 and since a0, . . . , an is regular we have Q = 0

Usually this argument is presented as a classical argument (we have applied a negative translation on this argument)

39 Constructive Homological Algebra McCoy’s Theorem

The same argument shows that

Theorem: (McCoy) If a0, . . . , an is regular in R then a0X0+a1X1+···+anXn is regular in R[X0,...,Xn]

We have replaced the ideal ha0, . . . , ani by the polynomial a0X0 + ··· + anXn H. Edwards Divisor Theory, Kronecker works with such polynomial

Use a0X0 + ··· + anXn instead of ha0, . . . , ani

40 Constructive Homological Algebra True Grade

We define Gr(a1, . . . , an) > 1 iff ha1, . . . , ani is regular

We define Gr(a1, . . . , an) > 2 iff ha1, . . . , ani is regular and Gr(a1, . . . , an) > 1 in the ring R[X1,...,Xn]/ha1X1 + . . . anXni

We define Gr(a1, . . . , an) > k + 1 iff ha1, . . . , ani is regular and Gr(a1, . . . , an) > k in the ring R[X1,...,Xn]/ha1X1 + . . . anXni

A sequence a1, . . . , an is regular iff Gr(a1, . . . , an) ≥ n

41 Constructive Homological Algebra True Grade

Gr(a1, . . . , an) > 2 iff ha1, . . . , ani is regular and regular mod. a1X1 + ··· + anXn

Proposition: Gr(a1, . . . , an) > 2 iff ha1, . . . , ani is regular and whenever b1, . . . , bn is proportional to a1, . . . , an, there exists an (unique) element t such that b1 = ta1, . . . , bn = tan

b1, . . . , bn proportional to a1, . . . , an means aibj = ajbi for all i, j

42 Constructive Homological Algebra 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 Here again, for a fixed size, this is a first-order statement

Logical form of the statement?

43 Constructive Homological Algebra Hilbert-Burch Theorem

  u0 v0 We prove it for n = 2 with A =  u1 v1  u2 v2 Question: how do we compute the gcd from the given data?

44 Constructive Homological Algebra Hilbert-Burch Theorem

Write ∆i = ujvk −ukvj we know that ∆0, ∆1, ∆2 is regular since A represents an injective map

Hence the element w = ∆0X0 +∆1X1 +∆2X2 is regular by McCoy’s Theorem

Notice that we have ai∆j = ai∆i since (a0 a1 a2)A = 0

45 Constructive Homological Algebra Hilbert-Burch Theorem

We change R to R[X0,X1,X2] we still have an exact sequence

2 A 3 (a0 a1 a2) 0 → R[X0,X1,X2] −−→R[X0,X1,X2] −−→ IR[X0,X1,X2] → 0

2 A 3 It follows from this that 0 → R[X0,X1,X2] −−→R[X0,X1,X2] is still exact modulo w, using the fact that w is regular

46 Constructive Homological Algebra Regular Element

ϕ ψ Lemma: If 0 → E −−→F −−→G is exact and a is regular for G then ϕ is still mono modulo a

a regular for G means az = 0 implies z = 0 for z in G

If we have ϕ(x) = ay then we have aψ(y) = 0 and hence ψ(y) = 0, since a is regular for G. Hence there exists x1 such that y = ϕ(x1) and we have ϕ(x − ax1) = 0 and hence x = 0 modulo a

47 Constructive Homological Algebra Hilbert-Burch Theorem

Hence ∆0, ∆1, ∆2 is still regular modulo w

Since we have ∆iaj = ∆jai it follows that

∆j(a0X0 + a1X1 + a2X2) = ajw = 0 modulo w. Hence a0X0 + a1X1 + a2X2 = 0 modulo w

Hence we have one element g such that ai = g∆i

By Vasconcellos Theorem, a0, a1, a2 is regular and so g is regular

48 Constructive Homological Algebra Hilbert-Burch Theorem

I claim that g is the GCD of a0, a1, a2

If we have ai = tbi then t is regular since a0, a1, a2 is regular

We have t(bi∆j − bj∆i) = 0 and hence bi∆j = bj∆i

Like before, we deduce that there exists s such that bi = s∆i

We then have ∆i(ts − g) = 0 and so g = ts

49 Constructive Homological Algebra Generalization of Hilbert-Burch Theorem

The algorithm to compute the gcd is

Cayley determinant of a complex

which is a generalization of the determinant of a linear map Rn → Rn

Theorem: If the ideal generated by ha1, . . . , ami has a finite free resolution then a1, . . . , am have a gcd, which is a regular element The proof being constructive it can be (and has been) implemented

Implementation in the system HOMALG as a bachelor thesis (F. Diebold supervised by M. Barakat)

50 Constructive Homological Algebra Generalization of Hilbert-Burch Theorem

The general algorithm can be used for the ring k[X1,...,Xn]

By Hilbert Syzygy Theorem, any finitely generated ideal hP1,...,Pmi has a finite free resolution and hence we get that P1,...,Pn have a gcd This is usually proved/implemented by reference to the result that R[X] is a GCD domain whenever R is a GCD domain

This gives an alternative algorithm for computation of GCD in which is sometimes more efficient than the direct algorithm (experimentally)

51 Constructive Homological Algebra Question

If F,G in R[X,Y ] let J be the Jacobian of F,G

It can be proved classically that Gr(F,G) > 2 if 1 = hF, G, Ji Intuitionistic proof?

52 Constructive Homological Algebra References

G. Kreisel and J.L. Krivine, Elements of Mathematical Logic

Th.C. and C. Quitt´e Constructive Finite Free Resolutions, Manuscripta Math., 2011

H. Lombardi and C. Quitt´e Alg`ebre Commutative, m´ethode constructive; modules projectifs de type fini

Northcott Finite Free Resolution

G. Wraith, Intuitionistic algebra, some recent development in topos theory, Proceeding of ICM, 1978

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