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 ∃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 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) ∨ 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 polynomial ring 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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