Noetherianity up to symmetry

Transcription

1 1 Noetherianity up to symmetry Jan Draisma TU Eindhoven and VU Amsterdam Singular Landscapes in honour of Bernard Teissier Aussois, June 2015

2 A landscape, and a disclaimer 2

3 # participants A landscape, and a disclaimer 2

4 # participants A landscape, and a disclaimer 2 Teissier #

5 A landscape, and a disclaimer 2 # participants Erdős # Teissier #

6 I. Equivariant Noetherianity 3

7 K[x 1, x 2, x 3,...] 4 For K a field, R := K[x 1, x 2,...] is not a Noetherian ring... but:

8 K[x 1, x 2, x 3,...] 4 For K a field, R := K[x 1, x 2,...] is not a Noetherian ring... but: Theorem [Cohen 1967/Aschenbrenner-Hillar 2007] Let Sym(N) act on R by πx i = x π(i). Every chain I 1 I 2... of Sym(N)-stable ideals of R stabilises, i.e., I n is constant for n 0.

9 K[x 1, x 2, x 3,...] 4 For K a field, R := K[x 1, x 2,...] is not a Noetherian ring... but: Theorem [Cohen 1967/Aschenbrenner-Hillar 2007] Let Sym(N) act on R by πx i = x π(i). Every chain I 1 I 2... of Sym(N)-stable ideals of R stabilises, i.e., I n is constant for n 0. Definition Given a commutative ring R, a monoid Π, and an action of Π on R by algebra homomorphisms, R is Π-Noetherian if every chain I 1 I 2... of Π-stable ideals stabilises.

10 K[x 1, x 2, x 3,...] 4 For K a field, R := K[x 1, x 2,...] is not a Noetherian ring... but: Theorem [Cohen 1967/Aschenbrenner-Hillar 2007] Let Sym(N) act on R by πx i = x π(i). Every chain I 1 I 2... of Sym(N)-stable ideals of R stabilises, i.e., I n is constant for n 0. Definition Given a commutative ring R, a monoid Π, and an action of Π on R by algebra homomorphisms, R is Π-Noetherian if every chain I 1 I 2... of Π-stable ideals stabilises. Equivalently: each Π-stable ideal I is generated by finitely many Π-orbits in R. R is a Noetherian R Π-module (multiplication: π r = π(r) π).

11 Proof of Cohen s theorem 5 Increasing maps Inc(N) := {π : N N π(1) < π(2) <...} is a monoid, and it acts on R = K[x 1, x 2,...] by πx i := x π(i). For example, if π : 1 2, 2 4, 3 5,..., then πx 2 1 x3 3 = x2 2 x2 5.

12 Proof of Cohen s theorem 5 Increasing maps Inc(N) := {π : N N π(1) < π(2) <...} is a monoid, and it acts on R = K[x 1, x 2,...] by πx i := x π(i). For example, if π : 1 2, 2 4, 3 5,..., then πx 2 1 x3 3 = x2 2 x2 5. Cohen s theorem follows from: Claim: K[x 1, x 2,...] is Inc(N)-Noetherian.

13 Proof of Cohen s theorem 5 Increasing maps Inc(N) := {π : N N π(1) < π(2) <...} is a monoid, and it acts on R = K[x 1, x 2,...] by πx i := x π(i). For example, if π : 1 2, 2 4, 3 5,..., then πx 2 1 x3 3 = x2 2 x2 5. Cohen s theorem follows from: Claim: K[x 1, x 2,...] is Inc(N)-Noetherian. Proof

14 Proof of Cohen s theorem 5 Increasing maps Inc(N) := {π : N N π(1) < π(2) <...} is a monoid, and it acts on R = K[x 1, x 2,...] by πx i := x π(i). For example, if π : 1 2, 2 4, 3 5,..., then πx 2 1 x3 3 = x2 2 x2 5. Cohen s theorem follows from: Claim: K[x 1, x 2,...] is Inc(N)-Noetherian. Proof reduce to monomial ideals (Inc(N) preserves monomial orders).

15 Proof of Cohen s theorem 5 Increasing maps Inc(N) := {π : N N π(1) < π(2) <...} is a monoid, and it acts on R = K[x 1, x 2,...] by πx i := x π(i). For example, if π : 1 2, 2 4, 3 5,..., then πx 2 1 x3 3 = x2 2 x2 5. Cohen s theorem follows from: Claim: K[x 1, x 2,...] is Inc(N)-Noetherian. Proof reduce to monomial ideals (Inc(N) preserves monomial orders). show that for any sequence m 1, m 2,... of monomials in x, there are i < j, π Inc(N) : (πm i ) m j (well-partial order).

16 Further examples: matrices 6 Theorem [Cohen 98/Hillar-Sullivant 09] K[x i j 1 i k, j N] is also Inc(N)-Noetherian (πx i j = x iπ( j) ).

17 Further examples: matrices 6 Theorem [Cohen 98/Hillar-Sullivant 09] K[x i j 1 i k, j N] is also Inc(N)-Noetherian (πx i j = x iπ( j) ). Unfortunately, K[x i j i, j N] with πx i j = x π(i),π( j) is not. But:

18 Further examples: matrices 6 Theorem [Cohen 98/Hillar-Sullivant 09] K[x i j 1 i k, j N] is also Inc(N)-Noetherian (πx i j = x iπ( j) ). Unfortunately, K[x i j i, j N] with πx i j = x π(i),π( j) is not. But: Proposition If chark = 0, then K[x i j i, j N]/((k + 1) (k + 1)-minors of x) is Inc(N)-Noetherian. (It is an invariant ring of GL k 1.)

19 Further examples: matrices 6 Theorem [Cohen 98/Hillar-Sullivant 09] K[x i j 1 i k, j N] is also Inc(N)-Noetherian (πx i j = x iπ( j) ). Unfortunately, K[x i j i, j N] with πx i j = x π(i),π( j) is not. But: Proposition If chark = 0, then K[x i j i, j N]/((k + 1) (k + 1)-minors of x) is Inc(N)-Noetherian. (It is an invariant ring of GL k 1.) Theorem [Sam-Snowden 15] If chark = 0, then K[x i j i, j N] is GL N GL N -Noetherian. Here GL N = { g } acts by left and right multiplication.

20 Topological Noetherianity 7 Definition A topological space X equipped with an action of a monoid Π by continuous maps is called Π-Noetherian if every chain X 1 X 2... of Π-stable closed subsets stabilises.

21 Topological Noetherianity 7 Definition A topological space X equipped with an action of a monoid Π by continuous maps is called Π-Noetherian if every chain X 1 X 2... of Π-stable closed subsets stabilises. If a K-algebra R is Π-Noetherian as a ring, then Hom(R, K) is a Π-Noetherian topological space. But there are many examples where the converse is unknown or false.

22 Topological Noetherianity 7 Definition A topological space X equipped with an action of a monoid Π by continuous maps is called Π-Noetherian if every chain X 1 X 2... of Π-stable closed subsets stabilises. If a K-algebra R is Π-Noetherian as a ring, then Hom(R, K) is a Π-Noetherian topological space. But there are many examples where the converse is unknown or false. Lemma Π-equivariant images and finite unions of Π-Noetherian spaces are Π-Noetherian. If a group G acts on X by homeo, and Z X is H-Noetherian for a subgroup H G, then GZ := g G gz is G-Noetherian.

23 Tuples of matrices 8 Theorem For any K and p, the space (K N N ) p is GL N GL N -Noetherian.

24 Tuples of matrices 8 Theorem For any K and p, the space (K N N ) p is GL N GL N -Noetherian. We don t know if this holds ring-theoretically.

25 Tuples of matrices 8 Theorem For any K and p, the space (K N N ) p is GL N GL N -Noetherian. We don t know if this holds ring-theoretically. Key notion The rank of a tuple (A 1,..., A p ) is min{rk i c i A i c P p 1 }.

26 Tuples of matrices 8 Theorem For any K and p, the space (K N N ) p is GL N GL N -Noetherian. We don t know if this holds ring-theoretically. Key notion The rank of a tuple (A 1,..., A p ) is min{rk i c i A i c P p 1 }. Dichotomy For X (K N N ) p closed and GL N GL N -stable, either: 1. sup A X rka < can do induction on p; or 2. sup A X rka = X = (K N N ) p.

27 II. Why? 9

28 Sequences of varieties 10 Motivating question X 1, X 2,... algebraic varieties X n A n closed embedding stabilise for n 0?

29 Sequences of varieties 10 Motivating question X 1, X 2,... algebraic varieties X n A n closed embedding stabilise for n 0? Running example A n = K n n (n n-matrices over a field K) X n = {x A n rank x 1} defined by equations x i j x kl x il x k j = 0 for all n 2

30 Passing to a limit 11 Set-up A n a finite-dimensional vector space, π n : A n+1 A n linear X n A n a closed subvariety, fitting in a commutative diagram π 1 π 2 π 3 A 1 A 2 A 3... X 1 X 2 X 3...

31 Passing to a limit 11 Set-up A n a finite-dimensional vector space, π n : A n+1 A n linear X n A n a closed subvariety, fitting in a commutative diagram A 1 π 1 A 2 π 2 A 3 π 3... A : dual of a countabledimensional space X 1 X 2 X 3... X : -dim variety

32 Passing to a limit 11 Set-up A n a finite-dimensional vector space, π n : A n+1 A n linear X n A n a closed subvariety, fitting in a commutative diagram A 1 π 1 A 2 π 2 A 3 π 3... A : dual of a countabledimensional space X 1 X 2 X 3... X : -dim variety Running example A n = K n n X n = {rank 1 matrices} π n forgets the last row and column A = K N N space with coordinates x i j, i, j N X = {N N rank 1 matrices}

33 Symmetry 12 Set-up π 1 π 2 π 3 A 1 A 2 A 3... A X 1 X 2 X 3... X

34 Symmetry 12 Set-up G 1 G 2 G 3... G Assumptions: G π 1 π 2 π n acts linearly 3 A 1 A 2 A 3... A G n preserves X n π n is G n -equivariant G acts on A X 1 X 2 X 3... X and preserves X

35 Symmetry 12 Set-up G 1 G 2 G 3... G Assumptions: G π 1 π 2 π n acts linearly 3 A 1 A 2 A 3... A G n preserves X n π n is G n -equivariant G acts on A X 1 X 2 X 3... X and preserves X Running example A n = K n n, G n = GL n (K) acting by (g, a) gag 1 G = GL N (K), preserves X 1 g 1...

36 Formalising stabilisation 13 Summary X is a variety in the vector space A with countably many coordinates. If f is a polynomial that vanishes everywhere on X, then so is g f := f g 1 for all g G.

37 Formalising stabilisation 13 Summary X is a variety in the vector space A with countably many coordinates. If f is a polynomial that vanishes everywhere on X, then so is g f := f g 1 for all g G. Question (with many variants) Is X the common zero set of finitely many orbits G f 1,...,G f s of polynomial equations? Typical proof strategy: find a G -Noetherian subvariety Y of A containing X.

38 Formalising stabilisation 13 Summary X is a variety in the vector space A with countably many coordinates. If f is a polynomial that vanishes everywhere on X, then so is g f := f g 1 for all g G. Question (with many variants) Is X the common zero set of finitely many orbits G f 1,...,G f s of polynomial equations? Typical proof strategy: find a G -Noetherian subvariety Y of A containing X. Example: rank-one matrices X is defined by the GL N (K)-orbit of x 11 x 22 x 12 x 21 so the family {X n } n stabilises.

39 III. Topics 14

40 Topic 1: bounded-rank tensors 15 Definition Rank of ω V 1 V n is the minimal k in any expression ω = k i=1 v i1 v in. (For n = 2 this is matrix rank.)

41 Topic 1: bounded-rank tensors 15 Definition Rank of ω V 1 V n is the minimal k in any expression ω = k i=1 v i1 v in. (For n = 2 this is matrix rank.) Theorem For any fixed k there is a d, independent of n and the V i, such that {ω rank ω k} is defined by polynomials of degree d. [D-Kuttler, 2014]

42 Topic 1: bounded-rank tensors 15 Definition Rank of ω V 1 V n is the minimal k in any expression ω = k i=1 v i1 v in. (For n = 2 this is matrix rank.) Theorem For any fixed k there is a d, independent of n and the V i, such that {ω rank ω k} is defined by polynomials of degree d. [D-Kuttler, 2014] Table k d [Landsberg-Manivel, 2004] [Qi, 2014] [Strassen,1983]

43 Topic 1: bounded-rank tensors 15 Definition Rank of ω V 1 V n is the minimal k in any expression ω = k i=1 v i1 v in. (For n = 2 this is matrix rank.) Theorem For any fixed k there is a d, independent of n and the V i, such that {ω rank ω k} is defined by polynomials of degree d. [D-Kuttler, 2014] Table k d [Landsberg-Manivel, 2004] [Qi, 2014] [Strassen,1983] Proof set-up A n = (K k+1 ) n X n = {rank k} G n = S n GL n k+1 π n : A n+1 A n, (v 1 v n+1 ) x 0 (v n+1 ) v 1 v n

44 Topic 2: Markov bases 16 Second hypersimplex P n := {v i j = e i + e j 1 i j n}

45 Topic 2: Markov bases 16 Second hypersimplex P n := {v i j = e i + e j 1 i j n} Theorem [De Loera-Sturmfels-Thomas 1995] P n has a Markov basis consisting of moves v i j + v kl v il + v k j and v i j v ji for i, j, k, l distinct; i.e., if i j c i j v i j = i j d i j v i j with c i j, d i j Z 0, then the expressions are connected by such moves without creating negative coefficients.

46 Topic 2: Markov bases 16 Second hypersimplex P n := {v i j = e i + e j 1 i j n} Theorem [De Loera-Sturmfels-Thomas 1995] P n has a Markov basis consisting of moves v i j + v kl v il + v k j and v i j v ji for i, j, k, l distinct; i.e., if i j c i j v i j = i j d i j v i j with c i j, d i j Z 0, then the expressions are connected by such moves without creating negative coefficients. Theorem [D-Eggermont-Krone-Leykin 2013] For any family (P n Z k n ), if P n = S n P n0 for n n 0, then n 1 : for n n 1 has a Markov basis M n with M n = S n M n1. we also have an algorithm for computing n 1 and M n1

47 Topic 3: Plücker varieties 17 Grassmannians Gr k (V) P( k V) is functorial in V, and the Hodge dual k V n k V with dim V = n maps Gr k (V) Gr n k (V ).

48 Topic 3: Plücker varieties 17 Grassmannians Gr k (V) P( k V) is functorial in V, and the Hodge dual k V n k V with dim V = n maps Gr k (V) Gr n k (V ). Definition A sequence (X k ) k of rules X k : V X k (V) P( k (V)) with these two properties is a Plücker variety.

49 Topic 3: Plücker varieties 17 Grassmannians Gr k (V) P( k V) is functorial in V, and the Hodge dual k V n k V with dim V = n maps Gr k (V) Gr n k (V ). Definition A sequence (X k ) k of rules X k : V X k (V) P( k (V)) with these two properties is a Plücker variety. Construction of new Plücker varieties tangential variety, secant variety, etc.

50 Topic 3: Plücker varieties 17 Grassmannians Gr k (V) P( k V) is functorial in V, and the Hodge dual k V n k V with dim V = n maps Gr k (V) Gr n k (V ). Definition A sequence (X k ) k of rules X k : V X k (V) P( k (V)) with these two properties is a Plücker variety. Construction of new Plücker varieties tangential variety, secant variety, etc. Theorem For a bounded Plücker variety X, (X k (K n )) k,n k is defined in bounded degree. [D-Eggermont 2014]

51 The infinite wedge 18 V :=..., x 3, x 2, x 1, x 1, x 2, x 3,... K V n,p := x n,..., x 1, x 1,..., x p V

52 The infinite wedge 18 V :=..., x 3, x 2, x 1, x 1, x 2, x 3,... K V n,p := x n,..., x 1, x 1,..., x p V Diagram 0 V 00 1 V 01 2 V 02 p V np p+1 V n,p+1 0 V 10 1 V 11 2 V 12 p V n+1,p

53 The infinite wedge 18 V :=..., x 3, x 2, x 1, x 1, x 2, x 3,... K V n,p := x n,..., x 1, x 1,..., x p V Diagram 0 V 00 1 V 01 2 V 02 p V np p+1 V n,p+1 t t x p+1 0 V 10 1 V 11 2 V 12 p V n+1,p

54 The infinite wedge 18 V :=..., x 3, x 2, x 1, x 1, x 2, x 3,... K V n,p := x n,..., x 1, x 1,..., x p V Diagram 0 V 00 1 V 01 2 V 02 p V np p+1 V n,p+1 t t x p+1 0 V 10 1 V 11 2 V 12 p V n+1,p Definition /2 V := lim p V n,p the infinite wedge (charge-0 part); basis {x I := x i1 x i2 } I, I = {i 1 < i 2 <...}, i k = k for k 0

55 The infinite wedge 18 V :=..., x 3, x 2, x 1, x 1, x 2, x 3,... K V n,p := x n,..., x 1, x 1,..., x p V Diagram 0 V 00 1 V 01 2 V 02 p V np p+1 V n,p+1 t t x p+1 0 V 10 1 V 11 2 V 12 p V n+1,p Definition /2 V := lim p V n,p the infinite wedge (charge-0 part); basis {x I := x i1 x i2 } I, I = {i 1 < i 2 <...}, i k = k for k 0 On /2 V acts GL := n,p GL(V n,p ).

56 The limit of a Plücker variety 19 Dual diagram 0 V00 1 V01 p V np p+1 V n,p+1 0 V 10 1 V 11 p V n+1,p

57 The limit of a Plücker variety 19 Dual diagram 0 V00 1 V01 p V np p+1 V n,p+1 0 V 10 1 V 11 p V n+1,p {X p } p 0 a Plücker variety varieties X n,p := X p (V n,p)

58 The limit of a Plücker variety 19 Dual diagram 0 V00 1 V01 p V np p+1 V n,p+1 0 V 10 1 V 11 p V n+1,p X n,p X n,p+1 X n+1,p {X p } p 0 a Plücker variety varieties X n,p := X p (V n,p)

59 The limit of a Plücker variety 19 Dual diagram 0 V00 1 V01 p V np p+1 V n,p+1 0 V 10 1 V 11 p V n+1,p X n,p X n,p+1 X n+1,p {X p } p 0 a Plücker variety varieties X n,p := X p (V n,p) X := lim X n,p is GL -stable subvariety of ( /2 V ) (Gr is Sato s Grassmannian)

60 The limit of a Plücker variety 19 Dual diagram 0 V00 1 V01 p V np p+1 V n,p+1 0 V 10 1 V 11 p V n+1,p X n,p X n,p+1 X n+1,p {X p } p 0 a Plücker variety varieties X n,p := X p (V n,p) X := lim X n,p is GL -stable subvariety of ( /2 V ) (Gr is Sato s Grassmannian) Theorem X bounded X cut out by finitely many GL -orbits of eqs.

61 Plücker varieties, proof sketch 20 By boundedness, X Y (k), where latter is defined in the dual infinite wedge by the orbit of a certain (2k 2k)-Pfaffian.

62 Plücker varieties, proof sketch 20 By boundedness, X Y (k), where latter is defined in the dual infinite wedge by the orbit of a certain (2k 2k)-Pfaffian. Prove by induction on k that Y (k) is GL -Noetherian, as follows:

63 Plücker varieties, proof sketch 20 By boundedness, X Y (k), where latter is defined in the dual infinite wedge by the orbit of a certain (2k 2k)-Pfaffian. Prove by induction on k that Y (k) is GL -Noetherian, as follows: Y (k) = Y (k 1) GL Z, where Z is an open subset where a specific (2k 2) (2k 2)-Pfaffian does not vanish.

64 Plücker varieties, proof sketch 20 By boundedness, X Y (k), where latter is defined in the dual infinite wedge by the orbit of a certain (2k 2k)-Pfaffian. Prove by induction on k that Y (k) is GL -Noetherian, as follows: Y (k) = Y (k 1) GL Z, where Z is an open subset where a specific (2k 2) (2k 2)-Pfaffian does not vanish. Z is stable under a subgroup GL N GL N H GL, and embeds equivariantly into some (K N N ) p.

65 Plücker varieties, proof sketch 20 By boundedness, X Y (k), where latter is defined in the dual infinite wedge by the orbit of a certain (2k 2k)-Pfaffian. Prove by induction on k that Y (k) is GL -Noetherian, as follows: Y (k) = Y (k 1) GL Z, where Z is an open subset where a specific (2k 2) (2k 2)-Pfaffian does not vanish. Z is stable under a subgroup GL N GL N H GL, and embeds equivariantly into some (K N N ) p. So Z is GL N GL N -Noetharian, and GL Z is GL -Noetherian, and so is Y (k), and hence X is defined by finitely many further GL -orbits of equations.

66 IV. Further areas 21

67 Stabilisation in other areas 22 Algebraic statistics families of graphical models where the graph grows [Hillar-Sullivant, Takemura, Yoshida, D-Eggermont,... ]

68 Stabilisation in other areas 22 Algebraic statistics families of graphical models where the graph grows [Hillar-Sullivant, Takemura, Yoshida, D-Eggermont,... ] Commutative algebra and representation theory higher syzygies, sequences of modules [Sam-Snowden, Church-Ellenberg-Farb,... ]

69 Stabilisation in other areas 22 Algebraic statistics families of graphical models where the graph grows [Hillar-Sullivant, Takemura, Yoshida, D-Eggermont,... ] Commutative algebra and representation theory higher syzygies, sequences of modules [Sam-Snowden, Church-Ellenberg-Farb,... ] Combinatorics matroid minor theory [Geelen-Gerards-Whittle,... ]

70 From very diverse areas of pure and applied mathematics large algebraic structures arise with remarkable finiteness properties. 23

71 23 From very diverse areas of pure and applied mathematics large algebraic structures arise with remarkable finiteness properties. Keywords include FI-modules (Church-Ellenberg-Farb), Delta-modules (Snowden), twisted commutative algebras (Sam-Snowden), equivariant Noetherianity, and equivariant Gröbner bases.

72 24 Thank you!