Application of non-commutative Gröbner basis to ...
Transcript of Application of non-commutative Gröbner basis to ...
![Page 1: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/1.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Application of non-commutative Gröbnerbasis to calculations of E2 terms of the
Adams spectral sequence
Tomohiro Fukaya
Department of MathematicsKyoto University
Free resolution of the Steenrod Algebra Kyoto university
![Page 2: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/2.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
OutlineIntroduction
Steenrod AlgebraThe Problem We Study and The Basic ToolAdams spectral sequence
Non-commutative Gröbner basisNon-commutative ringReview of Non-commutative Gröbner basisExample: Steenrod Algebra
Free resolutionSyzygy moduleSyzgy for Gröbner basisSyzygy for not Gröbner basis
Free resolution of the Steenrod Algebra Kyoto university
![Page 3: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/3.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
I I am interested in the application of Gröbner basis to thestudy of algebraic topology.
I The theory of Gröbner bases for polynomial rings wasdeveloped by Bruno Buchberger in 1965.
I One can view it as a multivariate, non-linear generalizationof:
I the Euclidean algorithm for computation of greatestcommon divisors,
I Gaussian elimination for linear systems, andI integer programming problems.
Free resolution of the Steenrod Algebra Kyoto university
![Page 4: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/4.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
I I am interested in the application of Gröbner basis to thestudy of algebraic topology.
I The theory of Gröbner bases for polynomial rings wasdeveloped by Bruno Buchberger in 1965.
I One can view it as a multivariate, non-linear generalizationof:
I the Euclidean algorithm for computation of greatestcommon divisors,
I Gaussian elimination for linear systems, andI integer programming problems.
Free resolution of the Steenrod Algebra Kyoto university
![Page 5: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/5.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
I I am interested in the application of Gröbner basis to thestudy of algebraic topology.
I The theory of Gröbner bases for polynomial rings wasdeveloped by Bruno Buchberger in 1965.
I One can view it as a multivariate, non-linear generalizationof:
I the Euclidean algorithm for computation of greatestcommon divisors,
I Gaussian elimination for linear systems, andI integer programming problems.
Free resolution of the Steenrod Algebra Kyoto university
![Page 6: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/6.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
I I am interested in the application of Gröbner basis to thestudy of algebraic topology.
I The theory of Gröbner bases for polynomial rings wasdeveloped by Bruno Buchberger in 1965.
I One can view it as a multivariate, non-linear generalizationof:
I the Euclidean algorithm for computation of greatestcommon divisors,
I Gaussian elimination for linear systems, andI integer programming problems.
Free resolution of the Steenrod Algebra Kyoto university
![Page 7: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/7.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
I I am interested in the application of Gröbner basis to thestudy of algebraic topology.
I The theory of Gröbner bases for polynomial rings wasdeveloped by Bruno Buchberger in 1965.
I One can view it as a multivariate, non-linear generalizationof:
I the Euclidean algorithm for computation of greatestcommon divisors,
I Gaussian elimination for linear systems, andI integer programming problems.
Free resolution of the Steenrod Algebra Kyoto university
![Page 8: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/8.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
I I am interested in the application of Gröbner basis to thestudy of algebraic topology.
I The theory of Gröbner bases for polynomial rings wasdeveloped by Bruno Buchberger in 1965.
I One can view it as a multivariate, non-linear generalizationof:
I the Euclidean algorithm for computation of greatestcommon divisors,
I Gaussian elimination for linear systems, andI integer programming problems.
Free resolution of the Steenrod Algebra Kyoto university
![Page 9: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/9.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
I I believe that Gröbner bases will become a powerfullmethod for the study of algebraiv topology.
I For example, I studied the ring structrure of thecohomology of the Oriented Grassmann manifolds usingGröbner basis. I obtaind the exact values of the cup-lengthfor an infinite family of the Oriented Grassmann manifolds.
I cup(G̃n,3) = n + 1 for n = 2m+1 − 4, m ≥ 1.I Today, I will talk on the algorithm of the calculation of the
E2 terms of the Adams spectral sequence usingnon-commutative Gröbner bases.
Free resolution of the Steenrod Algebra Kyoto university
![Page 10: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/10.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
I I believe that Gröbner bases will become a powerfullmethod for the study of algebraiv topology.
I For example, I studied the ring structrure of thecohomology of the Oriented Grassmann manifolds usingGröbner basis. I obtaind the exact values of the cup-lengthfor an infinite family of the Oriented Grassmann manifolds.
I cup(G̃n,3) = n + 1 for n = 2m+1 − 4, m ≥ 1.I Today, I will talk on the algorithm of the calculation of the
E2 terms of the Adams spectral sequence usingnon-commutative Gröbner bases.
Free resolution of the Steenrod Algebra Kyoto university
![Page 11: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/11.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
I I believe that Gröbner bases will become a powerfullmethod for the study of algebraiv topology.
I For example, I studied the ring structrure of thecohomology of the Oriented Grassmann manifolds usingGröbner basis. I obtaind the exact values of the cup-lengthfor an infinite family of the Oriented Grassmann manifolds.
I cup(G̃n,3) = n + 1 for n = 2m+1 − 4, m ≥ 1.I Today, I will talk on the algorithm of the calculation of the
E2 terms of the Adams spectral sequence usingnon-commutative Gröbner bases.
Free resolution of the Steenrod Algebra Kyoto university
![Page 12: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/12.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
I I believe that Gröbner bases will become a powerfullmethod for the study of algebraiv topology.
I For example, I studied the ring structrure of thecohomology of the Oriented Grassmann manifolds usingGröbner basis. I obtaind the exact values of the cup-lengthfor an infinite family of the Oriented Grassmann manifolds.
I cup(G̃n,3) = n + 1 for n = 2m+1 − 4, m ≥ 1.I Today, I will talk on the algorithm of the calculation of the
E2 terms of the Adams spectral sequence usingnon-commutative Gröbner bases.
Free resolution of the Steenrod Algebra Kyoto university
![Page 13: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/13.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Steenrod Algebra
OutlineIntroduction
Steenrod AlgebraThe Problem We Study and The Basic ToolAdams spectral sequence
Non-commutative Gröbner basisNon-commutative ringReview of Non-commutative Gröbner basisExample: Steenrod Algebra
Free resolutionSyzygy moduleSyzgy for Gröbner basisSyzygy for not Gröbner basis
Free resolution of the Steenrod Algebra Kyoto university
![Page 14: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/14.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Steenrod Algebra
Steenrod algebra, A2, which is generated by operations (i ≥ 0)
Sqi : H∗( ; F2) → H∗+i( ; F2),
I Sq0 = the identity homomorphism.I If x ∈ Hn(X ; F2), then Sqnx = x2.I x , y ∈ H∗(X ; F2),
Sqk (x ∪ y) =Pk
i=0 Sqix ∪ Sqk−iy , the Cartan formula.
I The following relations hold among the generators: if 0 < a < 2b
SqaSqb =
[a/2]Xi=0
b − 1− i
a− 2i
!Sqa+b−iSqi
These relations are called the Adem relations.
Free resolution of the Steenrod Algebra Kyoto university
![Page 15: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/15.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
The Problem We Study and The Basic Tool
OutlineIntroduction
Steenrod AlgebraThe Problem We Study and The Basic ToolAdams spectral sequence
Non-commutative Gröbner basisNon-commutative ringReview of Non-commutative Gröbner basisExample: Steenrod Algebra
Free resolutionSyzygy moduleSyzgy for Gröbner basisSyzygy for not Gröbner basis
Free resolution of the Steenrod Algebra Kyoto university
![Page 16: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/16.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
The Problem We Study and The Basic Tool
ProblemFor topological space X and Y , compute the groups ofmorphisms
{Y , X}k := limn→∞
[Σn+kY ,ΣnX ].
If Y = S0, then {Y , X}k = πSk (X ); the stable k-th homotopy group of X .
Free resolution of the Steenrod Algebra Kyoto university
![Page 17: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/17.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Adams spectral sequence
OutlineIntroduction
Steenrod AlgebraThe Problem We Study and The Basic ToolAdams spectral sequence
Non-commutative Gröbner basisNon-commutative ringReview of Non-commutative Gröbner basisExample: Steenrod Algebra
Free resolutionSyzygy moduleSyzgy for Gröbner basisSyzygy for not Gröbner basis
Free resolution of the Steenrod Algebra Kyoto university
![Page 18: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/18.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Adams spectral sequence
Theorem (Adams spectarl sequence)There is a spectral sequence, converging to (2){Y , X}∗, withE2-terms given by
Es,t2∼= Exts,t
A2(H∗(X ; F2), H∗(Y ; F2)) (1)
and differentials dr of bi-degree (r , r − 1).
Free resolution of the Steenrod Algebra Kyoto university
![Page 19: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/19.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Adams spectral sequence
E2-terms of Adams spectral sequence are given by A2-freeresolution of A2-module M = H∗(X ; F2)
· · · di+1−−→ Ridi−→ Ri−1
di−1−−→ · · · d1−→ R0d0−→ M → 0
where Ri is free A2-module (⊕A2) and A2 is the Steenrod algebra.For another A2-module N = H∗(Y ; F2), we obtain a cochaincomplex
Hom(M, N)d0−→ Hom(R0, N)
d1−→ Hom(R1, N)
d2−→ · · · .
The cohomology of this complex defines Ext∗,∗A2(M, N).
Free resolution of the Steenrod Algebra Kyoto university
![Page 20: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/20.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Adams spectral sequence
E2-terms of Adams spectral sequence are given by A2-freeresolution of A2-module M = H∗(X ; F2)
· · · di+1−−→ Ridi−→ Ri−1
di−1−−→ · · · d1−→ R0d0−→ M → 0
where Ri is free A2-module (⊕A2) and A2 is the Steenrod algebra.For another A2-module N = H∗(Y ; F2), we obtain a cochaincomplex
Hom(M, N)d0−→ Hom(R0, N)
d1−→ Hom(R1, N)
d2−→ · · · .
The cohomology of this complex defines Ext∗,∗A2(M, N).
Free resolution of the Steenrod Algebra Kyoto university
![Page 21: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/21.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Adams spectral sequence
E2-terms of Adams spectral sequence are given by A2-freeresolution of A2-module M = H∗(X ; F2)
· · · di+1−−→ Ridi−→ Ri−1
di−1−−→ · · · d1−→ R0d0−→ M → 0
where Ri is free A2-module (⊕A2) and A2 is the Steenrod algebra.For another A2-module N = H∗(Y ; F2), we obtain a cochaincomplex
Hom(M, N)d0−→ Hom(R0, N)
d1−→ Hom(R1, N)
d2−→ · · · .
The cohomology of this complex defines Ext∗,∗A2(M, N).
Free resolution of the Steenrod Algebra Kyoto university
![Page 22: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/22.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Adams spectral sequence
The key to calculate such a free resolution is
Non-commutative Gröbner basis and Syzygy module.
RemarkMr. Euiyong Park (Seoul National University) ,and his co-workers,are using the non-commutative Gröbner basis to study therepresentation of Lie algebras and their universal envelopingalgebras.
Free resolution of the Steenrod Algebra Kyoto university
![Page 23: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/23.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Adams spectral sequence
The key to calculate such a free resolution is
Non-commutative Gröbner basis and Syzygy module.
RemarkMr. Euiyong Park (Seoul National University) ,and his co-workers,are using the non-commutative Gröbner basis to study therepresentation of Lie algebras and their universal envelopingalgebras.
Free resolution of the Steenrod Algebra Kyoto university
![Page 24: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/24.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Adams spectral sequence
The key to calculate such a free resolution is
Non-commutative Gröbner basis and Syzygy module.
RemarkMr. Euiyong Park (Seoul National University) ,and his co-workers,are using the non-commutative Gröbner basis to study therepresentation of Lie algebras and their universal envelopingalgebras.
Free resolution of the Steenrod Algebra Kyoto university
![Page 25: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/25.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Non-commutative ring
OutlineIntroduction
Steenrod AlgebraThe Problem We Study and The Basic ToolAdams spectral sequence
Non-commutative Gröbner basisNon-commutative ringReview of Non-commutative Gröbner basisExample: Steenrod Algebra
Free resolutionSyzygy moduleSyzgy for Gröbner basisSyzygy for not Gröbner basis
Free resolution of the Steenrod Algebra Kyoto university
![Page 26: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/26.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Non-commutative ring
We consider the non-commutative polynomial ring.I k : field.I R = k < x1, x2, . . . , xi , · · · > non-commutative free
associative ringI the set of polynomials.
I B = {xi1xi2 · · · xik} ∪ {1}I the set of monomials.
Free resolution of the Steenrod Algebra Kyoto university
![Page 27: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/27.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Non-commutative ring
We consider the non-commutative polynomial ring.I k : field.I R = k < x1, x2, . . . , xi , · · · > non-commutative free
associative ringI the set of polynomials.
I B = {xi1xi2 · · · xik} ∪ {1}I the set of monomials.
Free resolution of the Steenrod Algebra Kyoto university
![Page 28: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/28.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Non-commutative ring
We consider the non-commutative polynomial ring.I k : field.I R = k < x1, x2, . . . , xi , · · · > non-commutative free
associative ringI the set of polynomials.
I B = {xi1xi2 · · · xik} ∪ {1}I the set of monomials.
Free resolution of the Steenrod Algebra Kyoto university
![Page 29: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/29.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Non-commutative ring
We consider the non-commutative polynomial ring.I k : field.I R = k < x1, x2, . . . , xi , · · · > non-commutative free
associative ringI the set of polynomials.
I B = {xi1xi2 · · · xik} ∪ {1}I the set of monomials.
Free resolution of the Steenrod Algebra Kyoto university
![Page 30: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/30.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Non-commutative ring
We consider the non-commutative polynomial ring.I k : field.I R = k < x1, x2, . . . , xi , · · · > non-commutative free
associative ringI the set of polynomials.
I B = {xi1xi2 · · · xik} ∪ {1}I the set of monomials.
Free resolution of the Steenrod Algebra Kyoto university
![Page 31: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/31.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Review of Non-commutative Gröbner basis
OutlineIntroduction
Steenrod AlgebraThe Problem We Study and The Basic ToolAdams spectral sequence
Non-commutative Gröbner basisNon-commutative ringReview of Non-commutative Gröbner basisExample: Steenrod Algebra
Free resolutionSyzygy moduleSyzgy for Gröbner basisSyzygy for not Gröbner basis
Free resolution of the Steenrod Algebra Kyoto university
![Page 32: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/32.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Review of Non-commutative Gröbner basis
Definition (Monomial order)Let ≤ be a well-ordering on B. ≤ is said to be a monomialordering on R if the following two conditions are satisfied.
I if u, v , w , s ∈ B with w ≤ u, then vws ≤ vus.I For u, w ∈ B, if u = vws for some v , s ∈ B with v 6= 1 or
s 6= 1, then w ≤ u.Hence 1 ≤ u for all u ∈ B.
Free resolution of the Steenrod Algebra Kyoto university
![Page 33: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/33.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Review of Non-commutative Gröbner basis
Definition (Monomial order)Let ≤ be a well-ordering on B. ≤ is said to be a monomialordering on R if the following two conditions are satisfied.
I if u, v , w , s ∈ B with w ≤ u, then vws ≤ vus.I For u, w ∈ B, if u = vws for some v , s ∈ B with v 6= 1 or
s 6= 1, then w ≤ u.Hence 1 ≤ u for all u ∈ B.
Free resolution of the Steenrod Algebra Kyoto university
![Page 34: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/34.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Review of Non-commutative Gröbner basis
Definition (Monomial order)Let ≤ be a well-ordering on B. ≤ is said to be a monomialordering on R if the following two conditions are satisfied.
I if u, v , w , s ∈ B with w ≤ u, then vws ≤ vus.I For u, w ∈ B, if u = vws for some v , s ∈ B with v 6= 1 or
s 6= 1, then w ≤ u.Hence 1 ≤ u for all u ∈ B.
Free resolution of the Steenrod Algebra Kyoto university
![Page 35: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/35.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Review of Non-commutative Gröbner basis
From here, We fix a monomial ordering.I For f ∈ R, f 6= 0, we may write
f = c1u1 + · · ·+ cmum
where ci ∈ k \ {0}, ui ∈ B, u1 > u2 > · · · > um.I lm(f ) = u1, the leading monomial of f .
Free resolution of the Steenrod Algebra Kyoto university
![Page 36: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/36.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Review of Non-commutative Gröbner basis
From here, We fix a monomial ordering.I For f ∈ R, f 6= 0, we may write
f = c1u1 + · · ·+ cmum
where ci ∈ k \ {0}, ui ∈ B, u1 > u2 > · · · > um.I lm(f ) = u1, the leading monomial of f .
Free resolution of the Steenrod Algebra Kyoto university
![Page 37: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/37.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Review of Non-commutative Gröbner basis
Theorem (Division on f by H)
INPUT f , H = {h1, . . . , hm . . . } ⊂ R
OUTPUT f =d∑
α=0
µαvαhiαsα + r
I µα ∈ k \ {0}, vα, sα ∈ B. lm(µαvαhiαsα) ≤ lm(f ).
I r = 0 or r =∑
j cj tj where cj ∈ k \ {0}, tj ∈ B,
I lm(r) ≤ lm(f ), none of the tj is divisible by any lm(hi).
I r is depend on the ordering on H.
Free resolution of the Steenrod Algebra Kyoto university
![Page 38: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/38.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Review of Non-commutative Gröbner basis
Theorem (Division on f by H)
INPUT f , H = {h1, . . . , hm . . . } ⊂ R
OUTPUT f =d∑
α=0
µαvαhiαsα + r
I µα ∈ k \ {0}, vα, sα ∈ B. lm(µαvαhiαsα) ≤ lm(f ).
I r = 0 or r =∑
j cj tj where cj ∈ k \ {0}, tj ∈ B,
I lm(r) ≤ lm(f ), none of the tj is divisible by any lm(hi).
I r is depend on the ordering on H.
Free resolution of the Steenrod Algebra Kyoto university
![Page 39: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/39.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Review of Non-commutative Gröbner basis
Theorem (Division on f by H)
INPUT f , H = {h1, . . . , hm . . . } ⊂ R
OUTPUT f =d∑
α=0
µαvαhiαsα + r
I µα ∈ k \ {0}, vα, sα ∈ B. lm(µαvαhiαsα) ≤ lm(f ).
I r = 0 or r =∑
j cj tj where cj ∈ k \ {0}, tj ∈ B,
I lm(r) ≤ lm(f ), none of the tj is divisible by any lm(hi).
I r is depend on the ordering on H.
Free resolution of the Steenrod Algebra Kyoto university
![Page 40: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/40.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Review of Non-commutative Gröbner basis
Theorem (Division on f by H)
INPUT f , H = {h1, . . . , hm . . . } ⊂ R
OUTPUT f =d∑
α=0
µαvαhiαsα + r
I µα ∈ k \ {0}, vα, sα ∈ B. lm(µαvαhiαsα) ≤ lm(f ).
I r = 0 or r =∑
j cj tj where cj ∈ k \ {0}, tj ∈ B,
I lm(r) ≤ lm(f ), none of the tj is divisible by any lm(hi).
I r is depend on the ordering on H.
Free resolution of the Steenrod Algebra Kyoto university
![Page 41: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/41.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Review of Non-commutative Gröbner basis
I The r appeared in above OUTPUT is called a remainder off on the division by G, dented f
H.
I Then fH
is depend on the order on H.I To overcome this difficulty, we need Gröbner basis.
Free resolution of the Steenrod Algebra Kyoto university
![Page 42: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/42.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Review of Non-commutative Gröbner basis
I The r appeared in above OUTPUT is called a remainder off on the division by G, dented f
H.
I Then fH
is depend on the order on H.I To overcome this difficulty, we need Gröbner basis.
Free resolution of the Steenrod Algebra Kyoto university
![Page 43: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/43.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Review of Non-commutative Gröbner basis
I The r appeared in above OUTPUT is called a remainder off on the division by G, dented f
H.
I Then fH
is depend on the order on H.I To overcome this difficulty, we need Gröbner basis.
Free resolution of the Steenrod Algebra Kyoto university
![Page 44: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/44.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Review of Non-commutative Gröbner basis
I The r appeared in above OUTPUT is called a remainder off on the division by G, dented f
H.
I Then fH
is depend on the order on H.I To overcome this difficulty, we need Gröbner basis.
Free resolution of the Steenrod Algebra Kyoto university
![Page 45: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/45.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Review of Non-commutative Gröbner basis
I The r appeared in above OUTPUT is called a remainder off on the division by G, dented f
H.
I Then fH
is depend on the order on H.I To overcome this difficulty, we need Gröbner basis.
Free resolution of the Steenrod Algebra Kyoto university
![Page 46: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/46.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Review of Non-commutative Gröbner basis
DefinitionLet I be two-side ideal of R. G = {g1, g2, . . . } ⊂ I is Gröbnerbasis, if and only if
(lm(g)|g ∈ G) = (lm(g)|g ∈ I)
I If G = {g1, g2, . . . } ⊂ R is Gröbner basis, thenI f
Gis independent of the order on G and f
Gis unique.
I f ∈ I = (g1, g2, . . . , gi , . . . ) ⇔ fG
= 0.I There exists an algorithm to compute a Gröbner basis ofH = {h1, . . . , hs}.
Free resolution of the Steenrod Algebra Kyoto university
![Page 47: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/47.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Review of Non-commutative Gröbner basis
DefinitionLet I be two-side ideal of R. G = {g1, g2, . . . } ⊂ I is Gröbnerbasis, if and only if
(lm(g)|g ∈ G) = (lm(g)|g ∈ I)
I If G = {g1, g2, . . . } ⊂ R is Gröbner basis, thenI f
Gis independent of the order on G and f
Gis unique.
I f ∈ I = (g1, g2, . . . , gi , . . . ) ⇔ fG
= 0.I There exists an algorithm to compute a Gröbner basis ofH = {h1, . . . , hs}.
Free resolution of the Steenrod Algebra Kyoto university
![Page 48: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/48.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Review of Non-commutative Gröbner basis
DefinitionLet I be two-side ideal of R. G = {g1, g2, . . . } ⊂ I is Gröbnerbasis, if and only if
(lm(g)|g ∈ G) = (lm(g)|g ∈ I)
I If G = {g1, g2, . . . } ⊂ R is Gröbner basis, thenI f
Gis independent of the order on G and f
Gis unique.
I f ∈ I = (g1, g2, . . . , gi , . . . ) ⇔ fG
= 0.I There exists an algorithm to compute a Gröbner basis ofH = {h1, . . . , hs}.
Free resolution of the Steenrod Algebra Kyoto university
![Page 49: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/49.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Review of Non-commutative Gröbner basis
DefinitionLet I be two-side ideal of R. G = {g1, g2, . . . } ⊂ I is Gröbnerbasis, if and only if
(lm(g)|g ∈ G) = (lm(g)|g ∈ I)
I If G = {g1, g2, . . . } ⊂ R is Gröbner basis, thenI f
Gis independent of the order on G and f
Gis unique.
I f ∈ I = (g1, g2, . . . , gi , . . . ) ⇔ fG
= 0.I There exists an algorithm to compute a Gröbner basis ofH = {h1, . . . , hs}.
Free resolution of the Steenrod Algebra Kyoto university
![Page 50: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/50.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Review of Non-commutative Gröbner basis
DefinitionLet I be two-side ideal of R. G = {g1, g2, . . . } ⊂ I is Gröbnerbasis, if and only if
(lm(g)|g ∈ G) = (lm(g)|g ∈ I)
I If G = {g1, g2, . . . } ⊂ R is Gröbner basis, thenI f
Gis independent of the order on G and f
Gis unique.
I f ∈ I = (g1, g2, . . . , gi , . . . ) ⇔ fG
= 0.I There exists an algorithm to compute a Gröbner basis ofH = {h1, . . . , hs}.
Free resolution of the Steenrod Algebra Kyoto university
![Page 51: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/51.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Example: Steenrod Algebra
OutlineIntroduction
Steenrod AlgebraThe Problem We Study and The Basic ToolAdams spectral sequence
Non-commutative Gröbner basisNon-commutative ringReview of Non-commutative Gröbner basisExample: Steenrod Algebra
Free resolutionSyzygy moduleSyzgy for Gröbner basisSyzygy for not Gröbner basis
Free resolution of the Steenrod Algebra Kyoto university
![Page 52: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/52.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Example: Steenrod Algebra
We can consider the Steenrod algebra as a quotient ring.I A = Z2〈Sq1, Sq2, . . . 〉.I G =
{SqaSqb −
∑[a/2]i=0
(b−1−ia−2i
)Sqa+b−iSqi |0 < a < 2b
}.
Set of Adem relations.I I = (G): A two-side Ideal generated by G.I A2 = A/I.
Free resolution of the Steenrod Algebra Kyoto university
![Page 53: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/53.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Example: Steenrod Algebra
We can consider the Steenrod algebra as a quotient ring.I A = Z2〈Sq1, Sq2, . . . 〉.I G =
{SqaSqb −
∑[a/2]i=0
(b−1−ia−2i
)Sqa+b−iSqi |0 < a < 2b
}.
Set of Adem relations.I I = (G): A two-side Ideal generated by G.I A2 = A/I.
Free resolution of the Steenrod Algebra Kyoto university
![Page 54: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/54.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Example: Steenrod Algebra
We can consider the Steenrod algebra as a quotient ring.I A = Z2〈Sq1, Sq2, . . . 〉.I G =
{SqaSqb −
∑[a/2]i=0
(b−1−ia−2i
)Sqa+b−iSqi |0 < a < 2b
}.
Set of Adem relations.I I = (G): A two-side Ideal generated by G.I A2 = A/I.
Free resolution of the Steenrod Algebra Kyoto university
![Page 55: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/55.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Example: Steenrod Algebra
We can consider the Steenrod algebra as a quotient ring.I A = Z2〈Sq1, Sq2, . . . 〉.I G =
{SqaSqb −
∑[a/2]i=0
(b−1−ia−2i
)Sqa+b−iSqi |0 < a < 2b
}.
Set of Adem relations.I I = (G): A two-side Ideal generated by G.I A2 = A/I.
Free resolution of the Steenrod Algebra Kyoto university
![Page 56: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/56.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Example: Steenrod Algebra
Definition (A monomial Ordering of Steenrod Algebra)Let u = Sqa1 · · ·Sqak , v = Sqb1 · · ·Sqbl ∈ B.Then, u ≥ v if and only if
I k > l or,I k = l and the right-most nonzero entry of
(a1 − b1, . . . , ak − bk ) is positive.
lm(SqaSqb −∑[a/2]
i=0
(b−1−ia−2i
)Sqa+b−iSqi) = SqaSqb.
PropositionG is a Gröbner basis of I.
Free resolution of the Steenrod Algebra Kyoto university
![Page 57: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/57.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Example: Steenrod Algebra
Definition (A monomial Ordering of Steenrod Algebra)Let u = Sqa1 · · ·Sqak , v = Sqb1 · · ·Sqbl ∈ B.Then, u ≥ v if and only if
I k > l or,I k = l and the right-most nonzero entry of
(a1 − b1, . . . , ak − bk ) is positive.
lm(SqaSqb −∑[a/2]
i=0
(b−1−ia−2i
)Sqa+b−iSqi) = SqaSqb.
PropositionG is a Gröbner basis of I.
Free resolution of the Steenrod Algebra Kyoto university
![Page 58: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/58.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Example: Steenrod Algebra
Definition (A monomial Ordering of Steenrod Algebra)Let u = Sqa1 · · ·Sqak , v = Sqb1 · · ·Sqbl ∈ B.Then, u ≥ v if and only if
I k > l or,I k = l and the right-most nonzero entry of
(a1 − b1, . . . , ak − bk ) is positive.
lm(SqaSqb −∑[a/2]
i=0
(b−1−ia−2i
)Sqa+b−iSqi) = SqaSqb.
PropositionG is a Gröbner basis of I.
Free resolution of the Steenrod Algebra Kyoto university
![Page 59: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/59.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Syzygy module
OutlineIntroduction
Steenrod AlgebraThe Problem We Study and The Basic ToolAdams spectral sequence
Non-commutative Gröbner basisNon-commutative ringReview of Non-commutative Gröbner basisExample: Steenrod Algebra
Free resolutionSyzygy moduleSyzgy for Gröbner basisSyzygy for not Gröbner basis
Free resolution of the Steenrod Algebra Kyoto university
![Page 60: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/60.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Syzygy module
Again, let R = k < x1, x2, . . . , xi , · · · >.We consider the algorithm of computing a free-resolution ofleft-R-module M,
· · · di+1−−→ Ridi−→ Ri−1
di−1−−→ · · · d1−→ R0d0−→ M → 0
where Ri ’s are free R-modules.
The key to compute the free resolutions is the
syzygy modules.
Free resolution of the Steenrod Algebra Kyoto university
![Page 61: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/61.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Syzygy module
Again, let R = k < x1, x2, . . . , xi , · · · >.We consider the algorithm of computing a free-resolution ofleft-R-module M,
· · · di+1−−→ Ridi−→ Ri−1
di−1−−→ · · · d1−→ R0d0−→ M → 0
where Ri ’s are free R-modules.
The key to compute the free resolutions is the
syzygy modules.
Free resolution of the Steenrod Algebra Kyoto university
![Page 62: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/62.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Syzygy module
Again, let R = k < x1, x2, . . . , xi , · · · >.We consider the algorithm of computing a free-resolution ofleft-R-module M,
· · · di+1−−→ Ridi−→ Ri−1
di−1−−→ · · · d1−→ R0d0−→ M → 0
where Ri ’s are free R-modules.
The key to compute the free resolutions is the
syzygy modules.
Free resolution of the Steenrod Algebra Kyoto university
![Page 63: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/63.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Syzygy module
DefinitionThe kernel of the map φ : Rs → M given by
(h1, . . . , hs) 7→s∑
i=1
hi fi , fi ∈ M
is called the syzygy module of the matrix t [f1 · · · fs]. It isdenoted Syz(f1, . . . , fs).
Free resolution of the Steenrod Algebra Kyoto university
![Page 64: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/64.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Syzgy for Gröbner basis
OutlineIntroduction
Steenrod AlgebraThe Problem We Study and The Basic ToolAdams spectral sequence
Non-commutative Gröbner basisNon-commutative ringReview of Non-commutative Gröbner basisExample: Steenrod Algebra
Free resolutionSyzygy moduleSyzgy for Gröbner basisSyzygy for not Gröbner basis
Free resolution of the Steenrod Algebra Kyoto university
![Page 65: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/65.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Syzgy for Gröbner basis
We consider only the case M = R.
Step 1: we consider
Syz(g1, . . . , gt) = Ker[Rt ×t [g1···gt ]−−−−−−→ R]
where {g1, . . . , gt} is Gröbner basis.
Free resolution of the Steenrod Algebra Kyoto university
![Page 66: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/66.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Syzgy for Gröbner basis
We consider only the case M = R.
Step 1: we consider
Syz(g1, . . . , gt) = Ker[Rt ×t [g1···gt ]−−−−−−→ R]
where {g1, . . . , gt} is Gröbner basis.
Free resolution of the Steenrod Algebra Kyoto university
![Page 67: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/67.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Syzgy for Gröbner basis
We let lm(gi) = Xi and Xij = lcm(Xi , Xj).Then the S-polynomialof gi and gj is, given by
S(gi , gj) =Xij
Xigi −
Xij
Xjgj , lp(S(gi , gj)) < lp(
Xij
Xigi), lp(
XijXj
gj),
S(gi , gj) =t∑
ν=t
hijνgν , lp(S(gi , gj)) ≥ (lp(hijνgν)),
Letsij :=
Xij
Xiei −
Xij
Xjej − (hij1, . . . , hijt) ∈ Rt .
Free resolution of the Steenrod Algebra Kyoto university
![Page 68: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/68.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Syzgy for Gröbner basis
We let lm(gi) = Xi and Xij = lcm(Xi , Xj).Then the S-polynomialof gi and gj is, given by
S(gi , gj) =Xij
Xigi −
Xij
Xjgj , lp(S(gi , gj)) < lp(
Xij
Xigi), lp(
XijXj
gj),
S(gi , gj) =t∑
ν=t
hijνgν , lp(S(gi , gj)) ≥ (lp(hijνgν)),
Letsij :=
Xij
Xiei −
Xij
Xjej − (hij1, . . . , hijt) ∈ Rt .
Free resolution of the Steenrod Algebra Kyoto university
![Page 69: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/69.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Syzgy for Gröbner basis
We let lm(gi) = Xi and Xij = lcm(Xi , Xj).Then the S-polynomialof gi and gj is, given by
S(gi , gj) =Xij
Xigi −
Xij
Xjgj , lp(S(gi , gj)) < lp(
Xij
Xigi), lp(
XijXj
gj),
S(gi , gj) =t∑
ν=t
hijνgν , lp(S(gi , gj)) ≥ (lp(hijνgν)),
Letsij :=
Xij
Xiei −
Xij
Xjej − (hij1, . . . , hijt) ∈ Rt .
Free resolution of the Steenrod Algebra Kyoto university
![Page 70: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/70.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Syzgy for Gröbner basis
We let lm(gi) = Xi and Xij = lcm(Xi , Xj).Then the S-polynomialof gi and gj is, given by
S(gi , gj) =Xij
Xigi −
Xij
Xjgj , lp(S(gi , gj)) < lp(
Xij
Xigi), lp(
XijXj
gj),
S(gi , gj) =t∑
ν=t
hijνgν , lp(S(gi , gj)) ≥ (lp(hijνgν)),
Letsij :=
Xij
Xiei −
Xij
Xjej − (hij1, . . . , hijt) ∈ Rt .
Free resolution of the Steenrod Algebra Kyoto university
![Page 71: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/71.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Syzgy for Gröbner basis
We let lm(gi) = Xi and Xij = lcm(Xi , Xj).Then the S-polynomialof gi and gj is, given by
S(gi , gj) =Xij
Xigi −
Xij
Xjgj , lp(S(gi , gj)) < lp(
Xij
Xigi), lp(
XijXj
gj),
S(gi , gj) =t∑
ν=t
hijνgν , lp(S(gi , gj)) ≥ (lp(hijνgν)),
Letsij :=
Xij
Xiei −
Xij
Xjej − (hij1, . . . , hijt) ∈ Rt .
Free resolution of the Steenrod Algebra Kyoto university
![Page 72: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/72.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Syzgy for Gröbner basis
We note that sij ∈ Syz(g1, . . . , gt), since
sij
g1...
gt
=(Xij
Xiei −
Xij
Xjej − (hij1, . . . , hijt)
) g1...
gs
= S(gi , gj)−
t∑ν=t
hijνgν = 0.
TheoremWith notation above, the collection {sij |1 ≤ i ≤ j ≤ t} isgenerating set for Syz(g1, . . . , gt).
Free resolution of the Steenrod Algebra Kyoto university
![Page 73: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/73.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Syzgy for Gröbner basis
TheoremThere exists an algorithm to compute a generating set of
Syz(g1, . . . , gt)
using S-polynomials and the division algorithm for {g1, . . . , gt}.
Free resolution of the Steenrod Algebra Kyoto university
![Page 74: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/74.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Syzygy for not Gröbner basis
OutlineIntroduction
Steenrod AlgebraThe Problem We Study and The Basic ToolAdams spectral sequence
Non-commutative Gröbner basisNon-commutative ringReview of Non-commutative Gröbner basisExample: Steenrod Algebra
Free resolutionSyzygy moduleSyzgy for Gröbner basisSyzygy for not Gröbner basis
Free resolution of the Steenrod Algebra Kyoto university
![Page 75: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/75.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Syzygy for not Gröbner basis
Step 2: we consider
Syz(f1, . . . , fs) = Ker[Rs ×t [f1···fs]−−−−−→ R]
where {f1, . . . , fs} is not Gröbner basis.
Free resolution of the Steenrod Algebra Kyoto university
![Page 76: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/76.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Syzygy for not Gröbner basis
I {f1, . . . , fs} ⊂ R: not Gröbner basis.I {g1, . . . , gt}: Gröbner basis of {f1, . . . , fs}.I F = t [f1 . . . fs], G = t [g1 . . . gt ].
I ∃S: s × t matrix, ∃T : t × s matrixs.t. F = SG and G = TF .
I compute a generating set {s1, . . . , sr} for Syz(g1, . . . , gt)I 0 = siG = si(TF ) = (siT )F ,
I 〈siT |i = 1, . . . , r〉 ⊂ Syz(f1, . . . , fs).I Is: s × s identity matrix,
(Is − ST )F = F − STF = F − SG = F − F = 0I the rows r1, . . . , rs of Is − TS are also in Syz(f1, . . . , fs).
Free resolution of the Steenrod Algebra Kyoto university
![Page 77: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/77.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Syzygy for not Gröbner basis
I {f1, . . . , fs} ⊂ R: not Gröbner basis.I {g1, . . . , gt}: Gröbner basis of {f1, . . . , fs}.I F = t [f1 . . . fs], G = t [g1 . . . gt ].
I ∃S: s × t matrix, ∃T : t × s matrixs.t. F = SG and G = TF .
I compute a generating set {s1, . . . , sr} for Syz(g1, . . . , gt)I 0 = siG = si(TF ) = (siT )F ,
I 〈siT |i = 1, . . . , r〉 ⊂ Syz(f1, . . . , fs).I Is: s × s identity matrix,
(Is − ST )F = F − STF = F − SG = F − F = 0I the rows r1, . . . , rs of Is − TS are also in Syz(f1, . . . , fs).
Free resolution of the Steenrod Algebra Kyoto university
![Page 78: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/78.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Syzygy for not Gröbner basis
I {f1, . . . , fs} ⊂ R: not Gröbner basis.I {g1, . . . , gt}: Gröbner basis of {f1, . . . , fs}.I F = t [f1 . . . fs], G = t [g1 . . . gt ].
I ∃S: s × t matrix, ∃T : t × s matrixs.t. F = SG and G = TF .
I compute a generating set {s1, . . . , sr} for Syz(g1, . . . , gt)I 0 = siG = si(TF ) = (siT )F ,
I 〈siT |i = 1, . . . , r〉 ⊂ Syz(f1, . . . , fs).I Is: s × s identity matrix,
(Is − ST )F = F − STF = F − SG = F − F = 0I the rows r1, . . . , rs of Is − TS are also in Syz(f1, . . . , fs).
Free resolution of the Steenrod Algebra Kyoto university
![Page 79: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/79.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Syzygy for not Gröbner basis
I {f1, . . . , fs} ⊂ R: not Gröbner basis.I {g1, . . . , gt}: Gröbner basis of {f1, . . . , fs}.I F = t [f1 . . . fs], G = t [g1 . . . gt ].
I ∃S: s × t matrix, ∃T : t × s matrixs.t. F = SG and G = TF .
I compute a generating set {s1, . . . , sr} for Syz(g1, . . . , gt)I 0 = siG = si(TF ) = (siT )F ,
I 〈siT |i = 1, . . . , r〉 ⊂ Syz(f1, . . . , fs).I Is: s × s identity matrix,
(Is − ST )F = F − STF = F − SG = F − F = 0I the rows r1, . . . , rs of Is − TS are also in Syz(f1, . . . , fs).
Free resolution of the Steenrod Algebra Kyoto university
![Page 80: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/80.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Syzygy for not Gröbner basis
I {f1, . . . , fs} ⊂ R: not Gröbner basis.I {g1, . . . , gt}: Gröbner basis of {f1, . . . , fs}.I F = t [f1 . . . fs], G = t [g1 . . . gt ].
I ∃S: s × t matrix, ∃T : t × s matrixs.t. F = SG and G = TF .
I compute a generating set {s1, . . . , sr} for Syz(g1, . . . , gt)I 0 = siG = si(TF ) = (siT )F ,
I 〈siT |i = 1, . . . , r〉 ⊂ Syz(f1, . . . , fs).I Is: s × s identity matrix,
(Is − ST )F = F − STF = F − SG = F − F = 0I the rows r1, . . . , rs of Is − TS are also in Syz(f1, . . . , fs).
Free resolution of the Steenrod Algebra Kyoto university
![Page 81: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/81.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Syzygy for not Gröbner basis
I {f1, . . . , fs} ⊂ R: not Gröbner basis.I {g1, . . . , gt}: Gröbner basis of {f1, . . . , fs}.I F = t [f1 . . . fs], G = t [g1 . . . gt ].
I ∃S: s × t matrix, ∃T : t × s matrixs.t. F = SG and G = TF .
I compute a generating set {s1, . . . , sr} for Syz(g1, . . . , gt)I 0 = siG = si(TF ) = (siT )F ,
I 〈siT |i = 1, . . . , r〉 ⊂ Syz(f1, . . . , fs).I Is: s × s identity matrix,
(Is − ST )F = F − STF = F − SG = F − F = 0I the rows r1, . . . , rs of Is − TS are also in Syz(f1, . . . , fs).
Free resolution of the Steenrod Algebra Kyoto university
![Page 82: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/82.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Syzygy for not Gröbner basis
I {f1, . . . , fs} ⊂ R: not Gröbner basis.I {g1, . . . , gt}: Gröbner basis of {f1, . . . , fs}.I F = t [f1 . . . fs], G = t [g1 . . . gt ].
I ∃S: s × t matrix, ∃T : t × s matrixs.t. F = SG and G = TF .
I compute a generating set {s1, . . . , sr} for Syz(g1, . . . , gt)I 0 = siG = si(TF ) = (siT )F ,
I 〈siT |i = 1, . . . , r〉 ⊂ Syz(f1, . . . , fs).I Is: s × s identity matrix,
(Is − ST )F = F − STF = F − SG = F − F = 0I the rows r1, . . . , rs of Is − TS are also in Syz(f1, . . . , fs).
Free resolution of the Steenrod Algebra Kyoto university
![Page 83: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/83.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Syzygy for not Gröbner basis
I {f1, . . . , fs} ⊂ R: not Gröbner basis.I {g1, . . . , gt}: Gröbner basis of {f1, . . . , fs}.I F = t [f1 . . . fs], G = t [g1 . . . gt ].
I ∃S: s × t matrix, ∃T : t × s matrixs.t. F = SG and G = TF .
I compute a generating set {s1, . . . , sr} for Syz(g1, . . . , gt)I 0 = siG = si(TF ) = (siT )F ,
I 〈siT |i = 1, . . . , r〉 ⊂ Syz(f1, . . . , fs).I Is: s × s identity matrix,
(Is − ST )F = F − STF = F − SG = F − F = 0I the rows r1, . . . , rs of Is − TS are also in Syz(f1, . . . , fs).
Free resolution of the Steenrod Algebra Kyoto university
![Page 84: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/84.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Syzygy for not Gröbner basis
I {f1, . . . , fs} ⊂ R: not Gröbner basis.I {g1, . . . , gt}: Gröbner basis of {f1, . . . , fs}.I F = t [f1 . . . fs], G = t [g1 . . . gt ].
I ∃S: s × t matrix, ∃T : t × s matrixs.t. F = SG and G = TF .
I compute a generating set {s1, . . . , sr} for Syz(g1, . . . , gt)I 0 = siG = si(TF ) = (siT )F ,
I 〈siT |i = 1, . . . , r〉 ⊂ Syz(f1, . . . , fs).I Is: s × s identity matrix,
(Is − ST )F = F − STF = F − SG = F − F = 0I the rows r1, . . . , rs of Is − TS are also in Syz(f1, . . . , fs).
Free resolution of the Steenrod Algebra Kyoto university
![Page 85: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/85.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Syzygy for not Gröbner basis
TheoremWith the notation above we haveSyz(f1, . . . , fs) = 〈s1T , . . . , sr T , r1, . . . , rs〉.
Using the Gröbner basis for module, we can extend above
theorem for Syz(f1, . . . , fs) = Ker[Rs ×t [f1···fs]−−−−−→ Rd ].
Free resolution of the Steenrod Algebra Kyoto university
![Page 86: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/86.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Syzygy for not Gröbner basis
TheoremWith the notation above we haveSyz(f1, . . . , fs) = 〈s1T , . . . , sr T , r1, . . . , rs〉.
Using the Gröbner basis for module, we can extend above
theorem for Syz(f1, . . . , fs) = Ker[Rs ×t [f1···fs]−−−−−→ Rd ].
Free resolution of the Steenrod Algebra Kyoto university
![Page 87: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/87.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Syzygy for not Gröbner basis
Using above theorem, we can compute the free resolution.
Kerφi
""DDDDDDDD
φi+2 // Ri+1φi+1 //
;;wwwwwwwwwRi
φi // Ri−1
Free resolution of the Steenrod Algebra Kyoto university
![Page 88: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/88.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
I By above method, we can compute the free resolution ofA2-module H∗(X ; Z/2).
I Moreover, using the division algorithm, we have theminimal resolution.
I I am writing a computer program to compute the E2-termsof Adams spectral sequence. The program will be availablefrom the web page(http://www.math.kyoto-u.ac.jp/˜tomo_xi).
Free resolution of the Steenrod Algebra Kyoto university
![Page 89: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/89.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
I By above method, we can compute the free resolution ofA2-module H∗(X ; Z/2).
I Moreover, using the division algorithm, we have theminimal resolution.
I I am writing a computer program to compute the E2-termsof Adams spectral sequence. The program will be availablefrom the web page(http://www.math.kyoto-u.ac.jp/˜tomo_xi).
Free resolution of the Steenrod Algebra Kyoto university
![Page 90: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/90.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
I By above method, we can compute the free resolution ofA2-module H∗(X ; Z/2).
I Moreover, using the division algorithm, we have theminimal resolution.
I I am writing a computer program to compute the E2-termsof Adams spectral sequence. The program will be availablefrom the web page(http://www.math.kyoto-u.ac.jp/˜tomo_xi).
Free resolution of the Steenrod Algebra Kyoto university
![Page 91: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/91.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Summary
Our method works for any space X , Y
Es,t2∼= Exts,t
A2(H∗(X ; F2), H∗(Y ; F2)) ⇒ (2){Y , X}∗
On the other hand,There are many calculations for the special case X , Y = S0
Es,t2∼= Exts,t
A2(F2, F2) ⇒ πS
∗
the stable homotopy groups of the sphere
Free resolution of the Steenrod Algebra Kyoto university
![Page 92: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/92.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Summary
Our method works for any space X , Y
Es,t2∼= Exts,t
A2(H∗(X ; F2), H∗(Y ; F2)) ⇒ (2){Y , X}∗
On the other hand,There are many calculations for the special case X , Y = S0
Es,t2∼= Exts,t
A2(F2, F2) ⇒ πS
∗
the stable homotopy groups of the sphere
Free resolution of the Steenrod Algebra Kyoto university
![Page 93: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/93.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
(a part of) Known results
0
1
2
3
4
5
6
7
8
9
10
11
12
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Free resolution of the Steenrod Algebra Kyoto university
![Page 94: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/94.jpg)
Introduction Non-commutative Gröbner basis Free resolution Application
Our results
0
1
2
3
4
5
6
7
8
9
10
11
12
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Free resolution of the Steenrod Algebra Kyoto university
![Page 95: Application of non-commutative Gröbner basis to ...](https://reader031.fdocuments.net/reader031/viewer/2022012109/61dc5e168b88fb25264c6283/html5/thumbnails/95.jpg)
Appendix
For Further Reading
For Further Reading I
William W. Adams and Philippe Loustaunau.An introduction to Gröbner bases, volume 3 of GraduateStudies in Mathematics.American Mathematical Society, Providence, RI, 1994.
Huishi Li.Noncommutative Gröbner bases and filtered-gradedtransfer, volume 1795 of Lecture Notes in Mathematics.Springer-Verlag, Berlin, 2002.
Free resolution of the Steenrod Algebra Kyoto university