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| * FIXME: summarize [[http://trac.sagemath.org/sage_trac/ticket/6491|#6491]] | * new optional package [[http://sage.math.washington.edu/home/SimonKing/Cohomology/|p_group_cohomology]] (Simon A. King, David J. Green) * Compute the cohomology ring with coefficients in GF(p) for any finite p-group, in terms of a minimal generating set a minimal set of algebraic relations. We use Benson's criterion to prove the completeness of the ring structure. * Compute depth, dimension, Poincare series and a-invariants of the cohomology rings. * Construct induced homomorphisms. * The package includes a list of cohomology rings for all groups of order 64. * With the package, the cohomology for all groups of order 128 and for the Sylow 2-subgroup of the third Conway group (order 1024) was computed for the first time. The result of these and many other computations (e.g., all but 6 groups of order 243) is accessible in a repository on sage.math. __Examples__: * Data that are included with the package: {{{ sage: from pGroupCohomology import CohomologyRing sage: H = CohomologyRing(64,132) # this is included in the package, hence, the ring structure is already there sage: print H Cohomology ring of Small Group number 132 of order 64 with coefficients in GF(2) Computation complete Minimal list of generators: [a_2_4, a 2-Cochain in H^*(SmallGroup(64,132); GF(2)), c_2_5, a 2-Cochain in H^*(SmallGroup(64,132); GF(2)), c_4_12, a 4-Cochain in H^*(SmallGroup(64,132); GF(2)), a_1_0, a 1-Cochain in H^*(SmallGroup(64,132); GF(2)), a_1_1, a 1-Cochain in H^*(SmallGroup(64,132); GF(2)), b_1_2, a 1-Cochain in H^*(SmallGroup(64,132); GF(2))] Minimal list of algebraic relations: [a_1_0*a_1_1, a_1_0*b_1_2, a_1_1^3+a_1_0^3, a_2_4*a_1_0, a_1_1^2*b_1_2^2+a_2_4*a_1_1*b_1_2+a_2_4^2+c_2_5*a_1_1^2] sage: H.depth() 2 sage: H.a_invariants() [-Infinity, -Infinity, -3, -3] sage: H.poincare_series() (-t^2 - t - 1)/(t^6 - 2*t^5 + t^4 - t^2 + 2*t - 1) }}} * Data from the repository on sage.math: {{{ sage: H = CohomologyRing(128,562) sage: len(H.gens()) 35 sage: len(H.rels()) 486 sage: H.depth() 1 sage: H.a_invariants() [-Infinity, -4, -3, -3] sage: H.poincare_series() (t^14 - 2*t^13 + 2*t^12 - t^11 - t^10 + t^9 - 2*t^8 + 2*t^7 - 2*t^6 + 2*t^5 - 2*t^4 + t^3 - t^2 - 1)/(t^17 - 3*t^16 + 4*t^15 - 4*t^14 + 4*t^13 - 4*t^12 + 4*t^11 - 4*t^10 + 4*t^9 - 4*t^8 + 4*t^7 - 4*t^6 + 4*t^5 - 4*t^4 + 4*t^3 - 4*t^2 + 3*t - 1) }}} * Some computation from scratch, involving different ring presentations and induced maps: {{{ sage: tmp_root = tmp_filename() sage: CohomologyRing.set_user_db(tmp_root) sage: H0 = CohomologyRing.user_db(8,3,websource=False) sage: print H0 Cohomology ring of Dihedral group of order 8 with coefficients in GF(2) Computed up to degree 0 Minimal list of generators: [] Minimal list of algebraic relations: [] sage: H0.make() sage: print H0 Cohomology ring of Dihedral group of order 8 with coefficients in GF(2) Computation complete Minimal list of generators: [c_2_2, a 2-Cochain in H^*(D8; GF(2)), b_1_0, a 1-Cochain in H^*(D8; GF(2)), b_1_1, a 1-Cochain in H^*(D8; GF(2))] Minimal list of algebraic relations: [b_1_0*b_1_1] sage: G = gap('DihedralGroup(8)') sage: H1 = CohomologyRing.user_db(G,GroupName='GapD8',websource=False) sage: H1.make() sage: print H1 # the ring presentation is different ... Cohomology ring of GapD8 with coefficients in GF(2) Computation complete Minimal list of generators: [c_2_2, a 2-Cochain in H^*(GapD8; GF(2)), b_1_0, a 1-Cochain in H^*(GapD8; GF(2)), b_1_1, a 1-Cochain in H^*(GapD8; GF(2))] Minimal list of algebraic relations: [b_1_1^2+b_1_0*b_1_1] sage: phi = G.IsomorphismGroups(H0.group()) sage: phi_star = H0.hom(phi,H1) sage: phi_star_inv = phi_star^(-1) # ... but the rings are isomorphic sage: [X==phi_star_inv(phi_star(X)) for X in H0.gens()] [True, True, True, True] sage: [X==phi_star(phi_star_inv(X)) for X in H1.gens()] [True, True, True, True] }}} * An example with an odd prime: {{{ sage: H = CohomologyRing(81,8) # this needs to be computed from scratch sage: H.make() sage: H.gens() [1, a_2_1, a 2-Cochain in H^*(SmallGroup(81,8); GF(3)), a_2_2, a 2-Cochain in H^*(SmallGroup(81,8); GF(3)), b_2_0, a 2-Cochain in H^*(SmallGroup(81,8); GF(3)), a_4_1, a 4-Cochain in H^*(SmallGroup(81,8); GF(3)), b_4_2, a 4-Cochain in H^*(SmallGroup(81,8); GF(3)), b_6_3, a 6-Cochain in H^*(SmallGroup(81,8); GF(3)), c_6_4, a 6-Cochain in H^*(SmallGroup(81,8); GF(3)), a_1_0, a 1-Cochain in H^*(SmallGroup(81,8); GF(3)), a_1_1, a 1-Cochain in H^*(SmallGroup(81,8); GF(3)), a_3_2, a 3-Cochain in H^*(SmallGroup(81,8); GF(3)), a_5_2, a 5-Cochain in H^*(SmallGroup(81,8); GF(3)), a_5_3, a 5-Cochain in H^*(SmallGroup(81,8); GF(3)), a_7_5, a 7-Cochain in H^*(SmallGroup(81,8); GF(3))] sage: len(H.rels()) 59 sage: H.depth() 1 sage: H.a_invariants() [-Infinity, -3, -2] sage: H.poincare_series() (t^4 - t^3 + t^2 + 1)/(t^6 - 2*t^5 + 2*t^4 - 2*t^3 + 2*t^2 - 2*t + 1) }}} |
Sage 4.1.1 Release Tour
Algebra
FIXME: summarize #6510
Basic Arithmetic
Calculus
FIXME: summarize #5996
Combinatorics
Cryptography
FIXME: summarize #6454
Documentation
FIXME: summarize #4460
Elliptic Curves
#6381 (bug in integral_points when rank is large):
The function integral_x_coords_in_interval() for finding all integral points on an elliptic curve defined over the rationals whose x-coordinate lies in an interval is now more efficient when the interval is large.
FIXME: summarize #6407
Graphics
Graph Theory
FIXME: summarize #6355
FIXME: summarize #6540
FIXME: summarize #6552
FIXME: summarize #6578
FIXME: summarize #5793
Interfaces
Linear Algebra
Modular Forms
Notebook
FIXME: summarize #5653
Number Theory
#6457 (Intersection of ideals in a number field)
Intersection of ideals in number fields is now implemented.
Numerical
Packages
FIXME: summarize #6558
FIXME: summarize #6380
FIXME: summarize #6443
FIXME: summarize #6445
FIXME: summarize #6451
FIXME: summarize #6453
FIXME: summarize #6528
FIXME: summarize #6143
FIXME: summarize #6438
FIXME: summarize #6493
FIXME: summarize #6563
FIXME: summarize #6602
FIXME: summarize #6302
new optional package p_group_cohomology (Simon A. King, David J. Green)
- Compute the cohomology ring with coefficients in GF(p) for any finite p-group, in terms of a minimal generating set a minimal set of algebraic relations. We use Benson's criterion to prove the completeness of the ring structure.
- Compute depth, dimension, Poincare series and a-invariants of the cohomology rings.
- Construct induced homomorphisms.
- The package includes a list of cohomology rings for all groups of order 64.
- With the package, the cohomology for all groups of order 128 and for the Sylow 2-subgroup of the third Conway group (order 1024) was computed for the first time. The result of these and many other computations (e.g., all but 6 groups of order 243) is accessible in a repository on sage.math.
Examples:
- Data that are included with the package:
sage: from pGroupCohomology import CohomologyRing sage: H = CohomologyRing(64,132) # this is included in the package, hence, the ring structure is already there sage: print H Cohomology ring of Small Group number 132 of order 64 with coefficients in GF(2) Computation complete Minimal list of generators: [a_2_4, a 2-Cochain in H^*(SmallGroup(64,132); GF(2)), c_2_5, a 2-Cochain in H^*(SmallGroup(64,132); GF(2)), c_4_12, a 4-Cochain in H^*(SmallGroup(64,132); GF(2)), a_1_0, a 1-Cochain in H^*(SmallGroup(64,132); GF(2)), a_1_1, a 1-Cochain in H^*(SmallGroup(64,132); GF(2)), b_1_2, a 1-Cochain in H^*(SmallGroup(64,132); GF(2))] Minimal list of algebraic relations: [a_1_0*a_1_1, a_1_0*b_1_2, a_1_1^3+a_1_0^3, a_2_4*a_1_0, a_1_1^2*b_1_2^2+a_2_4*a_1_1*b_1_2+a_2_4^2+c_2_5*a_1_1^2] sage: H.depth() 2 sage: H.a_invariants() [-Infinity, -Infinity, -3, -3] sage: H.poincare_series() (-t^2 - t - 1)/(t^6 - 2*t^5 + t^4 - t^2 + 2*t - 1)
- Data from the repository on sage.math:
sage: H = CohomologyRing(128,562) sage: len(H.gens()) 35 sage: len(H.rels()) 486 sage: H.depth() 1 sage: H.a_invariants() [-Infinity, -4, -3, -3] sage: H.poincare_series() (t^14 - 2*t^13 + 2*t^12 - t^11 - t^10 + t^9 - 2*t^8 + 2*t^7 - 2*t^6 + 2*t^5 - 2*t^4 + t^3 - t^2 - 1)/(t^17 - 3*t^16 + 4*t^15 - 4*t^14 + 4*t^13 - 4*t^12 + 4*t^11 - 4*t^10 + 4*t^9 - 4*t^8 + 4*t^7 - 4*t^6 + 4*t^5 - 4*t^4 + 4*t^3 - 4*t^2 + 3*t - 1)
- Some computation from scratch, involving different ring presentations and induced maps:
sage: tmp_root = tmp_filename() sage: CohomologyRing.set_user_db(tmp_root) sage: H0 = CohomologyRing.user_db(8,3,websource=False) sage: print H0 Cohomology ring of Dihedral group of order 8 with coefficients in GF(2) Computed up to degree 0 Minimal list of generators: [] Minimal list of algebraic relations: [] sage: H0.make() sage: print H0 Cohomology ring of Dihedral group of order 8 with coefficients in GF(2) Computation complete Minimal list of generators: [c_2_2, a 2-Cochain in H^*(D8; GF(2)), b_1_0, a 1-Cochain in H^*(D8; GF(2)), b_1_1, a 1-Cochain in H^*(D8; GF(2))] Minimal list of algebraic relations: [b_1_0*b_1_1] sage: G = gap('DihedralGroup(8)') sage: H1 = CohomologyRing.user_db(G,GroupName='GapD8',websource=False) sage: H1.make() sage: print H1 # the ring presentation is different ... Cohomology ring of GapD8 with coefficients in GF(2) Computation complete Minimal list of generators: [c_2_2, a 2-Cochain in H^*(GapD8; GF(2)), b_1_0, a 1-Cochain in H^*(GapD8; GF(2)), b_1_1, a 1-Cochain in H^*(GapD8; GF(2))] Minimal list of algebraic relations: [b_1_1^2+b_1_0*b_1_1] sage: phi = G.IsomorphismGroups(H0.group()) sage: phi_star = H0.hom(phi,H1) sage: phi_star_inv = phi_star^(-1) # ... but the rings are isomorphic sage: [X==phi_star_inv(phi_star(X)) for X in H0.gens()] [True, True, True, True] sage: [X==phi_star(phi_star_inv(X)) for X in H1.gens()] [True, True, True, True] - An example with an odd prime:
sage: H = CohomologyRing(81,8) # this needs to be computed from scratch sage: H.make() sage: H.gens() [1, a_2_1, a 2-Cochain in H^*(SmallGroup(81,8); GF(3)), a_2_2, a 2-Cochain in H^*(SmallGroup(81,8); GF(3)), b_2_0, a 2-Cochain in H^*(SmallGroup(81,8); GF(3)), a_4_1, a 4-Cochain in H^*(SmallGroup(81,8); GF(3)), b_4_2, a 4-Cochain in H^*(SmallGroup(81,8); GF(3)), b_6_3, a 6-Cochain in H^*(SmallGroup(81,8); GF(3)), c_6_4, a 6-Cochain in H^*(SmallGroup(81,8); GF(3)), a_1_0, a 1-Cochain in H^*(SmallGroup(81,8); GF(3)), a_1_1, a 1-Cochain in H^*(SmallGroup(81,8); GF(3)), a_3_2, a 3-Cochain in H^*(SmallGroup(81,8); GF(3)), a_5_2, a 5-Cochain in H^*(SmallGroup(81,8); GF(3)), a_5_3, a 5-Cochain in H^*(SmallGroup(81,8); GF(3)), a_7_5, a 7-Cochain in H^*(SmallGroup(81,8); GF(3))] sage: len(H.rels()) 59 sage: H.depth() 1 sage: H.a_invariants() [-Infinity, -3, -2] sage: H.poincare_series() (t^4 - t^3 + t^2 + 1)/(t^6 - 2*t^5 + 2*t^4 - 2*t^3 + 2*t^2 - 2*t + 1)