GA:010117On the three-element chain, Yoneda gives one natural map Hom(-,1) to Hom(-,2) and none in reverseGa1 Three Chain Yoneda Nat CountGa1 Representable ID ComponentPrivateLabels unavailablePrivate saving unavailableCategories, functors, and exact sequencesGraduate Algebra I · Mixed reviewProgress not loaded
GA:010118Over F_5 the maps f(x,y)=(x,y,x+y) and g=a+b-c form a split exact rank 2/1 ledgerGa1 F5 Split Exact RanksGa1 Section Complement FormulaPrivateLabels unavailablePrivate saving unavailableCategories, functors, and exact sequencesGraduate Algebra I · Mixed reviewProgress not loaded
GA:010127Over F2 tensor-Hom currying has dimension 12 and cardinality 4096, with matrix [[a,b,0],[b,0,a]] and B((1,1),(1,0,1))=(1,0)Ga1 F2 Tensor Hom AdjunctionGa1 Curry Matrix SamplePrivateLabels unavailablePrivate saving unavailableCategories, functors, and exact sequencesGraduate Algebra I · Mixed reviewProgress not loaded
GA:010219A Schreier transversal {1,a,a²} for F(a,b) onto C_3 yields free generators b, aba^{-1}, a²ba^{-2}, a³Ga1 Schreier Transversal Index ThreeGa1 Nielsen Schreier Rank FourPrivateLabels unavailablePrivate saving unavailableAdvanced finite and infinite group structureGraduate Algebra I · Mixed reviewProgress not loaded
GA:010228UT4(F2) has order 64, lower central series of class 3 with |gamma2|=8 on positions 13,14,24 and |gamma3|=2 on E14, center of order 2, and abelianization (C2)^3 of order 8Ga1 Ut4 Lower Central SeriesGa1 Ut4 Center AbelianizationPrivateLabels unavailablePrivate saving unavailableAdvanced finite and infinite group structureGraduate Algebra I · Mixed reviewProgress not loaded
GA:010320The crossing k[x,y]/(xy) is reduced of dimension one, with components meeting at (x,y) and tangent dimension twoGa1 Node Ring Minimal PrimesGa1 Connected Crossing TangentPrivateLabels unavailablePrivate saving unavailableRings, localization, and ideal structureGraduate Algebra I · Mixed reviewProgress not loaded
GA:010326F3[x,y]/(x^2,xy,y^3) has basis {1,x,y,y^2}, order 81, unique max ideal of order 27, layers of dimensions 1,2,1 and length 4, and socle span{x,y^2} of order 9Ga1 Artinian Local Associated GradedGa1 Socle Annihilator ComputationPrivateLabels unavailablePrivate saving unavailableRings, localization, and ideal structureGraduate Algebra I · Mixed reviewProgress not loaded
GA:010329In k[x,y] the ideal (x^2,xy) equals (x) intersect (x^2,y), with associated primes (x) and embedded (x,y), radical (x) of dimension 1, and colons I:y=(x), I:x=(x,y)Ga1 Embedded Primary DecompositionGa1 Colon Ideals Associated PrimesPrivateLabels unavailablePrivate saving unavailableRings, localization, and ideal structureGraduate Algebra I · Mixed reviewProgress not loaded
GA:010421Z-endomorphisms of Z/9 ⊕ Z/3 number 243, of which 108 are automorphisms and 135 are notGa1 Z9 Z3 End ParametrizationGa1 Z9 Z3 Aut CriterionPrivateLabels unavailablePrivate saving unavailableModules, chain conditions, and structure theoremsGraduate Algebra I · Mixed reviewProgress not loaded
GA:010431The Prüfer 2-group Z[1/2]/Z has unique finite submodules P[2^n], is Artinian not Noetherian, divisible hence injective, and has essential socle P[2] with injective hull PGa1 Prufer Artinian Not NoetherianGa1 Prufer Injective Hull SoclePrivateLabels unavailablePrivate saving unavailableModules, chain conditions, and structure theoremsGraduate Algebra I · Mixed reviewProgress not loaded
GA:010522S3 permutation character (3,1,0) splits as triv+std with inner products (1,0,1) and averaging projector J/3 sending (1,2,4) to (7/3,7/3,7/3)View access optionsGa1 S3 Permutation Character Inner ProductsGa1 Averaging Projector Invariant LinePrivateLabels unavailablePrivate saving unavailableIntroductory representation theoryGraduate Algebra I · Mixed reviewProgress not loaded
GA:010523C4 Fourier idempotents e_j=(1/4)sum i^{-jr}g^r are orthogonal, sum to 1, satisfy g e_j=i^j e_j, and the regular character is (4,0,0,0) with det L_g=-1View access optionsGa1 C4 Fourier IdempotentsGa1 Regular Representation DeterminantPrivateLabels unavailablePrivate saving unavailableIntroductory representation theoryGraduate Algebra I · Mixed reviewProgress not loaded
GA:010624Multiplication by 12 on Z/18 has kernel {0,3,6,9,12,15} and image {0,6,12}, so Hom(Z/12,Z/18) and Ext^1 are both Z/6 while Ext^{q>=2}=0View access optionsGa1 Hom Ext Cyclic GroupsGa1 Multiplication Map Kernel ImagePrivateLabels unavailablePrivate saving unavailableProjective, injective, and derived-functor motivationGraduate Algebra I · Mixed reviewProgress not loaded
GA:010625Q/Z is injective by divisibility and Baer, and the map 6Z→Q/Z with f(6)=1/4+Z has exactly six extensions β=1/24+k/6 for k=0..5View access optionsGa1 Baer Criterion Divisible GroupsGa1 Extension Count Along NZPrivateLabels unavailablePrivate saving unavailableProjective, injective, and derived-functor motivationGraduate Algebra I · Mixed reviewProgress not loaded
GA:010630C2 with the sign action on Z has periodic cochains multiplication by -2,0,-2,0, vanishing even cohomology including H0, and H^odd = Z/2, with H0 through H3 exactView access optionsGa1 C2 Sign Cochain ComplexGa1 Periodic Cohomology ParityPrivateLabels unavailablePrivate saving unavailableProjective, injective, and derived-functor motivationGraduate Algebra I · Mixed reviewProgress not loaded