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GA:010117On the three-element chain, Yoneda gives one natural map Hom(-,1) to Hom(-,2) and none in reverse

Ga1 Three Chain Yoneda Nat CountGa1 Representable ID Component

Categories, functors, and exact sequencesGraduate Algebra I · Mixed review

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 ledger

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Ga1 F5 Split Exact RanksGa1 Section Complement Formula

Categories, functors, and exact sequencesGraduate Algebra I · Mixed review

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)

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Ga1 F2 Tensor Hom AdjunctionGa1 Curry Matrix Sample

Categories, functors, and exact sequencesGraduate Algebra I · Mixed review

GA:010219A Schreier transversal {1,a,a²} for F(a,b) onto C_3 yields free generators b, aba^{-1}, a²ba^{-2}, a³

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Ga1 Schreier Transversal Index ThreeGa1 Nielsen Schreier Rank Four

Advanced finite and infinite group structureGraduate Algebra I · Mixed review

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 8

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Ga1 Ut4 Lower Central SeriesGa1 Ut4 Center Abelianization

Advanced finite and infinite group structureGraduate Algebra I · Mixed review

GA:010320The crossing k[x,y]/(xy) is reduced of dimension one, with components meeting at (x,y) and tangent dimension two

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Ga1 Node Ring Minimal PrimesGa1 Connected Crossing Tangent

Rings, localization, and ideal structureGraduate Algebra I · Mixed review

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 9

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Ga1 Artinian Local Associated GradedGa1 Socle Annihilator Computation

Rings, localization, and ideal structureGraduate Algebra I · Mixed review

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)

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Ga1 Embedded Primary DecompositionGa1 Colon Ideals Associated Primes

Rings, localization, and ideal structureGraduate Algebra I · Mixed review

GA:010421Z-endomorphisms of Z/9 ⊕ Z/3 number 243, of which 108 are automorphisms and 135 are not

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Ga1 Z9 Z3 End ParametrizationGa1 Z9 Z3 Aut Criterion

Modules, chain conditions, and structure theoremsGraduate Algebra I · Mixed review

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 P

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Ga1 Prufer Artinian Not NoetherianGa1 Prufer Injective Hull Socle

Modules, chain conditions, and structure theoremsGraduate Algebra I · Mixed review

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)

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Ga1 S3 Permutation Character Inner ProductsGa1 Averaging Projector Invariant Line

Introductory representation theoryGraduate Algebra I · Mixed review

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=-1

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Ga1 C4 Fourier IdempotentsGa1 Regular Representation Determinant

Introductory representation theoryGraduate Algebra I · Mixed review

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}=0

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Ga1 Hom Ext Cyclic GroupsGa1 Multiplication Map Kernel Image

Projective, injective, and derived-functor motivationGraduate Algebra I · Mixed review

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..5

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Ga1 Baer Criterion Divisible GroupsGa1 Extension Count Along NZ

Projective, injective, and derived-functor motivationGraduate Algebra I · Mixed review

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 exact

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Ga1 C2 Sign Cochain ComplexGa1 Periodic Cohomology Parity

Projective, injective, and derived-functor motivationGraduate Algebra I · Mixed review

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GA:010117On the three-element chain, Yoneda gives one natural map Hom(-,1) to Hom(-,2) and none in reverse

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 ledger

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)

GA:010219A Schreier transversal {1,a,a²} for F(a,b) onto C_3 yields free generators b, aba^{-1}, a²ba^{-2}, a³

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 8

GA:010320The crossing k[x,y]/(xy) is reduced of dimension one, with components meeting at (x,y) and tangent dimension two

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 9

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)

GA:010421Z-endomorphisms of Z/9 ⊕ Z/3 number 243, of which 108 are automorphisms and 135 are not

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 P

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)

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=-1

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}=0

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..5

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 exact

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GA:010117On the three-element chain, Yoneda gives one natural map Hom(-,1) to Hom(-,2) and none in reverse

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 ledger

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)

GA:010219A Schreier transversal {1,a,a²} for F(a,b) onto C_3 yields free generators b, aba^{-1}, a²ba^{-2}, a³

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 8

GA:010320The crossing k[x,y]/(xy) is reduced of dimension one, with components meeting at (x,y) and tangent dimension two

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 9

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)

GA:010421Z-endomorphisms of Z/9 ⊕ Z/3 number 243, of which 108 are automorphisms and 135 are not

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 P

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)

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=-1

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}=0

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..5

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 exact

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GA:010117On the three-element chain, Yoneda gives one natural map Hom(-,1) to Hom(-,2) and none in reverseView problem statement

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Problem results

GA:010117On the three-element chain, Yoneda gives one natural map Hom(-,1) to Hom(-,2) and none in reverseView problem statement

GA:010117On the three-element chain, Yoneda gives one natural map Hom(-,1) to Hom(-,2) and none in reverse

GA:010117On the three-element chain, Yoneda gives one natural map Hom(-,1) to Hom(-,2) and none in reverse