Part II
Mathematical Interpretation
Conventions Used in This Part
If not stated otherwise, all algebraic objects will be considered over a fixed field k
of characteristic zero. The symbol ⊗ will be reserved for the tensor product over k.
Given a set S, Span(S) will denote the k-vector space generated by S. We will
denote by 1 X or simply by 1 when X is understood, the identity endomorphism of
an object X (set, vector space, algebra, etc.).
By C 1 C 2 we denote the union of disjoint sets C 1 and C 2 . Notice that this
operation is strictly associative and symmetric, i.e.
C 1 C 2 = C 2 C 1 and (C 1 C 2 ) C 3 = C 1 (C 2 C 3 )
for each mutually disjoint sets C 1 , C 2 , and C 3 .
If not specified otherwise, by a grading we mean a Z grading. The degree of
a graded object will be denoted by |w| though we will sometimes omit the vertical
bars and write, e.g., (−1) a+b instead of (−1) |a|+|b| to save the space. We will use
the Koszul sign convention meaning that whenever we commute two “things” of
degrees p and q, respectively, we multiply the sign by (−1) pq .
By Σ n we denote, for n ≥ 1, the symmetric group of n elements realized, when
necessary, as the group of automorphism of the set {1, . . . , n}. The multiplication is
given by the composition of automorphisms, i.e. σ τ := σ ◦ τ , and the unit is the
identity. For graded indeterminates x 1 , . . . , x n and a permutation σ ∈ Σ n define the
Koszul sign ) = ; x 1 , . . . , x n ) by
x 1 · · · x n = (σ ; x 1 , . . . , x n ) · x σ (1) · · · x σ (n) ,
(1)
which has to be satisfied in the free graded commutative algebra k[x 1 , . . . , x n ].
Mathematical Interpretation
Conventions Used in This Part
If not stated otherwise, all algebraic objects will be considered over a fixed field k
of characteristic zero. The symbol ⊗ will be reserved for the tensor product over k.
Given a set S, Span(S) will denote the k-vector space generated by S. We will
denote by 1 X or simply by 1 when X is understood, the identity endomorphism of
an object X (set, vector space, algebra, etc.).
By C 1 C 2 we denote the union of disjoint sets C 1 and C 2 . Notice that this
operation is strictly associative and symmetric, i.e.
C 1 C 2 = C 2 C 1 and (C 1 C 2 ) C 3 = C 1 (C 2 C 3 )
for each mutually disjoint sets C 1 , C 2 , and C 3 .
If not specified otherwise, by a grading we mean a Z grading. The degree of
a graded object will be denoted by |w| though we will sometimes omit the vertical
bars and write, e.g., (−1) a+b instead of (−1) |a|+|b| to save the space. We will use
the Koszul sign convention meaning that whenever we commute two “things” of
degrees p and q, respectively, we multiply the sign by (−1) pq .
By Σ n we denote, for n ≥ 1, the symmetric group of n elements realized, when
necessary, as the group of automorphism of the set {1, . . . , n}. The multiplication is
given by the composition of automorphisms, i.e. σ τ := σ ◦ τ , and the unit is the
identity. For graded indeterminates x 1 , . . . , x n and a permutation σ ∈ Σ n define the
Koszul sign ) = ; x 1 , . . . , x n ) by
x 1 · · · x n = (σ ; x 1 , . . . , x n ) · x σ (1) · · · x σ (n) ,
(1)
which has to be satisfied in the free graded commutative algebra k[x 1 , . . . , x n ].