Bases: sage.homology.chain_complex.ChainComplex_class, sage.structure.unique_representation.UniqueRepresentation
A Koszul complex.
Let be a ring and consider
. The
Koszul complex
is given by defining a
chain complex structure on the exterior algebra
with
the basis
. The differential is
given by
where denotes the omitted factor.
Alternatively we can describe the Koszul complex by considering the
basic complex
Then the Koszul complex is given by
.
INPUT:
EXAMPLES:
sage: R.<x,y,z> = QQ[]
sage: K = KoszulComplex(R, [x,y])
sage: ascii_art(K)
[-y]
[x y] [ x]
0 <-- C_0 <------ C_1 <----- C_2 <-- 0
sage: K = KoszulComplex(R, [x,y,z])
sage: ascii_art(K)
[-y -z 0] [ z]
[ x 0 -z] [-y]
[x y z] [ 0 x y] [ x]
0 <-- C_0 <-------- C_1 <----------- C_2 <----- C_3 <-- 0
sage: K = KoszulComplex(R, [x+y*z,x+y-z])
sage: ascii_art(K)
[-x - y + z]
[ y*z + x x + y - z] [ y*z + x]
0 <-- C_0 <---------------------- C_1 <------------- C_2 <-- 0
REFERENCES: