Koszul Complexes¶
-
class
sage.homology.koszul_complex.
KoszulComplex
(R, elements)¶ 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:
R
– the base ringelements
– a tuple of elements ofR
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: