The Hom complex of curved differential graded right modules #
Let M and N be curved differential graded right modules over the same curved differential
graded algebra (A, d, w). As for ordinary DG modules, the degree-p cochains are the
right-module maps raising internal degree by p, and the differential is the graded commutator
δ(f) = d_N ∘ f - (-1)^p f ∘ d_M.
Individual curved modules have no cohomology, since d_M² x = x * w need not vanish. The Hom
differential nevertheless squares to zero: the mixed terms cancel by the sign rule, leaving
δ²(f)(x) = d_N² (f x) - f (d_M² x) = f x * w - f (x * w) = 0,
because both modules square to right multiplication by the same curvature and f is a
right-module map. Hence the cochains form an honest cochain complex of modules over the ground
ring, which is the input for the DG category of curved modules and its homotopy category. At
curvature zero it is the Hom complex of the underlying ordinary DG right modules.
Main definitions #
TauCeti.dgRightModuleCochains.curvedDifferential: the graded-commutator differential on cochains between two curved differential graded right modules.TauCeti.curvedDGRightModuleHomComplex: the Hom complex of two curved differential graded right modules.
Main results #
TauCeti.dgRightModuleCochains.curvedDifferential_comp_self: the Hom differential squares to zero.TauCeti.dgRightModuleCochains.curvedDifferential_zero_eq_zero_iff: the closed degree-zero cochains are the maps commuting with the module differentials.TauCeti.dgRightModuleCochains.curvedDifferential_neg_one_apply: the boundary of an odd homotopykisdN ∘ k + k ∘ dM.TauCeti.curvedDGRightModuleHomComplex_zero: at curvature zero the curved Hom complex is the ordinary DG Hom complex.
Implementation notes #
As for TauCeti.dgRightModuleHomComplex, the complex is exposed so that the component types in
its public differential application lemma reduce to the homogeneous-cochain modules.
References #
- L. Positselski, Differential graded Koszul duality: an introductory survey, Section 6.2.
- B. Keller, Deriving DG categories, Section 2, for the uncurved Hom complex.
The differential on homogeneous right-module cochains between two curved differential graded right modules over the same curved algebra: the graded commutator with the two module differentials.
Equations
Instances For
Evaluating the curved Hom differential gives the graded commutator with the module differentials.
The curved Hom differential squares to zero. Both modules square to right
multiplication by the same curvature w, and a cochain is a right-module map, so
f x * w - f (x * w) vanishes.
The curved Hom differential applied twice vanishes.
A degree-zero cochain is closed exactly when it commutes with the module differentials.
The curved Hom differential of a cochain k of degree -1, an odd homotopy, is the
boundary dN ∘ k + k ∘ dM: in the degree -1 case of the graded commutator, the Koszul sign
(-1) ^ (-1) = -1 turns the subtraction into an addition.
Zero curvature. For ordinary differential graded right modules, regarded as curved modules of curvature zero, the curved Hom differential is the ordinary DG Hom differential.
The Hom complex between two curved differential graded right modules over the same curved
algebra. Its degree-p term consists of the right-module linear maps raising internal degree by
p, and its differential is the graded commutator with the two module differentials.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The degree-p term of the curved Hom complex is the module of degree-p homogeneous
cochains.
The differential morphism of the curved Hom complex is induced by the graded commutator.
The differential of the curved Hom complex, evaluated on a homogeneous cochain, is the graded commutator with the module differentials.
Zero curvature. For ordinary differential graded right modules, regarded as curved modules of curvature zero, the curved Hom complex is the ordinary DG Hom complex.