nLab
minimal dg-module

Contents

Contents

Idea

A minimal dg-module (Roig 92, Roig 94, section 1) is a minimal model in the context of the homotopy theory of dg-modules.

Hence over dgc-algebras in non-positve degree, minimal dg-modules are the analogue of minimal Sullivan model as one passes from dg-algebras to (just) dg-modules. Minimal KS-extensions hence play the role of formal duals of minimal fibration in some applications of rational homotopy theory.

(In Halperin 83 it has “Koszul-Sullivan extensions” for relative Sullivan algebras, and “KS” in “KS-models” refers to that usage.)

Definition

Throughout, let AA be a dg-algebra. Eventually this is thought of as being a Sullivan model for the rationalization of the quotient of a topological space by a circle action.

We take all differentials to have degree +1. For VV a vector space and nn a natural number, we write V[n]V[n] for the chain complex concentrated on VV in degree nn.

Definition

(Hirsch extension of dg-modules)

Let NN be a dg-module over AA and let nn \in \mathbb{N} be a natural number. Then a degree nn Hirsch extension of NN is a monomorphisms of dg-modules of the form

N(N(AV[n]),d ϕ) N \hookrightarrow (N \oplus (A \otimes V[n]), d_{\phi})

given by a choice of

  1. a vector space VV

  2. a linear map d V:VN n+1d_V \;\colon\; V \to N_{n+1}

where the differential d ϕd_{\phi} is d Nd_N on NN, is d Ad_A on AA, and is given on VV by ϕ\phi followed by the action ρ\rho of AA on VV:

d ϕ(av)=(d Aa)v+(1) |a|ρ(a,ϕ(v)). d_\phi (a \otimes v) = (d_A a) \otimes v + (-1)^{\vert a \vert} \rho(a,\phi(v)) \,.

(Roig 94, def. 1.8)

Remark

It follows that for N 1,N 2N_1, N_2 two AA-dg-modules then homomorphisms ff out of a Hirsch extension of the former (def. )

f:(N 1olpus(AV[n]),d ϕ)N 2 f \;\colon\; (N_1 \olpus (A \otimes V[n]), d_\phi) \longrightarrow N_2

is equivalently

  1. a homomorphism of dg-modules h:N 1N 2h \colon N_1 \to N_2;

  2. a linear map g:V(N 2) ng \colon V \to (N_2)_{n}

such that

  • hd ϕ=d N 2gh \circ d_{\phi} = d_{N_2} \circ g.

The following defines a kind of minimal cofibrations of dg-modules.

Definition

(minimal KS-extension)

For NN a dg-module over AA, then a minimal KS-extension of NN is a certain transfinite composition of Hirsch extensions, namely a monomorphism

NN^ N \hookrightarrow \hat N

equipped with an exhaustive filtration {N^(n,q)} n,q\{\hat N(n,q)\}_{n,q \in \mathbb{N}} such that:

  1. N^(0,0)N\hat N(0,0) \simeq N;

  2. the inclusions N^(n,q)N^(n,q+1)\hat N(n,q) \hookrightarrow \hat N(n,q+1) are Hirsch extensions (def. ).

  3. N(n+1,0)=lim qN^(n,q)N(n+1,0) = \underset{\longrightarrow}{\lim}_q \hat N(n,q) (colimit over the sequence of Hirsch extensions in the previous degree).

Accordingly, a minimal KS-factorization of a morphism N 1N 2N_1 \to N_2 of AA-dg-modules is a factorization as a minimal KS-extension followed by a quasi-isomorphism

N 1 N 2 minimalKS-extension quasi-iso N 1(AV). \array{ N _1 && \longrightarrow && N_2 \\ & {}_{\mathllap{\text{minimal} \atop \text{KS-extension}}}\searrow && \nearrow_{\mathrlap{\text{quasi-iso}}} \\ && N_1 \oplus (A \otimes V) } \,.

Finally a minimal KS-model is a dg-module NN such that 0N0 \hookrightarrow N is a minimal KS-fibration.

(Roig 94, def. 1.9)

Proposition

Let the ground field be of characteristic zero.

Let f:N 1N 2f \colon N_1 \longrightarrow N_2 be a morphism of AA-dg-modules such that it induces a monomorphism in degree-0 cochain cohomology, H 0(f):H 0(N 1)H 0(N 2)H^0(f) \colon H^0(N_1) \hookrightarrow H^0(N_2) then it admits a minimal KS-factorization (def. ).

In particular, every dg-module has a minimal KS-model (def. ).

Roig & Saralegi-Aranguren 00, theorem 1.3.1

Examples

References

See also

  • Steve Halperin, Lectures on minimal models, Mem. Soc. Math. Franc. no 9/10 (1983) (web)

  • Flavio da Silveira, Rational homotopy theory of fibrations, Pacific Journal of Mathematics, Vol. 113, No. 1 (1984) (pdf)

Last revised on March 19, 2018 at 11:35:57. See the history of this page for a list of all contributions to it.