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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 $A$ 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 $V$ a vector space and $n$ a natural number, we write $V[n]$ for the chain complex concentrated on $V$ in degree $n$.

Definition

(Hirsch extension of dg-modules)

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

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

given by a choice of

1. a vector space $V$

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

where the differential $d_{\phi}$ is $d_N$ on $N$, is $d_A$ on $A$, and is given on $V$ by $\phi$ followed by the action $\rho$ of $A$ on $V$:

$d_\phi (a \otimes v) = (d_A a) \otimes v + (-1)^{\vert a \vert} \rho(a,\phi(v)) \,.$
Remark

It follows that for $N_1, N_2$ two $A$-dg-modules then homomorphisms $f$ out of a Hirsch extension of the former (def. )

$f \;\colon\; (N_1 \olpus (A \otimes V[n]), d_\phi) \longrightarrow N_2$

is equivalently

1. a homomorphism of dg-modules $h \colon N_1 \to N_2$;

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

such that

• $h \circ d_{\phi} = d_{N_2} \circ g$.

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

Definition

(minimal KS-extension)

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

$N \hookrightarrow \hat N$

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

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

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

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

Accordingly, a minimal KS-factorization of a morphism $N_1 \to N_2$ of $A$-dg-modules is a factorization as a minimal KS-extension followed by a quasi-isomorphism

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

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

Proposition

Let the ground field be of characteristic zero.

Let $f \colon N_1 \longrightarrow N_2$ be a morphism of $A$-dg-modules such that it induces a monomorphism in degree-0 cochain cohomology, $H^0(f) \colon H^0(N_1) \hookrightarrow H^0(N_2)$ 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