Coefficient-complete fixed-carrier codifferential deformation

Let $k$ be a characteristic-zero field, let $B$ be a commutative unital $k$-algebra concentrated in degree zero, and let $I$ be an ideal such that . Nilpotent ideals are included. Let $V$ be any graded $k$-vector space; no finite-dimensionality assumption is made. Use the graded tensor convention of Koszul sign rule for graded tensor products and permutations.

The carrier $C$ is either the coaugmented tensor coalgebra $T^c(V)={⨁}_{r\ge 0}{V}^{\otimes r}$ with deconcatenation, or the coaugmented graded symmetric coalgebra $S^c(V)={⨁}_{r\ge 0}{S}^r(V)$ with unshuffle coproduct. For the latter, $S^r$ means graded symmetric coinvariants, and the coproduct sums over all divisions of the inputs into two ordered sublists, with the Koszul sign. The empty word is $1_C$, the counit $ϵ$ is projection to length zero, and the coaugmentation is inclusion of scalars. These formulas specify the carrier without requiring a cofree universal property.

Define its coefficient completion, filtration, and tensor target by

All inverse limits in this definition are in graded modules: explicitly and $C_B={⨁}_j(C_B{)}^j$, and the tensor target is completed in each fixed total degree. All structure maps are extended degreewise through these inverse limits. This is coefficient completion of a direct sum: modulo each $I^n$ every element has finite length support. No unrestricted product over arities is included. Completed tensor powers and polynomial coefficients mean the analogous explicit inverse limits. A homogeneous map is understood in the grading induced by $C$, and continuity always means the displayed coefficient topology.

Let $E^d$ consist of continuous $B$-linear filtration-preserving homogeneous endomorphisms of degree $d$ on $C_B$, and set $E={⨁}_d{E}^d$, with composition $E^d{E}^e\subseteq E^{d+e}$. Completeness of this graded composition algebra always means completeness of each $E^d$ separately, not of its ungraded direct sum. All exponential and logarithm series below have inputs in $E^0$ and remain in $E^0$; their conjugation series on a homogeneous $T\in E^d$ remain in that same fixed $E^d$. Its order filtration is

$F^{p}E=\{T:T(F^q{C}_B)\subseteq F^{p+q}{C}_B\text{ for every }q\ge 0\}(p\ge 0)\mathrm{.}$

Let $\mathfrak{l}\subset E$ be the homogeneous coderivations satisfying $T(1_C)=0$, $ϵT=0$, and

$\mathrm{\Delta }T=(T\otimes 1+1\otimes T)\mathrm{\Delta }\mathrm{.}$

The sign in $1\otimes T$ is the graded tensor sign. In particular these coderivations have zero nullary component. Fix $D\in {\mathfrak{l}}^1$ with $D^2=0$. A coefficient-complete fixed-carrier codifferential deformation is $D+R$ with $R\in F^1{\mathfrak{l}}^1$ and $(D+R{)}^2=0$. Its allowed equivalences are degree-zero continuous $B$-linear counital coaugmented coalgebra automorphisms $G$ such that $G-\mathrm{id}\in F^{1}E$, acting by $D+R\mapsto G(D+R)G^{-1}$.

Here identity modulo $I$ can equivalently be stated as $(G-\mathrm{id})(C_B)\subseteq F^1{C}_B$. To see the filtration implication without assuming that $I$ is finitely generated, observe that $F^q{C}_B$ is the closure of $C\otimes I^q$: modulo $F^n$, $n>q$, every element lifts from $C\otimes I^q$. By $B$-linearity a continuous map with image in $F^1$ carries that dense submodule into $F^{q+1}$; closedness gives the same inclusion on all $F^q$.

This contract uses all continuous coefficient-complete Taylor operations. It does not identify them with a naive algebraic tensor ${\mathrm{Hom}}_k(W,U)\otimes I$ when the coefficient algebra is infinite dimensional. Over a finite-dimensional coefficient algebra, expanding in a finite coefficient basis does give the usual tensor identification, without any finite-dimensionality assumption on $W,U$.