Homogeneous Polynomial Laws

• Homogeneous polynomial laws are algebraic maps between modules defined by a scaling property (p_A(rx) = r^d p_A(x)) and uniquely decompose into homogeneous components.
• They leverage the universal properties of divided power algebras to connect symmetric multilinear forms with polynomial maps and enable algebraic adjunctions.
• The framework extends to weighted, bi-variant, and derived settings, underpinning applications in Clifford algebras and modern derived algebraic geometry.

A homogeneous polynomial law is an algebraic structure generalizing the notion of polynomial maps between modules over a commutative ring. In Roby’s formulation, a homogeneous polynomial law of degree $d$ between $R$-modules $M$ and $N$ is a family of maps $p_A: M \otimes_R A \to N \otimes_R A$ (where $A$ varies over all commutative $R$-algebras), which is functorial in $A$ and satisfies the key scaling property $p_A(r x) = r^d p_A(x)$ for all $r \in A$, $x \in M \otimes_R A$. Homogeneous polynomial laws play a critical role in the theory of divided power algebras, generalized Clifford algebra constructions, and the universal properties of algebraic adjunctions (Bach, 8 Oct 2025, Chambert-Loir et al., 5 Dec 2025, Velanga, 2017).

1. Definition and Foundational Properties

The formal definition is as follows: for a commutative ring $R$ and $R$-modules $M,N$, a polynomial law $f: M \rightsquigarrow N$ consists of set maps $f_S : S \otimes_R M \to S \otimes_R N$ for each commutative $R$-algebra $S$, natural in $S$. The law is homogeneous of degree $d$ if for all $r \in S$, $m \in S \otimes_R M$, $f_S(r m) = r^d f_S(m)$. The set of such degree-$d$ homogeneous laws is denoted $\mathscr{P}_R(M;N)_d$ or $P^d(M,N)$ (Chambert-Loir et al., 5 Dec 2025, Bach, 8 Oct 2025).

Homogeneous polynomial laws naturally generalize both algebraic (single-variable) polynomials and symmetric multilinear forms. Every $m$-linear map $A: M^m \to N$ induces an $m$-homogeneous polynomial law by evaluating on the diagonal: $P(x) = A(x,\ldots,x)$. Conversely, every homogeneous law arises from a suitable symmetric multilinear map via the polarization process ((Velanga, 2017), Theorem 2.1).

Homogeneous laws decompose uniquely into homogeneous pieces: any polynomial law $f$ admits a “locally finite” decomposition $f = \sum_{d \in \mathbb{N}} f_d$ with each $f_d \in \mathscr{P}_R(M;N)_d$ (Chambert-Loir et al., 5 Dec 2025).

2. Universal Properties via Divided Power Algebras

The divided power algebra $\Gamma_R(M)$, introduced by Roby, encodes the universal properties of homogeneous polynomial laws. For each $d$, $\Gamma^d_R(M)$ is the $d$th graded summand, and the assignment

$\mathrm{Hom}_R(\Gamma^d_R(M), N) \simeq \{ \text{degree-%%%%37%%%% homogeneous laws } M \to N \}$

is an $R$-linear isomorphism ((Chambert-Loir et al., 5 Dec 2025), Roby's Theorem). The map $m \mapsto m^{[d]}$ (the $d$th divided power) defines a degree-$d$ homogeneous law, and all such laws factor uniquely through the universal property of $\Gamma^d_R(M)$. This structure underpins the transfer of properties and constructions between polynomial laws and algebraic objects such as symmetric and exterior powers.

The divided power algebra itself is constructed via generators and relations: $$x^{[0]}=1,\quad (x+y)^{[n]}=\sum_{i+j=n}x^{[i]}y^{[j]},\quad (r x)^{[n]}=r^n x^{[n]},\quad x^{[m]} x^{[n]} = \binom{m+n}{m}x^{[m+n]}$$ and admits a grading $\Gamma_R(M) = \bigoplus_{d} \Gamma^d_R(M)$ (Chambert-Loir et al., 5 Dec 2025).

3. Weighted and Bi-Variant Polynomial Laws

Bach’s formulation generalizes classical polynomial laws to weighted polynomial laws, allowing simultaneous interaction with multiple target algebras in different degrees. A $d$-weighted polynomial law $\Psi$ consists of:

• $R$-modules $F$ and $G$,
• a sequence of associative $R$-algebras $A_0, \dots, A_d$,
• a (possibly inhomogeneous) law $\psi_{-1}: F \to G$,
• for each $i = 0, \dots, d$, a law $\psi_i : F \to A_i$.

When $G=0$ and all $A_i=N$ with $\psi_i=0$ for $i\ne d$, classical homogeneous laws are recovered as the special case $\psi_d \in P^d(F,N)$ (Bach, 8 Oct 2025). Bi-variant polynomial laws introduce additional components $\psi_{-2}$ for handling bilinear forms, as required in Clifford algebra constructions.

This categorical framework captures both classical and novel algebraic phenomena, unifying the treatment of objects such as generalized Clifford algebras, Weyl algebras, and projective restrictions.

4. Functoriality, Lifting, and Adjunctions

Homogeneous and weighted polynomial laws behave well with respect to base change and morphisms of schemes. The pull-back $f^*$ induces $P^d(X) \to P^d(Y)$ for any morphism $f:Y \to X$. If $f^*$ admits a left adjoint $f_\sharp$, and a law $\Psi_Y$ is a lift of $\Psi_X$ along $f$, there is an isomorphism of Clifford-type algebras: $$f_\sharp (\mathrm{Cl}(\Psi_Y)) \simeq \mathrm{Cl}(\Psi_X)$$ (Bach, 8 Oct 2025). This functorial adjunction structure is foundational to the modern perspective on Clifford algebras and their variants.

The Krashen–Lieblich projective restriction exemplifies this theory: for a locally free sheaf $F$ on $X$ and projective bundle $p: P(F) \to X$, the tautological homogeneous law on $F$ lifts uniquely via $\mathcal{O}_{P(F)}(-1)$, yielding explicit geometric models for Clifford-type constructs.

5. Derived and $\infty$-Categorical Extensions

Homogeneous polynomial laws and their associated algebraic constructions extend to derived and higher-categorical contexts. In the $\infty$-category of connective $R$-modules ($\mathrm{Mod}_R^{\mathrm{an}}$) and associative $E_1$-algebras ($\mathrm{Alg}_R^{\mathrm{an}}$), one considers derived homogeneous laws: triples $(K,A,\psi)$ with $\psi:L\Gamma^d(K)\to A$ and appropriate universal property. The derived Clifford algebra functor $C$ is the left adjoint to the forgetful functor, constructed via pushouts of free $\infty$-algebras ((Bach, 8 Oct 2025), Theorem 5.1.3).

This framework recovers classical results on $\pi_0$, and enables computation of derived or differential graded algebra (DGA) models with nontrivial higher homology, exhibiting base change compatibility: $$C(\Psi \otimes^L_R R') \simeq C(\Psi) \otimes^L_R R'$$ This categorical upgrade clarifies the nature of Clifford and related algebras in complex and derived algebraic contexts.

6. Examples and Applications

Table: Representative Constructions Involving Homogeneous Polynomial Laws

Example	Data	Resulting Algebraic Object
Weyl Algebra	Bilinear form $\psi: M \otimes M \to R$	$T(M)/(xy - yx - \psi(x,y))$
Quadratic Clifford	$q: M \to R$, $\psi_2 \in P^2(M, R)$	$T(M)/(m \otimes m - q(m))$
Roby Non-diagonal	Homog. polys $f_{im}$ on $R^n$	$R\langle a_J \rangle/($relations as in [KL A.1])
Projective Restriction	Tautological law on $\mathcal{O}(-1)$ over $P(F)$	$$\operatorname{Spec}(\mathrm{Cl}(\Psi|_{\mathcal{O}(-1)}))$$
Derived Clifford (Vezzosi)	Derived quadratic form	Recovers Vezzosi's derived Clifford algebra

Homogeneous polynomial laws fundamentally characterize the universal properties behind numerous algebraic and geometric constructions, including symmetric powers, binomial polynomials, monomial laws, and multipolynomials. Notably, every (multi)linear map between vector spaces (or Banach spaces) can be seen as a homogeneous polynomial via diagonalization, and polarization formulas extend to multipolynomial laws, with remainder terms precisely characterizable by combinatorial techniques (Velanga, 2017, Chambert-Loir et al., 5 Dec 2025).

7. Connections, Impact, and Current Directions

The theory of homogeneous polynomial laws serves as a connective tissue between algebraic geometry, representation theory, and homological algebra. Their categorical and universal properties undergird the development of crystalline cohomology, $p$-adic Hodge theory, and modern derived algebraic geometry (Bach, 8 Oct 2025, Chambert-Loir et al., 5 Dec 2025). Formalization efforts (e.g., in Lean/Mathlib) reinforce their foundational role and enable computational manipulation at the level of proof assistants (Chambert-Loir et al., 5 Dec 2025).

Current research directions include the systematic exploration of adjunctions between Clifford-type functors and their base categories, classification of weighted law lifts along morphisms of schemes, and explicit computation of derived Clifford algebras. Recent advances demonstrate that all classical and generalized Clifford constructions—ranging from Roby’s divided-power algebras to geometric and derived settings—fit naturally into the adjoint-colimit framework established by the universal properties of homogeneous (and weighted) polynomial laws (Bach, 8 Oct 2025).