Quadratic maps between modules

Abstract

Proposition 1.2. Suppose that R contains an element r such that r and r − 1 are invertible. Then any Rquadratic map f : M → N decomposes uniquely as a sum f = f 1 + f 2 of an R-linear map f 1 and a homoge-This criterion is improved in Example 4.5(2) below. Proof.We can takeby Remark 1.3 below. Uniqueness of f 1 and f 2 follows from the fact that under the hypothesis any map which is R-linear and homogenous R-quadratic is trivial; in fact,Note that the proposition applies whenever R is a field different from F 2 . If R = F 2 any Rquadratic map is homogenous.Finally, we discuss the case R = Z. Passi [15] defines a map f : G → A from a group G to an abelian group A to be (normalized) polynomial of degree. An inductive characterization of this property [9] shows that f is polynomial of degree 2 iff its cross-effect d f is homomorphic in each variable. If G is abelian, this is equivalent towhich is homogenous Z-quadratic; this suffices by Remark 1.3 below.Let us exhibit some elementary properties of R-quadratic maps. First note that if f is R-quadratic,Next for x, y, z in M and r, s in R the first condition in 1.1 can be written as(1.2) Additivity of f [r] then follows from (1.2) with s = r, and its R-linearity can be written as:(1.3) Remark 1.3. Relation (1.3) can be written as f s (rx) = r 2 f s (x), that is f s is a homogeneous R-quadratic map. Thus we see that f is R-quadratic iff its cross-effect is R-bilinear and its cross-actions are homogeneous R-quadratic.Clearly the set R-Quad(M, N) (resp. R-HQuad(M, N)) of the R-quadratic maps (resp. homogeneous R-quadratic maps) from M to N is an R-module, and pre-or postcomposition of an R-quadratic map (resp. homogeneous R-quadratic map) by an R-linear map is an R-quadratic (resp. homogeneous Rquadratic) map.Throughout this paper the tensor product of R-modules M and N is denoted by M ⊗ N instead of M ⊗ R N. Lemma 1.4. For any R-modules M, M , N one has a natural isomorphism R-Quad(M ⊕ M , N) ∼ = R-Quad(M, N) ⊕ R-Quad(M , N) ⊕ R-Hom(M ⊗ M , N).

Witt Vectors, Polynomial Maps, and Real Topological Hochschild Homology

“…v) Let T be a Tambara functor for a finite group G. The multiplicative transfer Not all polynomial maps can be decomposed into a sum of homogeneous maps. A well-known counterexample is the degree 2 map (see e.g., [GH09])…”