Supermodule - Homomorphisms


A homomorphism between supermodules is a module homomorphism that preserves the grading. Let E and F be right supermodules over A. A map

is a supermodule homomorphism if

for all aA and all x,yE. The set of all module homomorphisms from E to F is denoted by Hom(E, F).

In many cases, it is necessary or convenient to consider a larger class of morphisms between supermodules. Let A be a supercommutative algebra. Then all supermodules over A be regarded as superbimodules in a natural fashion. For supermodules E and F, let Hom(E, F) denote the space of all right A-linear maps (i.e. all module homomorphisms from E to F considered as ungraded right A-modules). There is a natural grading on Hom(E, F) where the even homomorphisms are those that preserve the grading

and the odd homomorphisms are those that reverse the grading

If φ ∈ Hom(E, F) and aA are homogeneous then

That is, the even homomorphisms are both right and left linear whereas the odd homomorphism are right linear but left antilinear (with respect to the grading automorphism).

The set Hom(E, F) can be given the structure of a bimodule over A by setting

begin{align}(acdotphi)(x) &= acdotphi(x)\
(phicdot a)(x) &= phi(acdot x).end{align}

With the above grading Hom(E, F) becomes a supermodule over A whose even part is the set of all ordinary supermodule homomorphisms

In the language of category theory, the class of all supermodules over A forms a category with supermodule homomorphisms as the morphisms. This category is a symmetric monoidal closed category under the super tensor product whose internal Hom functor is given by Hom.

Read more about this topic:  Supermodule

Other articles related to "homomorphisms, homomorphism":

Ringed Space - Morphisms
... data a continuous map f X → Y a family of ring homomorphisms φV OY(V) → OX(f -1(V)) for every open set V of Y which commute with the restriction maps ... Y, then the following diagram must commute (the vertical maps are the restriction homomorphisms) There is an additional requirement for morphisms between locally ...
Bimodule - Further Notions and Facts
... If M and N are R-S bimodules, then a map f M → N is a bimodule homomorphism if it is both a homomorphism of left R-modules and of right S-modules ... Bimodule homomorphisms are the same as homomorphisms of left modules ... if M is an R-S bimodule and L is an T-S bimodule, then the set HomS(M,L) of all S-module homomorphisms from M to L becomes a T-R module in a natural fashion ...
Semigroup Action - Formal Definitions - S-homomorphisms
... Then an S-homomorphism from X to X′ is a map such that for all and ... The set of all such S-homomorphisms is commonly written as ... M-homomorphisms of M-acts, for M a monoid, are defined in exactly the same way ...
Structure (mathematical Logic) - Homomorphisms and Embeddings - Homomorphisms
... Given two structures and of the same signature σ, a (σ-)homomorphism from to is a map which preserves the functions and relations ... The notation for a homomorphism h from to is ... category σ-Hom which has σ-structures as objects and σ-homomorphisms as morphisms ...
Category Of Rings - Properties - Morphisms
... This is a consequence of the fact that ring homomorphisms must preserve the identity ... Some special classes of morphisms in Ring include Isomorphisms in Ring are the bijective ring homomorphisms ... Monomorphisms in Ring are the injective homomorphisms ...