Editing Graded vector space (section)

==Homomorphisms==
{{Anchor|Graded linear map|Linear maps}}
{{redirect|Homogeneous linear map|the more general concept|Graded module homomorphism}}
For general index sets ''I'', a [[linear map]] between two ''I''-graded vector spaces {{nowrap|''f'' : ''V'' → ''W''}} is called a '''graded linear map''' if it preserves the grading of homogeneous elements. A graded linear map is also called a '''homomorphism''' (or '''morphism''') of graded vector spaces, or '''homogeneous linear map''':

:<math>f(V_i)\subseteq W_i</math>   for all ''i'' in ''I''.

For a fixed [[field (mathematics)|field]] and a fixed index set, the graded vector spaces form a [[category (mathematics)|category]] whose [[morphism]]s are the graded linear maps.

When ''I'' is a [[commutative monoid|commutative]] [[monoid]] (such as the natural numbers), then one may more generally define linear maps that are '''homogeneous''' of any degree ''i'' in ''I'' by the property

:<math>f(V_j)\subseteq W_{i+j}</math>   for all ''j'' in ''I'',

where "+" denotes the monoid operation. If moreover ''I'' satisfies the [[cancellation property]] so that it can be [[embedding|embedded]] into an [[abelian group]] ''A'' that it generates (for instance the integers if ''I'' is the natural numbers), then one may also define linear maps that are homogeneous of degree ''i'' in ''A'' by the same property (but now "+" denotes the group operation in ''A''). Specifically, for ''i'' in ''I'' a linear map will be homogeneous of degree −''i'' if

:<math>f(V_{i+j})\subseteq W_j</math>   for all ''j'' in ''I'', while
:<math>f(V_j)=0\,</math>   if {{nowrap|''j'' − ''i''}} is not in ''I''.

Just as the set of linear maps from a vector space to itself forms an [[associative algebra]] (the [[endomorphism algebra|algebra of endomorphisms]] of the vector space), the sets of homogeneous linear maps from a space to itself – either restricting degrees to ''I'' or allowing any degrees in the group ''A'' – form associative [[graded algebra]]s over those index sets.