Talk:Graded vector space

General I-graded vector spaces
TomyDuby (talk) 14:04, 12 September 2008 (UTC)
 * Is set I countable?
 * An example would be useful.

N-graded vector space
Does


 * $$V = \bigoplus_{n \in \mathbb{N}} V_n$$

mean


 * $$V = V_1 \bigoplus V_2 \bigoplus \cdots \bigoplus V_n \bigoplus \cdots  $$?

TomyDuby (talk) 14:23, 12 September 2008 (UTC)


 * No it means
 * $$V = V_0 \bigoplus V_1 \bigoplus V_2 \bigoplus \cdots \bigoplus V_n \bigoplus \cdots  $$.

Marc van Leeuwen (talk) 12:39, 25 November 2009 (UTC)

Grading linear maps over M×N is nonsense
The approach taken in the section "Linear maps" seems erroneous to me. After defining a homogeneous linear map of a given bi-degree, the notion of (general) graded linear map is not defined at all, but if one takes the following description (direct sum of spaces of homogeneous linear map) as definition, one gets way to few linear maps if the grading set is infinite (as is is in many important and very basic examples, like polynomial rings). This is because in a graded space elements must be finite sums of homogeneous elements. For instance in a polynomial ring K[X] (graded by degree) the identity map would fail to be graded linear, because it has "components" that are homogeneous linear of degree (n,n) for all n (but it is not the finite sum of those components).

Thus I also do not understand what is said in the paragraph containing graded map, since such maps are not in general (and even almost never) graded linear maps as defined above. Marc van Leeuwen (talk) —Preceding undated comment added 12:54, 25 November 2009 (UTC).