Let $U\subseteq \mathbb{R}^n$ be open and $W:=\oplus_{i\in \mathbb{Z}}W_i$ be a real graded vector space with $dim W_i<\infty$ for all $i$ and $W_0=\{0\}$. I would like to find a reasonable sheaf $\mathcal{O}$ of graded algebras which describes the "graded functions" on $U$. In this paper $\mathcal{O}$ is defined as
$\mathcal{O}(V)=C^{\infty}(V)\otimes S(W)$ for $V\subseteq U$ open, where $S(W)$
is the graded symmetric algebra of $W$, $S(W):=\oplus_{k\in \mathbb{N}_0}W^{\otimes k}/\langle\{u\otimes v-(-1)^{deg(u)deg(v)}v\otimes u\,|\,u,v\in W\:\text{homogenous vectors}\}$.
Clearly, $\mathcal{O}$ is a presheaf of graded algebras (where you use the grading coming from $W$, not the polynomial degree), but I don´t see why $\mathcal{O}$ is a sheaf!
As $S(W)$ can be viewed as the set of polynomials in a graded basis $(w_{\alpha})_{\alpha}$ of $W$, you can think of the sections of $\mathcal{O}$ as polynomials in $(w_{\alpha})_{\alpha}$ with smooth coefficients:
$\mathcal{O}(V)=C^{\infty}(V)[(w_{\alpha})_{\alpha}]$.
Given an open cover $\{U_i\}_i$ of $U$ with compatible sections $p_i\in \mathcal{O}(U_i)$, it can happen that there is no finite upper bound for the polynomial degree of the $p_i$, right?
So how can $S(W)$ can be modified? If you consider the sheafification of $\mathcal{O}$, i.e. formal power series in $(w_{\alpha})_{\alpha}$ that locally look like described above, I don´t know why this should be a sheaf of graded algebras. The same problem occurs if you work with more general power series.
I also considered finite sums of formal power series of the form $\sum_{I}f_I(x)w_{i_1}^{l_1}\cdots w_{i_k}^{l_l}$ where $l_1deg(w_{i_1})+\dots+l_kdeg(w_{i_k})=r$ for a fix $r\in \mathbb{Z}$. But again, why should this be a sheaf?
Many thanks for you help!
Best Answer
It sounds to me like you are using the wrong definition of "sheaf of graded algebras". A sheaf of graded algebras should not be a sheaf of algebras where each algebra happens to have a grading which is preserved by the restriction maps. Instead, a sheaf of graded algebras is a sheaf which takes values in the category of graded algebras. This means that the sheaf gluing condition is interpreted in terms of limits in the category of graded algebras, which are not the same as limits in the category of algebras. In particular, this makes it not a problem if you have sections $p_i\in\mathcal{O}(U_i)$ of unbounded degree: this would mean that no graded algebra $A$ can have an element that maps to all of the $p_i$ under morphisms $A\to\mathcal{O}(U_i)$, and so $\mathcal{O}(U)$ does not have to have such an element in order to satisfy the gluing condition (since $\mathcal{O}(U)$ just has to be the universal graded algebra with compatible morphisms to each $\mathcal{O}(U_i)$).
Generally, all of this is easier to understand if you define graded objects as sequences rather than single objects together with a direct sum decomposition. That is, a graded vector space should be defined as a sequence $(V_n)$ of vector spaces (rather than as a single vector space $\bigoplus V_n$ equipped with a direct sum decomposition). There is then a tensor product monoidal structure on the category of graded vector spaces, and a graded algebra is monoid object with respect to this monoidal structure. This perspective then has the advantage that the forgetful functor from graded algebras to graded sets (i.e., sequences of sets) preserves limits. So, limits in the category of graded algebras (which are what you care about for sheaves of graded algebras) are computed by just taking the ordinary limits in each graded piece separately. Concretely, this means that to check the gluing condition for a sheaf of graded algebras, you just have to check it for homogeneous elements of a fixed degree. (Or, if you like, this means that a sheaf of graded algebras is the same thing as a graded sheaf of algebras, i.e. a sequence of sheaves of vector spaces together with a monoid structure with respect to the tensor product of sequences of sheaves of vector spaces).
(In contrast, the forgetful functor taking a graded algebra to its single underlying set which is the direct sum of the graded pieces does not preserve limits! This is basically because infinite products do not play well with infinite direct sums, so an infinite cartesian product of direct sums does not have a natural direct sum decomposition. This means that limits in the category of graded algebras cannot be computed by thinking "elementwise", if you are considering elements of the direct sum.)