# Extensions of maps between graded modules

Let $R$ be a connected graded ring (like $R=\mathbb R[x_1,\dots,x_d]$ with the usual grading) and let $N \subset R^{\oplus n}$ be a graded submodule, i.e. $$N= \bigoplus_{i \in \mathbb N} (N \cap R^{\oplus n})_i,$$ which is also finitely generated free as an $R$-module.

I am thinking for example of the image $\mathbb R[x_1,\dots,x_d]^{\oplus k}$ in $\mathbb R[x_1,\dots,x_d]^{\oplus n}$ given by a suitable $k$-tuple of homogenous vectors in $\mathbb R[x_1,\dots,x_d]^{\oplus n}$ of various degrees.

Question: Let $\varphi \colon N \to R^{\oplus n}$ be a homomorphism of degree $1$. Under which conditions can we extend $\varphi$ to a homomorphism $\tilde \varphi\colon R^n \to R^n$ of degree $1$?

The obstruction corresponds to a class in ${\rm Ext}^1_{R}(R^{\oplus n}/N,R^{\oplus n})$ of degree zero.

Is there any general result about classes of degree zero in such such Ext-groups over graded rings? Is there a geometric meaning of classes in graded Ext-groups, such as ${\rm Ext}^1_{R}(R^{\oplus n}/N,R)$?

Is there a good reference for the special properties of graded Ext-groups?

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