Isomorphism of $S[t^2]$-modules

Question

I get lost reading a passage.

Let $S=k[x_1,\dots,x_n]$ be a standard graded polynomial ring and let $I$ be a homogeneous ideal of $S$. I would like to understand why there is an isomorphism $$S[It,t^2]\cong S[t^2]\oplus\Big(S[t^2]\otimes_S It\Big)$$ as $S[t^2]$-modules, where $t$ is a new variable and $\deg(t)=1$.

I'm not even sure of why $S[t^2]\otimes_S It$ is an $S[t^2]$-module. Given $$f\in S[t^2]\quad\text{ and}\quad\sum g_i\otimes h_i\in S[t^2]\otimes_S It,$$ is it right to define $$f\cdot\left(\sum g_i\otimes h_i\right):=\sum(fg_i)\otimes h_i\ ?$$ And if this is OK, could anyone give me a hint for the isomorphism above? Thank you very much!

Answer 1

For your second question, it is a general fact that if you have an R-algebra S (that is, a ring homomorphism R -> S), and an R-module M, then $S \otimes_R M$ is an $S$ module defined in a natural way $s \cdot (s' \otimes m) = (ss') \otimes m$.

Answer 2

Your description of the $S[t^2]$-module structure on $S[t^2]\otimes_S It$ is correct. More generally, is $B$ is an $A$-algebra and $M$ is an $A$-module, $B\otimes_A M$ is a $B$-module in the same way.

As a hint for the isomorphism, $S[It,t^2]$ is just those polynomials whose odd-degree coefficients are all in $I$. This is the direct sum of two submodules: one consisting of polynomials with only even-degree terms, and one consisting of polynomials with only odd-degree terms. Identify these submodules with the summands of $S[t^2]\oplus S[t^2]\otimes_S It$.