# If $R$ and $S$ are two graded rings, is there a name for the construction $\oplus_{n \geq 0} R_n \otimes_{\mathbb{Z}} S_n$?

If $R$ and $S$ are two graded rings, is there a name for the construction $\oplus_{n \geq 0} R_n \otimes_{\mathbb{Z}} S_n$?

This gives a graded ring, but it is not quite the tensor product since we are only allowing tensors between two elements of the same degree.

This comes up when twisting a sheaf of quasi-coherent graded algebras $A$ on a scheme by a line bundle $L$ : $A' = \oplus_{n \geq 0} A_n \otimes L^{\otimes n}$. (Ravi exercise 17.2.G)

I am wondering if this object has a standard name, mostly so that I can look up properties about it to sanity check if necessary.

• Segre product. ${}{}$ – user26857 Jan 17 '16 at 23:02
• @user26857 Oh, I see - presumably because of the Segre embedding. In particular if $L = O(1,1) = \pi_n^* (O(1)) \otimes \pi_m^*(O(1))$ on $P^n \times P^m$, the line bundle corresponding to the Segre embedding, then $\oplus L^{\otimes n}$ is the Segre product of $\oplus \pi_n^* (O(1))$ and $\oplus \pi_m^* (O(1))$. Thanks! – Lorenzo Jan 17 '16 at 23:17