Let I and J be nonzero ideals in two Noetherian algebras A and B over a field k. Let I+J\documentclass[12pt]{minimal}
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\begin{document}$$I+J$$\end{document} denote the ideal generated by I and J in A⊗kB\documentclass[12pt]{minimal}
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\begin{document}$$A\otimes _k B$$\end{document}. We prove the following expansion for the symbolic powers: (I+J)(n)=∑i+j=nI(i)J(j).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} (I+J)^{(n)} = \sum _{i+j = n} I^{(i)} J^{(j)}. \end{aligned}$$\end{document}If A and B are polynomial rings and if char(k)=0\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{char}\,}}(k) = 0$$\end{document} or if I and J are monomial ideals, we give exact formulas for the depth and the Castelnuovo–Mumford regularity of (I+J)(n)\documentclass[12pt]{minimal}
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\begin{document}$$(I+J)^{(n)}$$\end{document}, which depend on the interplay between the symbolic powers of I and J. The proof involves a result of independent interest which states that the induced map ToriA(k,I(n))→ToriR(k,I(n-1))\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Tor}\,}}_i^A(k,I^{(n)}) \rightarrow {{\,\mathrm{Tor}\,}}_i^R(k,I^{(n-1)})$$\end{document} is zero for any homogeneous ideal I and i≥0\documentclass[12pt]{minimal}
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\begin{document}$$i \ge 0$$\end{document}, n≥0\documentclass[12pt]{minimal}
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\begin{document}$$n \ge 0$$\end{document}. We also investigate other properties and invariants of (I+J)(n)\documentclass[12pt]{minimal}
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\begin{document}$$(I+J)^{(n)}$$\end{document} such as the equality between ordinary and symbolic powers, the Waldschmidt constant and the Cohen–Macaulayness.
