Symbolic powers of sums of ideals

By Huy Tài Hà, H. D. Nguyen, Ngô Viêt Trung, Tran Nam Trung. Mathematische Zeitschrift, 2017

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

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/033f02ffca7a0d41b6464f72682bb2ab318c27a0