Assume that all given rings are commutative, associative with identity, and Noetherian.
A recent result by L. Ein, R. Lazarsfeld and K. Smith [Invent. Math. 144, 241–252 (2001; Zbl 1076.13501)] established the following fact about the behavior of symbolic powers of ideals in affine regular rings of equal characteristic 0:
If \(h\) is the largest height of an associated prime of \(I\), then \(I^{(hn)}\subseteq I^n\) for all \(n\geq 0\). Here, if \(W\) is the complement of the union of the associated primes of \(I\), \(I^{(t)}\) denotes the contraction of \(I^tR_W\) to \(R\), where \(R_W\) is the localization of \(R\) at the multiplicative system \(W\).
The proof of this result needed the theory of multiplier ideals, including an asymptotic version and, in particular, required resolution of singularities as well as vanishing theorems.
In the present paper, Hochster and Huneke give stronger results that are proved by methods, in some ways, more elementary. The obtained results are valid in both equal characteristic 0 and in positive prime characteristic \(p\), but depend on reduction to characteristic \(p\). The authors use tight closure methods and, in consequence, they need neither resolution of singularities nor vanishing theorems that may fail in positive characteristic.
The main results in all characteristics obtained by the authors in this paper may be summarized in the following theorem. Denote by \(I^*\) the tight closure of the ideal \(I\):
Theorem. Let \(R\) be a Noetherian ring containing a field. Let \(I\) be any ideal of \(R\), and let \(h\) be the largest height of any associated prime of \(I\).
(a) If \(R\) is regular, \(I^{(hn+kn)}\subseteq (I^{(k+1)})^n\) for all positive \(n\) and nonnegative \(k\). In particular, \(I^{(hn)}\subseteq I^n\) for all positive integers \(n\).
(b) If \(I\) has finite projective dimension then \(I^{(hn)}\subseteq (I^n)^*\) for all positive integers \(n\).
(c) If \(R\) is finitely generated, geometrically reduced (in characteristic 0, this simply means that \(R\) is reduced) and equidimensional over a field \(K\), and locally \(I\) is either 0 or contains a nonzerodivisor (this is automatic if \(R\) is a domain), then, with \(J={\mathcal J}(R/K)\), for every nonnegative integer \(k\) and positive integer \(n\), we have that \(J^nI^{(hn+kn)}\subseteq ((I^{(k+1)})^n)^*\) and \(J^{n+1}I^{(hn+kn)}\subseteq (I^{(k+1)})^n\). In particular, we have that \(J^nI^{(hn)}\subseteq (I^n)^*\) and \(J^{n+1}I^{(hn)}\subseteq I^n\) for all positive integers \(n\).
The Jacobian ideal \({\mathcal J}(R/K)\) used in the last part of theorem is defined as follows: Let \(A\) be a reduced Noetherian ring with total quotient ring \(T={\mathcal T}(A)\), so that \(T\) is a finite product of fields. Let \(R\) be a finitely generated \(A\)-algebra such that \(R\) is torsion-free over \(A\) (i.e. nonzerodivisors of \(A\) are nonzerodivisors of \(R\)) and such that \(T\otimes_A R\) is equidimensional of dimension \(d\) and geometrically reduced. We then define the Jacobian ideal \({\mathcal J}(R/A)\) as follows. Choose a finite presentation of \(R\) over \(A\), say \(R\cong A[x_1,\ldots,x_n]/(f_1,\ldots,f_m)\), and let \({\mathcal J}(R/A)\) denote the ideal of \(R\) generated by the images of the \(n-d\) size minors of the Jacobian matrix \(\left(\partial f_j/\partial x_i \right)\).
The results stated in the theorem above are also valid if one defines \(h\) instead to be the largest analytic spread of \(IR_P\) for any associated prime \(P\) of \(I\). Recall that, in a Noetherian local ring \((R,m,K)\) with maximal ideal \(m\) and residue field \(K\), the analytic spread \(a(I)\) of an ideal \(I\subseteq m\) is the Krull dimension of the ring \[ K\otimes_R \text{gr}_I R \cong K \oplus I/mI \oplus I^2/mI^2 \oplus \cdots \oplus I^k/mI^k\oplus\cdots \]