Analytic Spread of Q-Divisorial Filtrations via Regular Alterations in Commutative Algebra
A criterion that identifies exactly when a Q-divisorial filtration on an equidimensional local ring has maximal analytic spread: the maximal ideal is an associated prime of R/I_n for some n. Students learn how the classical ideal-theoretic statement is lifted to whole graded filtrations, and how replacing resolution of singularities with projective regular alterations removes the equicharacteristic-zero hypothesis. It also covers why the first-difference length function of such a filtration is bounded but need not converge, and how symbolic powers of a prime can accumulate infinitely many distinct Rees valuations.
The Rees algebra and analytic spread of a divisorial filtration
A graded filtration I = {I_n} of a Noetherian local ring R satisfies I_m I_n contained in I_{m+n}; its Rees algebra is R[I] = the direct sum of the I_n, and its analytic spread is l(I) = dim R[I]/m_R…