arXiv subjects

Chris Skalit

Publications and source records attributed to Chris Skalit.

3 recordsLinked to original sources

arXiv2018

Positivity of Intersection Multiplicity Over a Two-Dimensional Base

We show that Serre's Intersection Multiplicity Conjecture holds for a formal power series ring A over a complete, two-dimensional regular local ring R. From this, we deduce the corresponding result for the local rings of any scheme X which is a smooth extension of a regular, two-dimensional base Y. We also investigate the connection between intersection multiplicity and transversality in the unramified setting via a local analysis on the blowup.

math.AC↗

arXiv2017

Koszul Factorization and the Cohen-Gabber Theorem

We present a sharpened version of the Cohen-Gabber theorem for equicharacteristic, complete local domains (A,m,k) with algebraically closed residue field and dimension d > 0. Namely, we show that for any prime number p, Spec(A) admits a dominant, finite map to Spec(k[[X_1,...,X_d]]) with generic degree relatively prime to p. Our result follows from Gabber's original theorem, elementary Hilbert-Samuel multiplicity theory, and a "factorization" of the map induced on the Grothendieck group G_0(A) by the Koszul complex.

math.AC↗

arXiv2014

Intersection Multiplicity of Serre in the Unramified Case

We describe here some recent progress pertaining to the Serre Intersection Multiplicity Conjecture. In particular, we show that if A is an unramified regular local ring, then just as in the equicharacteristic case, the intersection multiplicity of two complimentary-dimensional modules is bounded below by the product of their Hilbert-Samuel multiplicities. We also explain, in terms of the blowup of Spec(A) at the closed point, the geometric significance of achieving this lower bound.

math.AC↗