We present a sharpened version of the Cohen–Gabber theorem for equicharacteristic, complete local domains $(A,\mathfrak{m},k)$ with algebraically closed residue field and dimension $d>0$. Namely, we show that for any prime number $p$, $\operatorname{Spec}A$ admits a dominant, finite map to $\operatorname{Spec}k[[X_{1},\ldots,X_{d}]]$ 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 $\mathbf{G}_{0}(A)$ by the Koszul complex.
References
D. Eisenbud, Commutative algebra: With a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150, Springer, New York, 1995. MR1322960 10.1007/978-1-4612-5350-1 0819.13001 D. Eisenbud, Commutative algebra: With a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150, Springer, New York, 1995. MR1322960 10.1007/978-1-4612-5350-1 0819.13001
W. Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer, Berlin, 1998. MR1644323 10.1007/978-1-4612-1700-8 W. Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer, Berlin, 1998. MR1644323 10.1007/978-1-4612-1700-8
M. Hochster and C. Huneke, Tight closure, invariant theory, and the Briançon–Skoda theorem, J. Amer. Math. Soc. 3 (1990), no. 1, 31–116. MR1017784 10.2307/1990984 0701.13002 M. Hochster and C. Huneke, Tight closure, invariant theory, and the Briançon–Skoda theorem, J. Amer. Math. Soc. 3 (1990), no. 1, 31–116. MR1017784 10.2307/1990984 0701.13002
M. Hochster and C. Huneke, Localization and test exponents for tight closure, Michigan Math. J. 48 (2000), 305–329. Dedicated to William Fulton on the occasion of his 60th birthday. MR1786493 0993.13003 10.1307/mmj/1030132721 euclid.mmj/1030132721
M. Hochster and C. Huneke, Localization and test exponents for tight closure, Michigan Math. J. 48 (2000), 305–329. Dedicated to William Fulton on the occasion of his 60th birthday. MR1786493 0993.13003 10.1307/mmj/1030132721 euclid.mmj/1030132721
C. Huneke and I. Swanson, Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, vol. 336, Cambridge University Press, Cambridge, 2006. MR2266432 1117.13001 C. Huneke and I. Swanson, Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, vol. 336, Cambridge University Press, Cambridge, 2006. MR2266432 1117.13001
K. Kurano and K. Shimomoto, An elementary proof of Cohen–Gabber theorem in the equal characteristic $p > 0$ case; available at \arxivurlarXiv:1510.03573 [math.AC]. 1510.03573 K. Kurano and K. Shimomoto, An elementary proof of Cohen–Gabber theorem in the equal characteristic $p > 0$ case; available at \arxivurlarXiv:1510.03573 [math.AC]. 1510.03573
H. Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR0879273 H. Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR0879273
Moreau, V. Pilloni, M. Raynaud, J. Riou, B. Stroh, M. Temkin and W. Zheng, Travaux de Gabber sur l'uniformisation locale et la cohomologie étale des schémas quasi-excellents, Séminaire à l'École Polytechnique 2006–2008. [Seminar of the polytechnic school 2006–2008], Société Mathématique de France, Paris, 2014. With the collaboration of Frédéric Déglise, Alban Moreau, Vincent Pilloni, Michel Raynaud, Joël Riou, Benoît Stroh, Michael Temkin and Weizhe Zheng, Astérisque No. 363–364 (2014). MR3309086 Moreau, V. Pilloni, M. Raynaud, J. Riou, B. Stroh, M. Temkin and W. Zheng, Travaux de Gabber sur l'uniformisation locale et la cohomologie étale des schémas quasi-excellents, Séminaire à l'École Polytechnique 2006–2008. [Seminar of the polytechnic school 2006–2008], Société Mathématique de France, Paris, 2014. With the collaboration of Frédéric Déglise, Alban Moreau, Vincent Pilloni, Michel Raynaud, Joël Riou, Benoît Stroh, Michael Temkin and Weizhe Zheng, Astérisque No. 363–364 (2014). MR3309086
Z. Oscar and P. Samuel, Commutative algebra. Vol. II, Graduate Texts in Mathematics, vol. 29, Springer, New York, 1975. Reprint of the 1960 edition. MR0120249 Z. Oscar and P. Samuel, Commutative algebra. Vol. II, Graduate Texts in Mathematics, vol. 29, Springer, New York, 1975. Reprint of the 1960 edition. MR0120249
J.-P. Serre, Algèbre locale. Multiplicités, Cours au Collège de France, 1957–1958, rédigé par Pierre Gabriel, 2nd ed., Lecture Notes in Mathematics, vol. 11, Springer, Berlin, 1965. MR0201468 J.-P. Serre, Algèbre locale. Multiplicités, Cours au Collège de France, 1957–1958, rédigé par Pierre Gabriel, 2nd ed., Lecture Notes in Mathematics, vol. 11, Springer, Berlin, 1965. MR0201468
