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Singularities of ordinary deformation rings. (English) Zbl 1456.11096

Let \(F\) be a finite extension of \({\mathbb Q}_p\), \(k\) a finite field of characteristic \(p\) and \(V_0\) a finite dimensional \(k\)-vector space with a continuous representation of the absolute Galois group. Let \(\mathcal{O}\) be a finite totally ramified extension of \(W(k).\) In this situation there exists a universal ring \(R^{\mathrm{univ}}\) that parametizes framed deformations of \(V_0\) to Artinian \(\mathcal{O}\)-algebras. Let \(X\subset {\mathrm{MaxSpec}}(R^{\mathrm{univ}}[1/p])\) be the locus corresponding to ordinary representations, i.e., those that fulfill the following condition: \[ V|_{I_F}=\begin{pmatrix} \chi & *\\ 0 &1 \end{pmatrix}, \] where \(\chi\) is the cyclotomic character. Then it is known that \(X\) is Zariski closed and therefore equal to \( {\mathrm{MaxSpec}}(R[1/p])\) for a unique \(\mathcal O\)-flat reduced quotient of \(R^{\mathrm{univ}}.\) Some work devoted to understanding \(R\) was done by M. Kisin [Invent. Math. 178, No. 3, 587–634 (2009; Zbl 1304.11043)]. In the paper under review, the author presents an original method which allows him to describe the functor of points of \(R\) as well as obtain some new results about \(R.\) For example, he shows that a minor modification of \(R\) is normal and Cohen-Macaulay but usually not Gorenstein.

MSC:

11F80 Galois representations
11S23 Integral representations

Citations:

Zbl 1304.11043
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References:

[1] Bruns, W., Herzog, J.: Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics, vol. 39. Cambridge University Press, Cambridge (1993) · Zbl 0788.13005
[2] Bărcănescu, Ş., Manolache, N.: Betti numbers of Segre-Veronese singularities. Rev. Roumaine Math. Pures Appl. 26(4), 549-565 (1981) · Zbl 0465.13006
[3] Conrad, B.: Structure of ordinary-crystalline deformation ring for \[\ell =p\] ℓ=p. Unpublished notes. http://math.stanford.edu/ conrad/modseminar/pdf/L21.pdf · Zbl 0465.13006
[4] Calegari, F., Geraghty, D.: Modularity lifting beyond the Taylor-Wiles method, preprint · Zbl 1476.11078
[5] Dieudonné, J., Grothendieck, A.: Eléments de géométrie algébrique, Publ. Math. IHES, 4, 8, 11, 17, 20, 24, 28, 32, 1960-7 · Zbl 0203.23301
[6] Hartshorne, R.: Residues and Duality. Lecture Notes in Mathematics, vol. 20. Springer, New York (1966) · Zbl 0212.26101
[7] Kisin, M.: Moduli of finite flat group schemes, and modularity. Ann. Math. 170(3), 1085-1180 (2009) · Zbl 1201.14034 · doi:10.4007/annals.2009.170.1085
[8] Kisin, M.: Modularity of 2-adic Barsotti-Tate representations. Invent. Math. 178(3), 587-634 (2009) · Zbl 1304.11043 · doi:10.1007/s00222-009-0207-5
[9] Kisin, M.: Modularity of 2-dimensional Galois representations. Curr. Dev. Math. 2005, 191-230 (2005) · Zbl 1218.11056 · doi:10.4310/CDM.2005.v2005.n1.a7
[10] Khare, C., Wintenberger, J.-P.: Serre’s modularity conjecture. II. Invent. Math. 178(3), 505-586 (2009) · Zbl 1304.11042 · doi:10.1007/s00222-009-0206-6
[11] Weyman, J.: Cohomology of Vector Bundles and Syzygies, Cambridge Tracts in Mathematics, vol. 149. Cambridge University Press, Cambridge (2003) · Zbl 1075.13007 · doi:10.1017/CBO9780511546556
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