Criteria for complete intersections

We establish two criteria for certain local algebras to be complete intersections. These criteria play an important role in A. Wiles’s proof that all semi-stable elliptic curves over Q are modular. Introduction In this paper we discuss two results in commutative algebra that are used in A. Wiles’s proof that all semi-stable elliptic curves over Q are modular [11]. We first fix some notation that is used throughout this paper. Let O be a complete Noetherian local ring with maximal ideal mO and residue field k = O/mO. Suppose that we have a commutative triangle of surjective homomorphisms of complete Noetherian local O-algebras: R φ −→ T πR ↘ ↙ πT O. Assume that T is a finite flat O-algebra, i.e., that T is finitely generated and free as an O-module. In the applications in Wiles’s proof O is a discrete valuation ring, R is a deformation ring, T is a Hecke algebra and πT is the homomorphism associated to a certain eigenform. We prove two distinct criteria, formulated as Criterion I and Criterion II below, which give sufficient conditions to conclude that φ is an isomorphism and that R and T are complete intersections. We say that a local O-algebra that is finitely generated as an O-module is a complete intersection over O if it is of the form O[[X1, . . . , Xn]]/(f1, . . . , fn), with f1, . . . , fn ∈ O[[X1, . . . , Xn]]. We first state Criterion I. We put IR = kerπR and IT = kerπT . The congruence ideal of T is defined to be the O-ideal ηT = πT AnnT (IT ). Criterion I. Suppose that O is a complete discrete valuation ring and that ηT 6= 0. Then lengthO(IR/I 2 R) ≥ lengthO(O/ηT ). Moreover, equality holds if and only if φ is an isomorphism between complete intersections over O. 2 B. de Smit, K. Rubin, R. Schoof Wiles used a slightly weaker form of this criterion, where T is assumed to be Gorenstein, to show that certain “non-minimal” deformation rings are isomorphic to Hecke algebras [4]. The present version, without the Gorenstein condition, is due to H.W. Lenstra [6]. In Section 3 we give an alternative argument for Criterion I that was found by the first and the third author. Criterion I is an easy consequence of the following result, which holds without any conditions on O or ηT . Theorem. The map φ is an isomorphism between complete intersections over O if and only if φFitR(IR) 6⊂ mOT . Here FitR(IR) denotes the R-Fitting ideal of IR. Fitting ideals are instrumental in the proof of Criterion I. We recall their definition and basic properties in Section 1. A crucial special case of the theorem can already be found in a 1969 paper of H. Wiebe [10]; see also [1, Thm. 2.3.16]. More precisely, Wiebe’s result covers the case that O = k is a field, and φ is the identity on R = T . The statement is then that T is a complete intersection over k if and only if the Fitting ideal of its maximal ideal is non-zero. For the proof of Criterion I we need some properties of complete intersections that go back to J.T. Tate [8]. In Section 2 we formulate Tate’s result and prove it using Koszul complexes. These are discussed in Section 1. As a consequence we find that complete intersections have the Gorenstein property. The Gorenstein property does not occur in our proof of Criterion I, but we briefly discuss its significance in our context at the end of Section 2. In order to formulate Criterion II, assume that char(k) = p > 0, and let n ≥ 1. The ring O[[S1, . . . , Sn]] is filtered by the ideals Jm, with m ≥ 0, given by Jm = (ωm(S1), . . . , ωm(Sn)), where ωm(S) denotes the polynomial (1 + S) m − 1. Note that J0 = (S1, . . . , Sn). Criterion II. Suppose that for every m > 0 there is a commutative diagram of O-algebras O[[S1, . . . , Sn]] −→ Rm φm −→ Tm y y