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Let $h^{n,ord}(Np^\infty)$ be the cuspidal nearly ordinary Hecke algebra of tame level $N$. For $N \geq 4$, we know that the Hecke algebra is the generic fibre of the Hecke-Hilbert Eigenvariety and so $h^{n,ord}(Np^\infty)$ is necessary an equidimensional ring (since the Eigenvariety is equidimensional by construction).

Is there an equivalent result in the case where the tame level is one ??

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  • $\begingroup$ It is absurd to suggest that equidimensionality of the nearly ordinary Hecke algebra is somehow a consequence of the theory of eigenvarieties. This follows directly from Hida's construction (for GL2 over any totally real field) and was known a good decade or more before eigenvarieties came along. $\endgroup$
    – David Loeffler
    Commented Sep 12, 2016 at 6:38
  • $\begingroup$ For $GL_2$ over $\mathbb{Q}$, the ordinary Hecke algebra $h^{ord}(Np^{\infty})$ is finite and free-torsion over the Iwasawa algebra and hence it is flat over the Iwasawa algebra and using the Going down we deduce that the ordinary Hecke algebra is equidimensional of dimension $2$ but for $GL_2$ over a totally real field, we know that $h^{n,ord}(Np^{\infty})$ is finite and torsion-free over the Iwasawa algebra of $[F:\mathbb{Q}]+1$ variables but we don't know if $h^{n,ord}(Np^{\infty})$ is flat over the Iwasawa algebra. $\endgroup$
    – Adel BETINA
    Commented Sep 12, 2016 at 12:33
  • $\begingroup$ Isn't a finite torsion-free algebra over the Iwasawa algebra necessarily equidimensional? $\endgroup$
    – David Loeffler
    Commented Sep 12, 2016 at 13:16

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