I’m writing to ask a question about Mazur’s dimensional conjecture in Lemma 7.5 of the paper [Galatius S, Venkatesh A. Derived Galois deformation rings. Advances in Mathematics. 2018 Mar 17;327:470-623].
Let $p$ be a prime, $K$ a number field, $S$ a finite set of primes of $K$ containing any places above $p$ and infinite places, and $G_{K,S}$ the Galois group of the maximal extension of $K$ unramified outside $S$. Then Mazur's dimension conjecture on universal deformation ring states the following: Let $\bar{\rho}:G_{K,S}\to \text{GL}_{n}(\mathbf{F}_p)$ be an absolutely irreducible mod-$p$ representation, then the (usual) universal deformation ring $R_{\bar{\rho}}$ is complete intersection ring of dimension
$$ h^{1}-h^{2}$$ over $\mathbb{Z}_p$, where $h^{i}:=\dim_{\mathbf{F}_p}~H^{i}(G_{K,S},\text{ad}(\bar{\rho}))$ for $ i=1,2, \text{ad}(\bar{\rho})$ is the adjoint representation of $\bar{\rho}$ and $\mathbf{Z}_p$ is the ring of $p$-adic integers. My questions are the following:
By Lemma 7.5 (and it proof) in the paper, it seems that the above conjecture is equivalent to the following weaker statement: the minimal number of relations in a minimal presentation of $R_{\bar{\rho}} $ is equal to $h^{2}$. Is it correct?
This surprised me, and I may have understood something wrong. Thank you for correcting me.
