14 problems

Let $A\to B$ be a local homomorphism of artinian local rings, and let $\operatorname{edim} A$ and $\operatorname{edim} B$ denote their embedding dimensions. de Smit's conjecture. I…

Let $h:[n]\to[n]$ be a Hessenberg function, let $\mathsf{x}\in\mathfrak{gl}_n(\mathbb{C})$, and let $\varphi_{\mathsf{x},h}$ be the one-parameter family constructed from the associ…

Let $D$ be either of the domains … or … Here ${\operatorname{Int}}(D)$ denotes the ring of integer-valued polynomials on $D$, regarded as a $D$-module. Non-flatness conjecture. For…

Let $\fg$ be a Kac–Moody algebra, $\fb$ a Borel subgroup, $\bC$ the curve, and $\mu$ a dominant coweight. Let $\on{Maps}^\mu(\bC,\CG_{\fg,\fb})$ be the corresponding space of maps,…

Let $W$ be the affine Weyl group, let $x\in W$, and let $[w,\mathbf{r}]\in\frak{P}$. Let $\calI(x)^+_{w,\mathbf{r}}$ be the corresponding space and let $\frak{c}_{w,\mathbf{r}}$ be…

Formal K-flatness conjecture. Formal K-flatness is equivalent to K-flatness.

Let $k$ be a field, let $\bb V$ and $\bb W$ be $k$-vector spaces with $\dim(\bb W)\leq m$, and let $P:\bb V\to k$ be a polynomial of degree $d$. For an affine map…

Let $A\to B$ be a morphism of Artin local rings with the same embedding dimension, and let $M$ be a $B$-module that is flat over $A$. de Smit's conjecture. The $B$-module $M$ is fl…

Let $A$ be the base ring and let $G\in\operatorname{Gr}^{+}_{A}$ be a group scheme of additive type. Write $G=\operatorname{Spec}(B_G)$ and let $\operatorname{Prim}(B_G)$ denote th…

Pappas–Rapoport flatness conjecture. The scheme $P$ is flat over

Spin local-model flatness conjecture. The spin local model $M^{\mathrm{spin}}_{G,\{\mu\},{\mathcal L}}$ is flat over its base.

Let $f\colon X\to Y$ be a morphism essentially of finite type, where $Y$ is smooth over a field. For a positive integer $d$, let … be the $d$-fold fiber product, let…

Let $D$ be an integral domain. An H-domain is a domain in which every $t$-maximal ideal is divisorial, and an ideal is faithfully flat when it is faithfully flat as a $D$-module. G…

Vasconcelos's conjecture. $A$ is $R$-flat if and only if its $n$-fold tensor power