The node is standard syntomic

The local model R[x, y] ⧸ (xy - a) of a node is a standard syntomic R-algebra of relative dimension one, for every commutative ring R and every a ∈ R. Its presentation has two generators and the single relation xy - a; it is free, hence flat, over R; and its fibre over a prime p of R is the node κ(p)[x, y] ⧸ (xy - ā) over the residue field, which has Krull dimension one whether or not ā vanishes. This is the local shape of a family of curves with at worst nodal singularities over a discrete valuation ring with uniformizer π, where a = πⁿ.

Main results

• TauCeti.NodeAlgebra.isStandardSyntomicOfRelativeDimension: R[x, y] ⧸ (xy - a) is standard syntomic of relative dimension one over R.

References

• The Stacks Project, Example 55.14.1, Tag 0CDC

instance TauCeti.NodeAlgebra.isStandardSyntomicOfRelativeDimension {R : Type u_1} [CommRing R] (a : R) : Algebra.IsStandardSyntomicOfRelativeDimension 1 R (NodeAlgebra R a)

The local model R[x, y] ⧸ (xy - a) of a node is standard syntomic of relative dimension one over R.