Connectedness in Codimension One and the Non-$S_2$ Locus

Abstract

We formulate a structural principle for finite $S_2$-objects: coherent $S_2$-sheaves and finitely generated graded $S_2$-modules decompose canonically according to the connected components in codimension $1$ of their support. This gives criteria relating indecomposability of $S_2$-objects to connectedness in codimension $1$ of their supports, and extends the Hochster--Huneke correspondences for complete local rings between connectedness in codimension $1$, indecomposability of canonical modules, and localness of the $S_2$-ifications. As a consequence, if $A$ is a local ring admitting a canonical module $ω_A$, there are canonical decompositions of both $ω_A$ and the $S_2$-ification $\operatorname{End}_A(ω_A)$ whose indecomposable summands are the canonical modules and $S_2$-ifications of the quotient rings associated to the connected components in codimension $1$. We then apply this viewpoint to the non-$S_2$ locus. For $A$ equidimensional and unmixed, this locus is naturally realized as $\operatorname{Supp}_A C$ via the $S_2$-ification sequence $0 \to A \to \operatorname{End}_A(ω_A) \to C \to 0$. The natural map between deficiency modules $K^{\dim C+1}(A)\to K^{\dim C}(C)$ identifies the canonical module $K^{\dim C}(C)$ with the $S_2$-hull of $K^{\dim C+1}(A)$. Under suitable conditions, this allows codimension-$1$ connectedness of the non-$S_2$ locus to be detected by the deficiency module $K^{\dim C+1}(A)$. We illustrate the theory with examples and apply it to codimension $2$ lattice ideals, obtaining connectedness-in-codimension-$1$ results for the non-$S_2$ loci of certain toric and lattice rings.