Continuous closure, axes closure, and natural closure

Abstract

Let R R be a reduced affine C \mathbb {C} -algebra with corresponding affine algebraic set X X . Let C ( X ) \mathcal {C}(X) be the ring of continuous (Euclidean topology) C \mathbb {C} -valued functions on X X . Brenner defined the continuous closure I c o n t I^{\mathrm {cont}} of an ideal I I as I C ( X ) ∩ R I\mathcal {C}(X) \cap R . He also introduced an algebraic notion of axes closure I a x I^{\mathrm {ax}} that always contains I c o n t I^{\mathrm {cont}} , and asked whether they coincide. We extend the notion of axes closure to general Noetherian rings, defining f ∈ I a x f \in I^{\mathrm {ax}} if its image is in I S IS for every homomorphism R → S R \to S , where S S is a one-dimensional complete seminormal local ring. We also introduce the natural closure I ♮ I^{\natural } of I I . One of many characterizations is I ♮ = I + { f ∈ R : ∃ n > 0 w i t h f n ∈ I n + 1 } I^{\natural } = I + \{f \in R: \exists n >0 \mathrm {\ with\ } f^n \in I^{n+1}\} . We show that I ♮ ⊆ I a x I^{\natural } \subseteq I^{\mathrm {ax}} and that when continuous closure is defined, I ♮ ⊆ I c o n t ⊆ I a x I^{\natural } \subseteq I^{\mathrm {cont}} \subseteq I^{\mathrm {ax}} . Under mild hypotheses on the ring, we show that I ♮ = I a x I^{\natural } = I^{\mathrm {ax}} when I I is primary to a maximal ideal and that if I I has no embedded primes, then I = I ♮ I = I^{\natural } if and only if I = I a x I = I^{\mathrm {ax}} , so that I c o n t I^{\mathrm {cont}} agrees as well. We deduce that in the polynomial ring C [ x 1 , … , x n ] \mathbb {C} \lbrack x_1, \ldots , x_n \rbrack , if f = 0 f = 0 at all points where all of the ∂ f ∂ x i {\partial f \over \partial x_i} are 0, then f ∈ ( ∂ f ∂ x 1 , … , ∂ f ∂ x n ) R f \in ( {\partial f \over \partial x_1}, \, \ldots , \, {\partial f \over \partial x_n})R . We characterize I c o n t I^{\mathrm {cont}} for monomial ideals in polynomial rings over C \mathbb {C} , but we show that the inequalities I ♮ ⊆ I c o n t I^{\natural } \subseteq I^{\mathrm {cont}} and I c o n t ⊆ I a x I^{\mathrm {cont}} \subseteq I^{\mathrm {ax}} can be strict for monomial ideals even in dimension 3. Thus, I c o n t I^{\mathrm {cont}} and I a x I^{\mathrm {ax}} need not agree, although we prove they are equal in C [ x 1 , x 2 ] \mathbb {C}[x_1, x_2] .