Vijaylaxmi Trivedi
TIFR, Mumbai
January 2, 2025
Numerical characterizations for integral dependence of graded ideals.: Two nonzero ideals $\mathcal{I} \subseteq \mathcal{J} \subset \O_X$, where ${X}$ denotes an affine normal variety, are \textit{ integral dependent} (or have the same integral closure) if \(\mathcal{I} \mathcal{O}_{X^+} = \mathcal{J} \mathcal{O}_{X^+}\), where $X^+ \longrightarrow X$ is the normalization of the blow ups along $\mathcal{J}$. This notion is used in singularity theory. In dealing with questions which depend only on the integral closure of an ideal, it is often convenient to replace $\mathcal{J}$ by a smaller ideal $\mathcal{I}$ which has the same integral closure. The notion of integral closure makes sense in any Noetherian commutative ring. The integral dependence of ideals of finite colength in a local ring has a well known numerical characterization due to D. Rees, using Hilbert-Samuel multiplicities. Attempts to give a numerical characterization for ideals which are not necessarily of finite colength led to numerical invariants like $j$-multiplicity, $\varepsilon$-multiplicity etc., which require several localizations, and hence are not amenable to computations. Also there exists a notion of ?multiplicity sequence? which gives a numerical characterization of integral dependence in terms of a sequence of invariants defined inductively. In this talk we give several new numerical characterizations of the integral dependence of ${I}$ and ${J}$ in a graded setup. A novelty of this approach is that it does not involve localization and only requires checking computable and well-studied invariants like Hilbert-Samuel multiplicities. Apart from several well-established results, the proofs of these results involve establishing the existence of density functions to study the asymptotic growth of the ideals arising from the powers of graded ideals. This talk is based on some joint works with Suprajo Das and Sudeshna Roy.
TIFR, Mumbai
January 2, 2025
Numerical characterizations for integral dependence of graded ideals.: Two nonzero ideals $\mathcal{I} \subseteq \mathcal{J} \subset \O_X$, where ${X}$ denotes an affine normal variety, are \textit{ integral dependent} (or have the same integral closure) if \(\mathcal{I} \mathcal{O}_{X^+} = \mathcal{J} \mathcal{O}_{X^+}\), where $X^+ \longrightarrow X$ is the normalization of the blow ups along $\mathcal{J}$. This notion is used in singularity theory. In dealing with questions which depend only on the integral closure of an ideal, it is often convenient to replace $\mathcal{J}$ by a smaller ideal $\mathcal{I}$ which has the same integral closure. The notion of integral closure makes sense in any Noetherian commutative ring. The integral dependence of ideals of finite colength in a local ring has a well known numerical characterization due to D. Rees, using Hilbert-Samuel multiplicities. Attempts to give a numerical characterization for ideals which are not necessarily of finite colength led to numerical invariants like $j$-multiplicity, $\varepsilon$-multiplicity etc., which require several localizations, and hence are not amenable to computations. Also there exists a notion of ?multiplicity sequence? which gives a numerical characterization of integral dependence in terms of a sequence of invariants defined inductively. In this talk we give several new numerical characterizations of the integral dependence of ${I}$ and ${J}$ in a graded setup. A novelty of this approach is that it does not involve localization and only requires checking computable and well-studied invariants like Hilbert-Samuel multiplicities. Apart from several well-established results, the proofs of these results involve establishing the existence of density functions to study the asymptotic growth of the ideals arising from the powers of graded ideals. This talk is based on some joint works with Suprajo Das and Sudeshna Roy.