# Multiplicity Conjecture for Standard Determinantal Ideals

Part of the Progress in Mathematics book series (PM, volume 264)

## Abstract

Let $$I \subset R = K\left[ {x_1 , \ldots x_n } \right]$$ be a graded ideal of arbitrary codimension c. Consider the minimal graded free R-resolution of R/I,
$$0 \to \oplus _{j \in \mathbb{Z}} R\left( { - j} \right)^{\beta _{p,j} \left( {{R \mathord{\left/ {\vphantom {R I}} \right. \kern-\nulldelimiterspace} I}} \right)} \to \cdots \to \oplus _{j \in \mathbb{Z}} R\left( { - j} \right)^{\beta _{1,j} \left( {{R \mathord{\left/ {\vphantom {R I}} \right. \kern-\nulldelimiterspace} I}} \right)} \to R \to {R \mathord{\left/ {\vphantom {R I}} \right. \kern-\nulldelimiterspace} I} \to 0,$$
where, as usual, we denote by $$\beta _{i,j} \left( {{R \mathord{\left/ {\vphantom {R I}} \right. \kern-\nulldelimiterspace} I}} \right)$$ = dim $$Tor_i^R \left( {{R \mathord{\left/ {\vphantom {R {I,K}}} \right. \kern-\nulldelimiterspace} {I,K}}} \right)_j$$ the (i, j)th graded Betti number of R/I and by $$\beta _i \left( {{R \mathord{\left/ {\vphantom {R I}} \right. \kern-\nulldelimiterspace} I}} \right) = \sum _{j \in \mathbb{Z}} \beta _{i,j} \left( {{R \mathord{\left/ {\vphantom {R I}} \right. \kern-\nulldelimiterspace} I}} \right)$$ the ith total Betti number of R/I.

## Keywords

Betti Number Pure Resolution Homogeneous Ideal Maximal Minor Homogeneous Matrix