# 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
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.