Heegaard Floer correction term

Let $Y$ be a closed connected oriented rational homology three-sphere, let $\mathfrak{t}$ be a Spin${}^c$ structure, and suppose the Heegaard Floer groups, absolute rational grading, and natural map ${\iota }_{\ast }:H{F}^{\mathrm{\infty }}(Y,\mathfrak{t})\to H{F}^{+}(Y,\mathfrak{t})$ are defined and independent of the construction choices. View $H{F}^{+}$ as an ${\mathbb{F}}_2[U]$-module. An element is $U$-torsion if some positive power of $U$ annihilates it. The correction term $d(Y,\mathfrak{t})$ is the minimum absolute grading of a nonzero homogeneous element in $\mathrm{im}{\iota }_{\ast }$. In the standard plus tower every element is $U$-torsion, although every element is divisible by every positive power of $U$; these two properties are compatible. The definition uses the nonzero homogeneous image elements, rather than imposing $U$-nontorsion. The standard normalization is $d(S^3,{\mathfrak{t}}_0)=0$ for the unique Spin${}^c$ structure ${\mathfrak{t}}_0$ on $S^3$. Existence of the minimum and independence of choices are separate proof obligations.

The Floer groups and absolute grading convention

Ozsvath and Szabo, Absolutely Graded Floer Homologies and Intersection Forms for Four-Manifolds with Boundary, Definition 4.1

Hom, Lecture Notes on Heegaard Floer Homology, Section 2.2