Multigraded Local Cohomology Categories: Homological Structure and Canonical Sources
Abstract
Let S=kk[x1,…,xn] with its standard ℕn-grading, and let m=(x1,…,xn). We study the essential images of the functors M↦ Him(M(−1)) on categories of positively determined S-modules, where 1=(1,…,1). For a fixed determining bound t∈ℕn and 0≤ i<n, we identify the image in degree i with the category of finite-dimensional representations of the subposet of [0,t] consisting of multidegrees with at least n−i positive coordinates. These images are Serre subcategories of the category of negatively t-determined modules. The top-degree image consists precisely of second cosyzygies in that category and is extension-closed, but is not abelian when t≠0. For the non-top image categories, we determine attached primes from projective covers and compute the Bass numbers at m and the injective dimensions over S of projective and injective objects. In every non-top degree, we construct an exact functorial section selecting, for each nonzero target, a source of minimal projective dimension n−i and maximal depth i among all its sources. If n≥2 and t≥1, no fully faithful section exists in degrees $1$ and n−1, even without requiring exactness. In top degree, graded Matlis duality followed by S-duality gives a fully faithful functorial section. We also describe the global images obtained by varying the determining bound. The non-top global categories have enough injectives and no nonzero projective objects, while the top category corresponds to second cosyzygies in the category of finite-support representations of ℕn.
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BibTeX
@misc{lu2026multigraded,
title = {{Multigraded Local Cohomology Categories: Homological Structure and Canonical Sources}},
author = {Dancheng Lu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.11219},
eprint = {2609.11219},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11219},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Multigraded Local Cohomology Categories: Homological Structure and Canonical Sources
AU - Dancheng Lu
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2609.11219
UR - https://arxiv.org/abs/2609.11219
AB - Let S=kk[x_1,…,x_n] with its standard ℕ^n-grading, and let m=(x_1,…,x_n). We study the essential images of the functors M↦ H^i_m(M(−1)) on categories of positively determined S-modules, where 1=(1,…,1). For a fixed determining bound t∈ℕ^n and 0≤ i<n, we identify the image in degree i with the category of finite-dimensional representations of the subposet of [0,t] consisting of multidegrees with at least n−i positive coordinates. These images are Serre subcategories of the category of negatively t-determined modules. The top-degree image consists precisely of second cosyzygies in that category and is extension-closed, but is not abelian when t≠0. For the non-top image categories, we determine attached primes from projective covers and compute the Bass numbers at m and the injective dimensions over S of projective and injective objects. In every non-top degree, we construct an exact functorial section selecting, for each nonzero target, a source of minimal projective dimension n−i and maximal depth i among all its sources. If n≥2 and t≥1, no fully faithful section exists in degrees $1$ and n−1, even without requiring exactness. In top degree, graded Matlis duality followed by S-duality gives a fully faithful functorial section. We also describe the global images obtained by varying the determining bound. The non-top global categories have enough injectives and no nonzero projective objects, while the top category corresponds to second cosyzygies in the category of finite-support representations of ℕ^n.
ER -