Multigraded Local Cohomology Categories: Homological Structure and Canonical Sources

Abstract

Let $S=\kk[x_1,\ldots,x_n]$ with its standard ${\mathbb{N}}^n$-grading, and let $\mathfrak{m}=(x_1,\dots ,x_n)$. We study the essential images of the functors $M\mapsto H_{\mathfrak{m}}^i(M(-1))$ on categories of positively determined $S$-modules, where $1=(1,\dots ,1)$. For a fixed determining bound $\mathbf{t}\in {\mathbb{N}}^n$ and $0\le i<n$, we identify the image in degree $i$ with the category of finite-dimensional representations of the subposet of $[0,\mathbf{t}]$ consisting of multidegrees with at least $n-i$ positive coordinates. These images are Serre subcategories of the category of negatively $\mathbf{t}$-determined modules. The top-degree image consists precisely of second cosyzygies in that category and is extension-closed, but is not abelian when $\mathbf{t}\ne 0$. For the non-top image categories, we determine attached primes from projective covers and compute the Bass numbers at $\mathfrak{m}$ 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 $n\ge 2$ and $\mathbf{t}\ge 1$, 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 ${\mathbb{N}}^n$.

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