Some extensions of theorems of Kn\'orrer and Herzog-Popescu
| Autor: | Dugas, Alex S., Leuschke, Graham J. |
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| Rok vydání: | 2017 |
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| Druh dokumentu: | Working Paper |
| Popis: | A construction due to Kn\"orrer shows that if $N$ is a maximal Cohen-Macaulay module over a hypersurface defined by $f+y^2$, then the first syzygy of $N/yN$ decomposes as the direct sum of $N$ and its own first syzygy. This was extended by Herzog-Popescu to hypersurfaces $f+y^n$, replacing $N/yN$ by $N/y^{n-1}N$. We show, in the same setting as Herzog-Popescu, that the first syzygy of $N/y^{k}N$ is always an extension of $N$ by its first syzygy, and moreover that this extension has useful approximation properties. We give two applications. First, we construct a ring $\Lambda^\#$ over which every finitely generated module has an eventually $2$-periodic projective resolution, prompting us to call it a "non-commutative hypersurface ring". Second, we give upper bounds on the dimension of the stable module category (a.k.a. the singularity category) of a hypersurface defined by a polynomial of the form $x_1^{a_1} + \dots + x_d^{a_d}$. Comment: 20 pages, comments welcome. v2 repairs a small gap in section 4 |
| Databáze: | arXiv |
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