Gorenstein homomorphisms of noncommutative rings
Gorenstein homomorphisms of noncommutative rings
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非交换环的 Gorenstein 同态
DOI：
10.1006/jabr.1998.7608
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发表时间：
1999
影响因子：
0.9
通讯作者：
Peter Jørgensen
中科院分区：
文献类型：
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作者：
Peter Jørgensen
令 R 和 S 通过平衡对偶复形连接分级诺以太代数（kis 是一个域）。设 ρ: R→ S 为分级代数同态，使得 fd R (S)<∞ 且 fd R opp (S)< Infini。然后我们定义了 ρ 是 Gorenstein 同态的含义，并为这种同态建立了一个理论，使 Gorenstein 性的经典理论相对化；也就是说，将其从代数的水平提升到同态的水平。关于 Gorenstein 同态的一点是，它们以良好的方式在域和目标之间转移同源特性。例如，我们得到： 定理（“上升-下降”）。[方程] 我们也将该理论应用于规范同态 k→ S，结果令人惊喜地给出： 定理。左姐姐 AS-Gorenstein⇔ 右姐姐 AS-Gorenstein。这些结果和其他结果的存在归功于 Gorenstein 同态下巴斯数的良好行为，例如：定理（“基数变化”）。如果 ρ 是 Gorenstein 同态，且 X∈ D b fg (GrR)，则存在 Bass 数等式：μ i+ 深度 (S) S (S L⊗ R X)+ μ i+ 深度 (R) R (X)。这里μ j R (X)是XoverR的第j个低音数，定义为μ j R (X)+dim i Ext j R (k, X)。如前所述，除了上述定理之外，我们还表明 Gorenstein 同态理论还包含 Gorenstein 代数的一些经典结果的推广。
Let R and S be connected graded noetheriank-algebras (kis a field) with balanced dualizing complexes. Let ρ: R→ Sbe a graded algebra homomorphism such that fd R (S)<∞ and fd R opp (S)<∞. We then define what it means for ρ to be a Gorenstein homomorphism, and set up a theory for such homomorphisms which relativizes the classical theory of Gorenstein-ness; that is, lifts it from the level of algebras to the level of homomorphisms. One point about Gorenstein homomorphisms is that they transfer homological properties between domain and target in a good way. For instance, we obtain: Theorem (" Ascent-descent").[equation] We also apply the theory to the canonical homomorphismk→ S, and here it turns out by a pleasant surprise to give: Theorem. Sis left AS-Gorenstein⇔ Sis right AS-Gorenstein. These and other results owe their existence to the good behavior of Bass-numbers under Gorenstein homomorphisms, exemplified by: Theorem (" Base-change"). If ρ is a Gorenstein homomorphism, andX∈ D b fg (GrR), then there are equalities of Bass-numbers, μ i+ depth (S) S (S L⊗ R X)+ μ i+ depth (R) R (X). Here μ j R (X) is thejth Bass-number ofXoverR, defined byμ j R (X)+ dim i Ext j R (k, X). As mentioned, we show in addition to the above theorems that the theory of Gorenstein homomorphisms contains generalizations of some classical results on Gorenstein algebras.