论文标题
关于不变元素的子代数：有限和浸入
On the subalgebra of invariant elements: finiteness and immersions
论文作者
论文摘要
让$ r $成为环上的代数$ \ bbbk $，$ t $ an $ r $ -Algebra，$ m $ a $ a $ a $ r $ r $ -module和$ n $ a $ t $ module。让$ g $成为$ \ bbbk $的线性还原组方案，配备了代表$ρ：\ undesline {g} _ {r} \ rightArrow \ undesline {\ textrm {aut}} _ {\ textrm {aut}} _ {\ textrm {mod}（mod}（r）}（r）}（r）}（m）$。对于分级的$ t $ -Algebra $ a $，定义为$ a：= \ left（s_ {t}^{\ bullet}（m^{\ vee} \ otimes_ {r} n）\ right）此外，我们建立了$ \ textrm {proj} \ a $的封闭嵌入到一个投射空间中的条件。由于我们没有提出任何Noetherian的假设，因此我们的结果将文献中的假设推广，提供了有关模量问题的新工具。
Let $R$ be an algebra over a ring $\Bbbk$, $T$ an $R$-algebra, $M$ a finitely generated projective $R$-module, and $N$ a $T$-module. Let $G$ be a linearly reductive group scheme over $\Bbbk$ equipped with a representation $ρ:\underline{G}_{R}\rightarrow \underline{\textrm{Aut}}_{\textrm{Mod}(R)}(M)$. For the graded $T$-algebra $A$, defined as $ A := \left( S_{T}^{\bullet} (M^{\vee} \otimes_{R}N )\right)^{G}, $ we determine the conditions under which the graded $T$-algebra $A$ is finitely generated, finitely presented, or flat. Furthermore, we establish the conditions under which a closed embedding of $\textrm{Proj} \ A$ into a projective space exists. Since we do not impose any Noetherian hypotheses, our results generalize those in the literature, providing new powerful tools regarding moduli problems.
