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        Questions tagged [ac.commutative-algebra]

        Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.

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        Power series ring $R[[X_1,\ldots,X_d]]$ over a U.F.D. $R$

        Let $R$ be a U.F.D. and \begin{align*} T \,\colon= R[[X_1,\ldots,X_d]]. \end{align*} Suppose that we have $d$ elements $f_1,\ldots,f_d \in T$ and let us consider an ideal $J$ of $T$ such that $(f_1,\...
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        Can any one help me with question 8? [on hold]

        enter image description here can anyone help me with question 8?
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        136 views

        Equivalence of definitions of Cohen-Macaulay type

        I know that the Cohen-Macaulay type has this two definitions: Let $(R,\mathfrak{m},k)$ be a Cohen-Macaulay (noetherian) local ring; $M$ a finite $R$-module of depth t. The number $r(M) = dim_k Ext_R^...
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        Invariants of linear endomorphisms of tensor products

        Let $V$ and $W$ be two finite dimensional vector spaces over an algebraically closed field $K$ of characteristic zero. Consider the coordinate ring $K[\mathrm{End}(V\otimes W)]$ of the affine space ...
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        146 views

        Injective map on spectra [on hold]

        I know that if we have a surjection $f:B\rightarrow A$, this induces an injection on the spectra $f^* \colon \operatorname{Spec} A\hookrightarrow \operatorname{Spec} B.$ What about the opposite? ...
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        Standard reference/name for “initial ideals $\Leftrightarrow$ associated graded rings”

        Let $R$ be a $\mathbb Z$-graded commutative ring and let $I\subset R$ be an ideal. On one hand, we may consider the initial ideal $\mathrm{in}_{\deg}(I)$. That is the space spanned by elements of the ...
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        Symmetric polynomials in two sets of variables

        Suppose $f(x_1,...,x_m,y_1,...,y_n)$ is a polynomial with coefficients in some field which is invariant under permuting the $x$'s and the $y$'s. Then $f$ can be generated elementary functions $e_k(x_1,...
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        Does $\mathrm{grade}(J,M) = \mathrm{depth} M$?

        Let $(R,\mathfrak{m})$ be a Cohen–Macaulay ring, $J$ an ideal of $R$ such that $\dim R/J >0$ and $M$ a finitely generated $R$-module. Is $\mathrm{grade}(J,M) =\mathrm{depth} M$ true? Here $\...
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        1answer
        90 views

        For a holonomic $D_X$-module $M$, can $gr M$ have embedded primes?

        Let $M$ be a holonomic $D_X$ module. This means that the minimal primes in $\sqrt{Ann(gr M)}$ are $n=\dim X$ dimensional, for some (and any) good filtration on $M$. But what about the embedded primes? ...
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        Noether’s “set theoretic foundations” of algebra. Reference

        In [C Mclarty] we read [Noether] project was to get abstract algebra away from thinking about operations on elements, such as addition or multiplication of elements in groups or rings. Her algebra ...
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        1answer
        188 views

        Étale fibration for $K[[X_1,…,X_n]]$

        Let us consider a formal power series ring $A_n \colon= K[[X_1,\ldots,X_n]]$ with $0 \ll n < \infty$ and we shall consider a prime ideal ${\frak P}$ of $A_n$ such that $1 < {\mathrm{ht}}({\frak ...
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        1answer
        145 views

        Maximize $L^p$ norm over sphere

        For $p \in \mathbb{R}$, consider the function $$F_p(\lambda_1, \dots, \lambda_n) = \lambda_1^p + \dots + \lambda_n^p.$$ My goal is to maximize this function under the constraints that $$ \lambda_1^2 +...
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        1answer
        129 views

        Bound on number of proper ideals of norm equal to n

        I have read in the paper by Einsiedler, Lindenstrauss, Michel and Venkatesh on Duke's Theorem the following bound that I don't understand: Let $d$ be a positive non-square interger and set let $K = \...
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        155 views

        Primes of the power series rings

        Let $A_n \colon= K[[X_1,\ldots,X_n]]$ be a $n$-variable formal power series ring. By setting $X_n \mapsto 0$, we obtain a natural surjection \begin{equation*} \psi_{n,n-1} \colon A_n \...
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        58 views

        A presentation for a subalgebra

        Let $K$ be a field, and let $I=(g_1,\ldots, g_r)$ be an ideal in $A:=K[X_1,\ldots ,X_n]$. Let $\{f_1,\ldots f_m\}$ be a subset of $A$, and let $B$ be the $K$-subalgebra of $A$ generated by $f_1,\...

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        山西福彩快乐十分钟
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