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

        4
        votes
        0answers
        57 views

        Support of Matlis dual

        Let $(A,m)$ be a commutative noetherian local ring, $E$ the injective hull of $A/m$, and $M$ a finitely generated $A$-module. What is the connection between the support of $M$ and the support of the ...
        5
        votes
        0answers
        93 views
        +100

        Divided power algebra is artinian as a module over the polynomial ring

        I already asked this on math.stackexchange.com, but did not receive much responses. I hope this is also appropriate for mathoverflow. In the paper Homological algebra on a complete intersection, with ...
        4
        votes
        1answer
        160 views

        $f,g \in \mathbb{Z}[x,y]$ satisfying: $\operatorname{Jac}(f,g)=0$ and $f,g \notin \mathbb{Z}[h]$ for every $h \in \mathbb{Z}[x,y]$?

        Is it possible to find $f,g \in \mathbb{Z}[x,y]$ (with $\deg(f),\deg(g) \geq 1$) such that the following two conditions are satisfied: (1) $\operatorname{Jac}(f,g)=f_xg_y-f_yg_x = 0$. (2) ...
        7
        votes
        2answers
        180 views

        Noetherian local ring and the growth of $\dim_k \operatorname{Ext}^i(k,k)$

        Let $A$ be a noetherian local ring with residue field $k$, one can consider $\operatorname{Ext}^i(k,k)$ for every natural number $i$. If it is zero for large $i$, then $A$ is regular and the converse ...
        2
        votes
        1answer
        223 views

        A peculiar operation on $M_2(\mathbb Z)$ which along with the usual matrix addition, makes $M_2(\mathbb Z)$ into a commutative ring with unity [closed]

        For $A=\begin{pmatrix} a_1 & b_1 \\ c_1&d_1 \end{pmatrix}, B=\begin{pmatrix} a_2 & b_2 \\ c_2&d_2 \end{pmatrix}\in M_2(\mathbb Z)$, define $A*B:=a_1L_1BR_1+b_1L_1BR_2+c_1L_2BR_1+...
        7
        votes
        0answers
        83 views
        +50

        Cellular and primary binomial ideals

        An ideal $I$ is called cellular if every variable is either a nonzerodivisor modulo $I$ or is nilpotent modulo $I$. An ideal $I$ is called primary if whenever $fg \in I$ then or $f \in I$ or $g$ is ...
        8
        votes
        0answers
        87 views

        Is there a homological interpretation for the cokernel of the kernel of a map between complexes induced by tensor product?

        Let $A$ be a free abelian group of rank 2, and let $S = \mathbb{Z}[A]\cong\mathbb{Z}[a_1^{\pm1},a_2^{\pm1}]$ the group algebra for $A$. Let $t : S\times S\rightarrow S$ be the $S$-module map given by ...
        0
        votes
        0answers
        100 views

        Is it possible to generalize a result of Katsylo-Zhang concerning the two-dimensional JC?

        Let $p=p(x,y), q=q(x,y) \in \mathbb{C}[x,y]$ be a Jacobian pair, namely, $Jac(p,q)=p_xq_y-p_yq_x \in \mathbb{C}^{\times}$. Denote the total degrees of $p$ and $q$ by $\deg(p)$ and $\deg(q)$, and ...
        8
        votes
        1answer
        83 views

        Injective indecomposable modules over Laurent polynomial rings

        What does the injective envelope of $\mathbb C[x,x^{-1}]/(p(x,x^{-1}))$ as a $\mathbb C[x,x^{-1}]$-module look like where $p(x,x^{-1})$ is an irreducible element? I’m sure this is well known, but ...
        37
        votes
        2answers
        842 views

        The roots of unity in a tensor product of commutative rings

        For $i\in\{1,2\}$ let $A_i$ be a commutative ring with unity whose additive group is free and finitely-generated. Assume that $A_i$ is connected in the sense that $0$ and $1$ are unique solutions of ...
        2
        votes
        1answer
        111 views

        $(x + y + z)…(x + y\omega_n^{n-1} + z\omega_n) = x^n + y^n + z^n - P(x,y,z)$ To find $P$

        $$(x + y + z)(x + y\omega_n + z\omega_n^{n-1})(x + y\omega_n^2 + z\omega_n^{n-2})....(x + y\omega_n^{n-1} + z\omega_n) = x^n + y^n + z^n - P(x,y,z)$$ where $\omega_n$ is an nth root of unity. The ...
        4
        votes
        0answers
        83 views

        Compute the closure of graph of function from complement of hypersurface in $\mathbb{A}^n$

        I'm hoping someone can give me some tips to help speed up computation on the following problem: Suppose I have a map $G=(g_1/f,\dots,g_m/f):\mathbb{A}^n\setminus{V(f)}\to \mathbb{A}^m$. I'm ...
        0
        votes
        0answers
        84 views

        gcd of polynomial values

        Suppose that $f$ and $g$ are two coprime polynomials in $\mathbb Z[x]$. I'm interested in any sort of upper bound on $gcd(f(a),g(a))$, in terms of the integer $a$. Are there any results of this type?...
        8
        votes
        1answer
        375 views

        Do commutative rings without unity have the IBN property?

        Let $R$ be a commutative rng, i.e. a commutative ring without an identity element. Does $R$ still have the Invariant Basis Number (IBN) property? Recall that a ring is said to have the IBN ...
        6
        votes
        0answers
        142 views

        Fraction fields of strict henselizations of DVRs

        Let $A_{1},A_{2}$ be discrete valuation rings whose fraction fields are isomorphic. Let $A_{i}^{\mathrm{sh}}$ be the strict henselization of $A_{i}$, and let $K_{i}$ be the fraction field of $A_{i}^{\...

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