# Tagged Questions

For questions about modules over rings, concerning either their properties in general or regarding specific cases.

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### Why are Grothendieck's and Hartshorne's definitions of quasi-coherence equivalent?

Hartshorne's Algebraic Geometry defines an $\mathcal O_X$-module $\mathscr F$ to be quasi-coherent if there is an open affine cover $(U_i=\operatorname{Spec} A_i)_{i\in\mathcal I}$ of $X$ such that ...
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### An odd expression appearing in proof that kernel and image of certain map on group ring $\mathbb Z[G]$ are equal

Let $G = \langle g \rangle$ be a cyclic group of order $n$. Consider the free ring $\mathbb Z[G]$ of all formal sums of elements from $G$ with coefficients from $\mathbb Z$, with multiplication given ...
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### Every finitely generated flat module over a ring with a finite number of minimal primes is projective

Over a commutative ring $R$, a finite type locally free (weak sense) module for which the rank function is locally constant is projective. If we notice that for each minimal prime $p$ of the ring, ...
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### Is $A / \mathfrak{m}$ flat if $A$ is a local ring?

I'd like to prove the following: if $A$ is a local ring and $\mathfrak{m} \subset A$ its maximal ideal, then $A / \mathfrak{m}$ is a flat $A$-module. How can I do this? I've tried to find a suitable ...
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### Problem with the ring $R=\begin{bmatrix}\Bbb Z & 0\\ \Bbb Q &\Bbb Q\end{bmatrix}$ and its ideal $D=\begin{bmatrix}0&0\\ \Bbb Q & \Bbb Q\end{bmatrix}$

Let us consider the ring $R:=\begin{bmatrix}\Bbb Z & 0\\ \Bbb Q & \Bbb Q\end{bmatrix}$ and its two-sided ideal $D:=\begin{bmatrix}0 & 0\\ \Bbb Q & \Bbb Q\end{bmatrix}$. Let then ...
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### Ring of infinite global dimension which does not have a finitely generated module of infinite projective dimension

Let $R$ be a ring of infinite global dimension. A priori we can't immediately conclude that $R$ has a module of infinite projective dimension, since it could be the case that $R$ only has a sequence ...
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### Constructing a ring consisting of formal infinite series from a given ring

Let $A$ be an $\mathbb{N}$-graded $\Bbbk$-algebra, where $\Bbbk$ is a field, and where $\dim_\Bbbk A_n < \infty$ for all $n \in \mathbb{N}$. I can't see anything preventing me from constructing a ...
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### Prove that $\operatorname{Ext}^{d+1}(A, B)\cong \operatorname{Ext}^1(M_d,B)$

So, given a resolution, with $P_{i}$ projective modules: $$0\longrightarrow M_d\longrightarrow P_{d-1} \longrightarrow \cdots \longrightarrow P_0 \longrightarrow A\longrightarrow 0,$$ I'm trying to ...
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### Kernel of a natural map is a direct summand of the covariant extension

I am reading chapter 2 of 'Homological Algebra' by Cartan and Eilenberg. 1/ Given a ring homomorphism $\varphi: \Lambda \rightarrow \Gamma$ and a right $\Gamma$-module $A$, we can treat $A$ and ...
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### Is an injective map of $B$-modules also injective as an $A$-linear map if $B$ is an $A$-algebra?

I've been going through my submitted exercises again of my Commutative Algebra-class and I have the following question: Let $A$ be a commutative ring with unity. Given any injective homomorphism of ...
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### Noetherian rings and modules

A ring is left-Noetherian if every left ideal of R is finitely generated. Suppose R is a left-Noetherian ring and M is a finitely generated R-module. Show that M is a Noetherian R-module. I'm ...
For $A$-modules and homomorphisms $0\to M'\stackrel{u}{\to}M\stackrel{v}{\to}M''\to 0$ is exact. Prove if $M'$ and $M''$ are fintely generated then $M$ is finitely generated.