I have a commutative algebra class and I heard the theorem from the professor: Let $R$ be a Noetherian ring and $\{E_i : i\in I\}$ be a collection of injective $R$-modules then $\bigoplus_{i\in I} E_i$ is also injective. My questions are: How to prove that? My professor does not talk about the proof of it […]

In order to test injectivity of a module $M$ it suffices to check if every linear map from an arbitrary ideal extends to the ring or not. Similarly in order to check the flatness of a module $M$ it suffices to check whether tensoring with it preserves injectivity of $0 \to I \to R$. Is […]

The question is basically like this: Prove that if $S_{\cdot}$ is a finitely generated (in degree 1) graded ring over a field $k$ and $M_{\cdot}$ is finitely generated, then the saturation map $M_{n}\rightarrow\Gamma(\text{Proj}S_{\cdot},\widetilde{M(n)_{\cdot}})$ is an isomorphism for large $n$. The hint in [Hartshorne] says we can follow the hint in the proof of Theorem 5.19. […]

Let $R$ be a commutative unital ring and $M$ a finitely generated projective $R$-module. My question is: if $N_1$ and $N_2$ are f.g. projective submodules of $M$, is $N_1 \cap N_2$ f.g.? Is it projective? (Surely the answer is no but I haven’t been able to find a counterexample. Also, sorry for the lack of […]

Let $A^{\bullet}$ be a graded commutative algebra. Denote by $A^{\bullet}$-mod the category of graded modules over $A^{\bullet}$. Let $A$ be $A^{\bullet}$ considered as an algebra (we forgot grading). Finally let $A$-mod be category of modules over $A$. So we have an oblivion functor $$ Obl: A^{\bullet}-{\rm mod} \rightarrow A-{\rm mod}.$$ Consider $P^{\bullet} \in A^{\bullet}$-mod such […]

In “Cover of a direct summand” it is asked to show that if a direct sum has a projective cover, and if one of the summands has a projective cover, then so does the other. I gave a solution that works for any class $\mathcal{X}$ closed under direct sums and direct summands, however I use […]

Let $L,N$ be $R$-modules. If $L$ and $L \oplus N$ have projective covers, is it true that $N$ admits a projective cover?

I have found a result concerning projectivity of a certain ring extension: Lemma 2.64. This says the following: Let $A$ be an integral domain or a noetherian ring, $B$ an $A$-algebra, $C$ an $A$-subalgebra of $B$. Assume that $B$ is a f.g. flat $A$-module and $C$ is a f.g. free $A$-module. Then $B$ is a […]

For commutative rings $R \subseteq S$, recall that $S$ is separable over $R$, if $S$ is a projective $S \otimes_R S$-module. (via $f: S \otimes_R S \to S$ given by: $f(s_1 \otimes_R s_2)=s_1s_2$). My question: Assume $A \subseteq B \subseteq C$ are commutative rings, such that $C$ is separable over $A$. Is $C$ separable over […]

Let $P$ be a finitely generated projective module. Prove that $P\cong \operatorname{Hom}_{R}(\operatorname{Hom}_{R}(P,R),R)$.

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