# Questions tagged [projective-modules]

For questions about projective modules over a ring and projective objects in related categories.

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### Are finitely presented algebras over VNRs projective?

Question: Let $A$ be a commutative von Neumann regular ring, and $B$ an $A$-algebra of finite presentation, i.e. $B = A[x_1, \ldots, x_n]/(f_1, \ldots, f_m)$. Is $B$ a projective $A$-module? In the ...
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### Rings such that torsion-free/flat/projective modules are flat/projective/free

While thinking about this question (and specifically YCor's remarks), I tried to remember what can be said about rings such that every torsion-free module is free, and I could not; and such things, ...
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This is from Hartshrone exercise 6.6 part (a). Let $A$ be a regular local ring and $M$ be a finitely generated $A$-module, prove the following $M$ is projective $\iff$ $\operatorname{Ext}^{i}(M,A)=\{... 2answers 252 views ### Projective modules restricted to smooth curves I asked this question on Stack Exchange, but no one answered this. I want to prove a coherent sheaf$M$on$X$is locally free if and only if this is true for$M|_{X'}$, for all smooth curves$X'$... 1answer 140 views ### Is a tower of locally-free modules locally a tower of free modules? Suppose we have a (commutative, unital) ring$R$and a (commutative, unital)$R$-algebra$A$such that$A$is projective of constant rank$n$as an$R$-module. This condition is equivalent to there ... 2answers 283 views ### tangent bundle on noncommutative manifold Using the Serre-Swan's theorem, one can do vector bundle theory on noncommutative manifold$(A,H,D)$, by replacing vector bundle by finitely generated projectve module$M$. For the construction of ... 2answers 222 views ### When splitting of short exact sequence preserves the kernels This is a problem that I thought at first was obvious but that became less clear the more I thought about it. Assume we have a finitely generated algebra$A$over a field$k$, and a short exact ... 1answer 101 views ### On the finiteness of an Auslander-Reiten component I am reading a paper called A NOTE ON THE RADICAL OF A MODULE CATEGORY by CLAUDIA CHAIO AND SHIPING LIU. This is Theorem 2.7: And this is part of it's proof, in which the direction (2)$\Rightarrow $... 1answer 117 views ### infinite left degrees I am reading a paper called A NOTE ON THE RADICAL OF A MODULE CATEGORY by CLAUDIA CHAIO AND SHIPING LIU. This is a part of the paper: Definition: Let$f: X \rightarrow Y$be an irreducible morphism ... 1answer 108 views ### About composition factors [closed] I am reading a paper called A NOTE ON THE RADICAL OF A MODULE CATEGORY by CLAUDIA CHAIO AND SHIPING LIU. This is part of the proof of Lemma 2.3$A$is assumed to be an Artin algebra and mod(A) the ... 1answer 146 views ### Question on simple modules and projective covers I have the following question: Let$A$be an Artin algebra. Let$S_1$and$S_2$be simple modules in$\text{mod}(A)$and let$P(S_1)$be the projective cover of$S_1$. Let$f: P(S_1) \rightarrow S_2$... 2answers 146 views ### Question on injective hulls How can I show the following: Let$f: M \rightarrow N$be a morphism in$\text{mod}(A)$, where$A$is an Artin algebra. Suppose$f \neq 0$. Then there exists a simple module$S$with its injective ... 1answer 81 views ### injective hull and projective cover of simple modules are indecomposable Let$A$be an Artinian algebra. Let$S$be a simple module over$A$. Let$\pi: S \rightarrow I$be the injective hull and$\tau: P \rightarrow S$be the projective cover of$S$. Then$I$and$P$must ... 1answer 107 views ### Finitely presented modules admitting projective covers A ring$R$is called semi-perfect if every finitely generated$R$-module has a projective cover, and it can be proved that this is equivalent to say that the category consisting of the finitely ... 0answers 59 views ### When is a height one prime ideal projective? Let$A$be a ring and$(0) \neq P\subset A$be a prime ideal. If$A$is a noetherian domain and if$P,$seen as an$A$-module, is projective then the height of$P$is 1. (Matsumura, Commutative ring ... 1answer 384 views ### Separability of$\mathbb{C}[x]$over its$\mathbb{C}$-subalgebras For commutative rings$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$. ... 1answer 186 views ### Trace ideal of a projective module In his 1969 paper "On projective modules of finite rank", Wolmer Vasconcelos writes Let$M$be a projective$R$-module... The trace of$M$is defined to be the image of the map$M \otimes_R ...
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Let $A$ be a $\mathbb N$-graded ring. One can consider the two categories $M_A^g$ and $M_A^u$ of graded and ungraded modules over $A$. Both have, say, enough projectives, hence one can compute various ...
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### Second summand to make projective module free

Suppose there's a projective $R$-module $P$ (non-free). We know that there is another $R$-module $M$ such that $P\oplus M$ is free over $R$. Is there a way to write down such an $M$ in terms of $P$? ...
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### Projective (or injective) object in a subcategory

Let $\mathcal{A}$ be an abelian category and $\mathcal{B}$ be a full subcategory of $\mathcal{A}$. Suppose that $\mathcal{B}$ is abelian and that the inclusion of $\mathcal{B}$ in $\mathcal{A}$ is ...
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### If $M \otimes -$ is continuous, why is $M$ f.g. projective? Alternative proof

Let $R$ be a commutative ring and $M$ be some $R$-module such that $M \otimes -$ is continuous (i.e. preserves all limits). Then one can show that $M$ is f.g. projective. One way to prove this is to ...
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### Role of stably free modules in algebraic geometry

For any ring $R$, a non-zero module $S$ is stably free if $S\oplus R^a$ is free ($a\geq 1$). This may be an overly vague question, but I am wondering in what contexts do stably free modules arise in ...
### Does $A\oplus M_n(R)\cong B\oplus M_n(R)$ imply $A\cong B$? $R$ Dedekind domain
Let $R$ be a Dedekind domain and $A, B$ be finitely generated projective $M_n(R)$-modules. Is it true that $A\oplus M_n(R)\cong B\oplus M_n(R)\:\:\Rightarrow\:\:A\cong B$? Here, the isomorphism is ...
The injective hull for a module always exists, however over certain rings modules may not have projective covers. I have a question. If $A$ is an Artinian module on a Noetherian local ring $R$ then \$...