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accepted Exact functors and “short” exact functors
Sep
9
answered Exact functors and “short” exact functors
Sep
9
comment Exact functors and “short” exact functors
I wonder if there is a concrete proof that shows "every exact functor $A\operatorname{Mod}\to A\operatorname{Mod}$ sends a zero module to a zero module," without any reference to the category theory? Here by an "exact functor" I mean it sends every exact sequence to an exact sequence.
Sep
9
comment Exact functors and “short” exact functors
Why is an exact functor additive?
Sep
9
asked Exact functors and “short” exact functors
Sep
7
comment Additive functor over a short split exact sequence.
Perhaps not "precisely" in the literal sense; it seems that $r\circ s=0$ or $g\circ f=0$ (but not both) can be omitted.
Jul
22
accepted Is the colimit of finite tensor products a tensor product?
Jul
22
asked Is the colimit of finite tensor products a tensor product?
Jul
21
comment Example of strict inclusion for the localization of associated primes
Sorry, I meant $\operatorname{Spec}R/\mathbb{F}_2^{(\mathbb{N})}\cap\operatorname{Ass}R= \varnothing$ for the first equality. I reviewed your answer more carefully, and I agree it is a correct answer!
Jul
20
accepted Example of strict inclusion for the localization of associated primes
Jul
20
comment Example of strict inclusion for the localization of associated primes
Thanks for the answer. I'm having two issues: 1.Why does $\mathbb{F}_2^{(\mathbb{N})}\subset P$ imply $e=1$? 2.Why does $\operatorname{Spec}R/\mathbb{F}_2\cap\operatorname{Ass}R=\varnothing$ imply $\operatorname{Ass}R\cap\operatorname{Spec}S^{-1}R=\varnothing$?
Jul
20
comment Example of strict inclusion for the localization of associated primes
What did you mean by the parentheses in the notation $\mathbb{F}_2^{(\mathbb{N})}$ ?
Jul
16
revised Example of strict inclusion for the localization of associated primes
added 1 character in body
Jul
16
comment Example of strict inclusion for the localization of associated primes
You are right; I took the liberty of abusing notation and considered $\operatorname{Spec}S^{-1}A$ to be a subset of $\operatorname{Spec}A$.
Jul
16
asked Example of strict inclusion for the localization of associated primes
Jul
16
accepted A question related to associated prime ideals
Jul
16
comment Associate prime ideals and exact sequences of $R$-modules
@tlquyen : I think "finite module" usually means finitely generated module, not module of finite length.
Jul
10
awarded  Benefactor
Jul
10
accepted Pseudo associated primes and short exact sequences
Jul
7
accepted Does maximal Cohen-Macaulay modules localize?