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Hi everyone!

Let $X$ be Hausdorff, locally compact, paracompact. Consider $\mathcal{F}$ a soft sheaf on $X$: as there are a variety of definitions of soft sheaf, let me emphasize my definition (taken from Gelfand-Manin, Methods of Homological Algebra, p.38): $\mathcal{F}$ is soft if for any closed $Z \subset X$, the restriction map $$\mathcal{F}(X) \rightarrow \varinjlim_{U \supset Z} \mathcal{F}(U)$$is a surjection.

For an open embedding $j: U \rightarrow X$, is it true that $j^* \mathcal{F}$ is again soft?

Thanks so much!

What I understand/motivation: I can prove this when $U$ is again paracompact - while (Hausdorff) and (Hausdorff + locally compact) both are inherited by open subsets, my impression was that paracompactness does not always descend, hence my question. Also, to be fair Gelfand-Manin also ask that $X$ be separable - will this affect the answer?

By the way, the motivation for my question is in trying to understand their proof of Verdier duality; on p. 231 they state that if $L$ is soft, then so is $L \otimes j_!j^*\mathbb{Z}$. My approach would be $$L \otimes j_! j^* \mathbb{Z} \simeq j_! j^* L$$ and Gelfand-Manin state in an exercise (which I'm still puzzling out) that $f_!$ between locally compact Hausdorff spaces sends soft sheaves to soft sheaves.

One last soft (hah!) question: if I use the definition of a soft sheaf found here: http://amathew.wordpress.com/2011/06/10/soft-sheaves/ literally everything I'm doing becomes much easier, and the way Gelfand-Manin use the word 'deduce' in Exercise 1(b) on p. 236 makes me think this was in fact the definition they want to have (otherwise the 'if' statement as far as I can tell requires a reasonably involved paracompactness argument - and they don't define a soft sheaf on a not paracompact space, yet don't ask that the spaces in that exercise be paracompact, which adds to the confusion!). Has anyone run into problems with GM's notion of softness before when trying to read this section/does anyone have a reason not to simply supplant their definition with Akhil's?

Hi everyone! Answered to my satisfaction in the comments - thanks nosr and Jacob Bell! :)

Let $X$ be Hausdorff, locally compact, paracompact. Consider $\mathcal{F}$ a soft sheaf on $X$: as there are a variety of definitions of soft sheaf, let me emphasize my definition (taken from Gelfand-Manin, Methods of Homological Algebra, p.38): $\mathcal{F}$ is soft if for any closed $Z \subset X$, the restriction map $$\mathcal{F}(X) \rightarrow \varinjlim_{U \supset Z} \mathcal{F}(U)$$is a surjection.

For an open embedding $j: U \rightarrow X$, is it true that $j^* \mathcal{F}$ is again soft?

Thanks so much!

What I understand/motivation: I can prove this when $U$ is again paracompact - while (Hausdorff) and (Hausdorff + locally compact) both are inherited by open subsets, my impression was that paracompactness does not always descend, hence my question. Also, to be fair Gelfand-Manin also ask that $X$ be separable - will this affect the answer?

By the way, the motivation for my question is in trying to understand their proof of Verdier duality; on p. 231 they state that if $L$ is soft, then so is $L \otimes j_!j^*\mathbb{Z}$. My approach would be $$L \otimes j_! j^* \mathbb{Z} \simeq j_! j^* L$$ and Gelfand-Manin state in an exercise (which I'm still puzzling out) that $f_!$ between locally compact Hausdorff spaces sends soft sheaves to soft sheaves.

One last soft (hah!) question: if I use the definition of a soft sheaf found here: http://amathew.wordpress.com/2011/06/10/soft-sheaves/ literally everything I'm doing becomes much easier, and the way Gelfand-Manin use the word 'deduce' in Exercise 1(b) on p. 236 makes me think this was in fact the definition they want to have (otherwise the 'if' statement as far as I can tell requires a reasonably involved paracompactness argument - and they don't define a soft sheaf on a not paracompact space, yet don't ask that the spaces in that exercise be paracompact, which adds to the confusion!). Has anyone run into problems with GM's notion of softness before when trying to read this section/does anyone have a reason not to simply supplant their definition with Akhil's?

Hi everyone!

Let $X$ be Hausdorff, locally compact, paracompact. Consider $\mathcal{F}$ a soft sheaf on $X$: as there are a variety of definitions of soft sheaf, let me emphasize my definition (taken from Gelfand-Manin, Methods of Homological Algebra, p.38): $\mathcal{F}$ is soft if for any closed $Z \subset X$, the restriction map $$\mathcal{F}(X) \rightarrow \varinjlim_{U \supset Z} \mathcal{F}(U)$$is a surjection.

For an open embedding $j: U \rightarrow X$, is it true that $j^* \mathcal{F}$ is again soft?

Thanks so much!

What I understand/motivation: I can prove this when $U$ is again paracompact - while (Hausdorff) and (Hausdorff + locally compact) both are inherited by open subsets, my impression was that paracompactness does not always descend, hence my question. Also, to be fair Gelfand-Manin also ask that $X$ be separable - will this affect the answer?

By the way, the motivation for my question is in trying to understand their proof of Verdier duality; on p. 231 they state that if $L$ is soft, then so is $L \otimes j_!j^*\mathbb{Z}$. My approach would be $$L \otimes j_! j^* \mathbb{Z} \simeq j_! j^* L$$ and Gelfand-Manin state in an exercise (which I'm still puzzling out) that $f_!$ between locally compact Hausdorff spaces sends soft sheaves to soft sheaves.

Hi everyone!

Let $X$ be Hausdorff, locally compact, paracompact. Consider $\mathcal{F}$ a soft sheaf on $X$: as there are a variety of definitions of soft sheaf, let me emphasize my definition (taken from Gelfand-Manin, Methods of Homological Algebra, p.38): $\mathcal{F}$ is soft if for any closed $Z \subset X$, the restriction map $$\mathcal{F}(X) \rightarrow \varinjlim_{U \supset Z} \mathcal{F}(U)$$is a surjection.

For an open embedding $j: U \rightarrow X$, is it true that $j^* \mathcal{F}$ is again soft?

Thanks so much!

What I understand/motivation: I can prove this when $U$ is again paracompact - while (Hausdorff) and (Hausdorff + locally compact) both are inherited by open subsets, my impression was that paracompactness does not always descend, hence my question. Also, to be fair Gelfand-Manin also ask that $X$ be separable - will this affect the answer?

By the way, the motivation for my question is in trying to understand their proof of Verdier duality; on p. 231 they state that if $L$ is soft, then so is $L \otimes j_!j^*\mathbb{Z}$. My approach would be $$L \otimes j_! j^* \mathbb{Z} \simeq j_! j^* L$$ and Gelfand-Manin state in an exercise (which I'm still puzzling out) that $f_!$ between locally compact Hausdorff spaces sends soft sheaves to soft sheaves.

One last soft (hah!) question: if I use the definition of a soft sheaf found here: http://amathew.wordpress.com/2011/06/10/soft-sheaves/ literally everything I'm doing becomes much easier, and the way Gelfand-Manin use the word 'deduce' in Exercise 1(b) on p. 236 makes me think this was in fact the definition they want to have (otherwise the 'if' statement as far as I can tell requires a reasonably involved paracompactness argument - and they don't define a soft sheaf on a not paracompact space, yet don't ask that the spaces in that exercise be paracompact, which adds to the confusion!). Has anyone run into problems with GM's notion of softness before when trying to read this section/does anyone have a reason not to simply supplant their definition with Akhil's?

Hi everyone!

Let $X$ be Hausdorff, locally compact, paracompact. Consider $\mathcal{F}$ a soft sheaf on $X$: as there are a variety of definitions of soft sheaf, let me emphasize my definition (taken from Gelfand-Manin, Methods of Homological Algebra, p.38): $\mathcal{F}$ is soft if for any closed $Z \subset X$, the restriction map $$\mathcal{F}(X) \rightarrow \varinjlim_{U \supset Z} \mathcal{F}(U)$$is a surjection.

For an open embedding $j: U \rightarrow X$, is it true that $j^* \mathcal{F}$ is again soft?

Thanks so much!

What I understand/motivation: I can prove this when $U$ is again paracompact - while (Hausdorff) and (Hausdorff + locally compact) both are inherited by open subsets, my impression was that paracompactness does not always descend, hence my question. Also, to be fair Gelfand-Manin also ask that $X$ be separable - will this affect the answer?

By the way, the motivation for my question is in trying to understand their proof of Verdier duality; on p. 231 they state that if $L$ is soft, then so is $L \otimes j_!j^*\mathbb{Z}$. My approach would be $$L \otimes j_! j^* \mathbb{Z} \simeq j_! j^* L$$ and Gelfand-Manin state in an exercise (which I'm still puzzling out) that any $f_!$ sends soft sheaves to soft sheaves.

Hi everyone!

Let $X$ be Hausdorff, locally compact, paracompact. Consider $\mathcal{F}$ a soft sheaf on $X$: as there are a variety of definitions of soft sheaf, let me emphasize my definition (taken from Gelfand-Manin, Methods of Homological Algebra, p.38): $\mathcal{F}$ is soft if for any closed $Z \subset X$, the restriction map $$\mathcal{F}(X) \rightarrow \varinjlim_{U \supset Z} \mathcal{F}(U)$$is a surjection.

For an open embedding $j: U \rightarrow X$, is it true that $j^* \mathcal{F}$ is again soft?

Thanks so much!

What I understand/motivation: I can prove this when $U$ is again paracompact - while (Hausdorff) and (Hausdorff + locally compact) both are inherited by open subsets, my impression was that paracompactness does not always descend, hence my question. Also, to be fair Gelfand-Manin also ask that $X$ be separable - will this affect the answer?

By the way, the motivation for my question is in trying to understand their proof of Verdier duality; on p. 231 they state that if $L$ is soft, then so is $L \otimes j_!j^*\mathbb{Z}$. My approach would be $$L \otimes j_! j^* \mathbb{Z} \simeq j_! j^* L$$ and Gelfand-Manin state in an exercise (which I'm still puzzling out) that $f_!$ between locally compact Hausdorff spaces sends soft sheaves to soft sheaves.