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The Stacks project

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\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

On left comment #10081 on Item 50

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On René Bruin left comment #10080 on Lemma 10.112.7 in Commutative Algebra

Am I right that there is a typo in the sentence And then we may choose lying over ?

If I understand it correctly, then it should be , since by the going down property, one defines a chain of prime ideals lying over the chain of prime ideals in the sense that the preimage of under the map equals .


On Junyan Xu left comment #10079 on Lemma 10.30.5 in Commutative Algebra

Let me present a proof that does not rely on localizations or quotients (so it applies to semirings as well). This lemma is a special case (where is injective and ) of the following more general going-up result:

If is a homomorphism of commutative semirings and is an ideal in , then for any prime ideal , there exists a prime ideal such that . If is minimal over , then of course .

This result directly follows from the following fact: if is a submonoid of a commutative semiring and is an ideal disjoint from it, then there exists a maximal such ideal containing , and any such ideal is prime. A formalized proof is available at https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/Maximal.html#Ideal.isPrime_of_maximally_disjoint. To obtain the result, just notice that the submonoid is disjoint from an ideal (or subset) iff .

The implication (1) ⇒ (2) in https://stacks.math.columbia.edu/tag/00FL also follows from this general result by taking , so is the kernel of . (1) says that the kernel is contained in the nilradical of , but every prime contains the nilradical, so a minimal prime over the nilradical is the same as a minimal prime of .

In https://stacks.math.columbia.edu/tag/00HY, one can directly show the image is equal to the zero locus of . Every prime ideal in clearly contains , and on the other hand, if a prime ideal contains then there is a prime such that by the general result, so because and is closed under specialization.


On left comment #10078 on Section 5.5 in Topology

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On left comment #10077 on Remark 13.19.5 in Derived Categories

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On left comment #10076 on Remark 13.19.5 in Derived Categories

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On left comment #10075 on Remark 13.19.5 in Derived Categories

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On LZW left comment #10074 on Lemma 10.99.1 in Commutative Algebra

Krull intersection requires the local ring to be noetherian. See: https://stacks.math.columbia.edu/tag/00IP


On alberto tonolo left comment #10073 on Section 4.24 in Categories

As you observed at the end of the proof of Lemma 24.4, one can join Lemmas 24.3 and 24.4 in a unique result proving that TFAE:


On Joshua Mundinger left comment #10072 on Lemma 10.94.1 in Commutative Algebra

Lemma 05A3 (2) is also immediate from https://stacks.math.columbia.edu/tag/059M and that restriction of scalars along commutes with arbitrary products.


On Barbara Fantechi left comment #10071 on Section 96.17 in Sheaves on Algebraic Stacks

I think Lemma 06WX (2) needs the assumption that f is flat, since is a functor to modules and tensoring to get is only exact when f is flat. For instance if f is the inclusion of a closed point in then is injective but is not.


On Barbara Fantechi left comment #10070 on Section 96.17 in Sheaves on Algebraic Stacks

I think Lemma 06WX (2) needs the assumption that f is flat, since is a functor to modules and tensoring to get is only exact when f is flat. For instance if f is the inclusion of a closed point in then is injective but is not.


On left comment #10069 on Section 29.10 in Morphisms of Schemes

In the definition of "purely inseparable extension", it is written that "these extensions only show up in positive characteristic". Hence, using the definitions in this section, it would follow that radicial morphisms also only show up in positive characteristic, which is not what we want, right? Perhaps one should change the definition of "purely inseparable extension" to include the characteristic zero case, by saying that is purely inseparable if each is purely inseparable over .


On left comment #10068 on Section 25.3 in Hypercoverings

From Street Art to Canvas

In the vibrant tapestry of contemporary art, the evolution of street art has transcended its urban origins, finding a prominent place within the hallowed halls of galleries and auction houses. This metamorphosis from ephemeral graffiti on the wall to coveted canvas pieces illustrates not only a shift in artistic expression but also a burgeoning recognition of street art as a legitimate form of fine art.

Graffiti on The Wall Understanding Street Art

Street art, often characterized by its bold colors and provocative themes, serves as a powerful medium for social commentary and cultural reflection. It challenges the boundaries of traditional art forms, inviting viewers to engage with the narratives woven into the urban landscape. As society increasingly embraces this genre, understanding its nuances becomes essential for both artists and collectors alike.

Learning to Spray Painting

For those aspiring to delve into the world of street art, mastering the technique of spray painting is paramount. This dynamic form of expression requires not only technical skill but also an innate understanding of color theory and composition. Workshops and tutorials abound, providing budding artists with the tools necessary to transform their visions into tangible works of art.

Drawing to Spray Painting

Transitioning from drawing to spray painting can be both exhilarating and daunting. Artists often find that their foundational skills in sketching enhance their ability to manipulate spray cans, allowing for intricate designs and bold statements. This synergy between traditional drawing techniques and modern spray painting opens new avenues for creativity, enabling artists to explore their unique styles.

Instigating the ‘Shift’ to Bankable Art

As street art garners recognition within the mainstream art market, a palpable shift occurs—one that transforms once-underground artists into bankable commodities. This transition not only elevates the status of street art but also prompts discussions about authenticity, ownership, and the commercialization of what was once deemed transgressive. As we navigate this evolving landscape, it becomes imperative to celebrate the roots of street art while acknowledging its potential as a lucrative investment.


On Dat Pham left comment #10067 on Section 59.58 in Étale Cohomology

Perhaps one should also mention that the isomorphisms have been known also for locally profinite by \ref{https://arxiv.org/pdf/2106.04473}. (This should also follow from the two facts: (i) the functor taking to its associated condensed abelian group is exact for discrete , (ii) condensed group cohomology agrees with continous group cohomology for locally profinite and discrete (or more generally if is solid). See the notes \ref{https://janschuetz.perso.math.cnrs.fr/skripte/homology_profinite.pdf}.)


On Joe Lamond left comment #10066 on Lemma 10.37.5 in Commutative Algebra

If is a normal domain, and is a multiplicative set containing , then is the zero ring, which is not even an integral domain. I suppose the zero ring is, strictly speaking, a normal ring, since it is vacuously true that the localizations at prime ideals are normal domains, but I wonder if this was really intended.


On Lucas Henrique left comment #10065 on Lemma 10.119.11 in Commutative Algebra

Why are there finitely many maximals containing ? This seems to assume that is Noetherian -- it is equivalent to say that has finitely many minimal primes, and I think this already requires the Noetherian hypothesis, which is only stated later.


On Anonymous left comment #10064 on Lemma 21.17.17 in Cohomology on Sites

Typo: distinghuised


On left comment #10063 on Proposition 66.14.1 in Properties of Algebraic Spaces

I presume that (ii) should be "equivalence relation" or, equivalently, "monomorphism". But I don't understand how (iii) is supposed to be interpreted at all. If the condition holds for all points u, then it's of course fine. But if it holds for all but one orbit, it is not fine. Take for example Hironaka's example of a proper smooth algebraic space of dimension 3 which is not a scheme. It can be obtained as the quotient of a free involution on a proper scheme . There is a closed orbit in which has no affine neighborhood and the complement of is quasi-projective. So is a scheme where corresponds to . Any dense open affine in pulls-back to a dense -stable affine open in . Does dense have another meaning in this context?


On some guy left comment #10062 on Section 10.9 in Commutative Algebra

@10056: per the conventions, all rings in the Stacks Project are commutative with 1.