The Stacks project

Comments 1 to 20 out of 9220 in reverse chronological order.

\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 #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.


On Sinnaruil left comment #10061 on Section 6.7 in Sheaves on Spaces

I wonder could we add some categorical definition here? It seems convenient to define a sheaf as a special contravariant functor.


On ZL left comment #10060 on Lemma 101.21.1 in Morphisms of Algebraic Stacks

Typo : "...wee see that there exists an open subspace of containing <<>> such that the interesction is flat and ..."


On left comment #10059 on Remark 13.19.5 in Derived Categories

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On Ulrich Görtz left comment #10058 on Lemma 5.20.4 in Topology

In the definition of , the signs should be reversed (and accordingly in the computation at the end). As it is now, is the negative of a dimension function.


On David Epstein left comment #10057 on Definition 10.9.6 in Commutative Algebra

The notation for scalar multiplication is wrong.


On David Epstein left comment #10056 on Section 10.9 in Commutative Algebra

Throughout this section, the rings are assumed to be commutative, but this is not stated, except in the title. This is not enough.


On left comment #10055 on Section 15.1 in More on Algebra

Mr. Brainwash on Wikipedia

Mr. Brainwash su Wikipedia

Mr. Brainwash sur Wikipedia

Andy Warhol on Wikipedia

Coco Chanel on Wikipedia


On José Burgos left comment #10054 on Lemma 40.14.10 in More on Groupoid Schemes

A typo: constuctible should be constructible


On Branislav Sobot left comment #10053 on Lemma 15.9.7 in More on Algebra

I believe this lemma is wrong as stated. I think you need an additional condition that and are at most the degree of . The counterexample I have in mind is: Take ring , , and .


On left comment #10052 on Proposition 15.48.7 in More on Algebra

In the last sentence of the proof of case (2), an explicit reference to Lemma 07P9 can improve the clarity.


On left comment #10051 on Lemma 15.48.6 in More on Algebra

I think replacing with also works, and it is better to say instead of .


On François Loeser left comment #10050 on Lemma 97.12.6 in Criteria for Representability

Line -2 : "is representable by an algebraic space over ", the use of here is confusing, since it has already another meaning. .