The Stacks project

I think Lemma 06WX (2) needs the assumption that f is flat, since $f^{-1}$ is a functor to $f^{-1}O_Y$ modules and tensoring to get $f^\star$ is only exact when f is flat. For instance if f is the inclusion of a closed point in $A^1$ then $O_X$ is injective but $f_*O_X$ is not.

I think Lemma 06WX (2) needs the assumption that f is flat, since $f^{-1}$ is a functor to $f^{-1}O_Y$ modules and tensoring to get $f^\star$ is only exact when f is flat. For instance if f is the inclusion of a closed point in $A^1$ then $O_X$ is injective but $f_*O_X$ is not.

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 $K/F$ is purely inseparable if each $\alpha \in K - F$ is purely inseparable over $F$.

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.

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

If $R$ is a normal domain, and $S\subset R$ is a multiplicative set containing $0$, then $S^{-1}R$ 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.

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

Typo: distinghuised

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 $X$ of dimension 3 which is not a scheme. It can be obtained as the quotient of a free involution on a proper scheme $U$. There is a closed orbit $\lbrace u_1, u_2 \rbrace$ in $U$ which has no affine neighborhood and the complement of $\lbrace u_1, u_2 \rbrace$ is quasi-projective. So $X\smallsetminus \lbrace x\rbrace$ is a scheme where $x$ corresponds to $\lbrace u_1, u_2 \rbrace$. Any dense open affine in $X\smallsetminus \lbrace x\rbrace$ pulls-back to a dense $R$-stable affine open in $U$. Does dense have another meaning in this context?

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

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

Typo : "...wee see that there exists an open subspace $U'$ of $U \times _ \mathcal {X} W$ containing <<$p'(z')$>> such that the interesction $U' \cap (V(f _1, \ldots , f_ r) \times _ \mathcal {X} W)$ is flat and ..."

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

The notation for scalar multiplication is wrong.

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

Mr. Brainwash on Wikipedia

Mr. Brainwash su Wikipedia

Mr. Brainwash sur Wikipedia

Andy Warhol on Wikipedia

Coco Chanel on Wikipedia

A typo: constuctible should be constructible

I believe this lemma is wrong as stated. I think you need an additional condition that $\bar{g}$ and $\bar{h}$ are at most the degree of $f$. The counterexample I have in mind is: Take ring $A:=\mathbb{Z}$, $I:=4\mathbb{Z}$, $f=1$ and $\bar{g}=\bar{h}=2x^2+2x+1$.

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