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If any counterexamples (that is, local set-theoretic complete intersections of small codimension that are not complete intersections) are known, I would like to know their lower cohomology. …

text we have several statements of the following sort: for a certain morphism $f:X\to Y$ of varieties over an (algebraically closed) field of characteristic $p$ and some $c>0$ the corresponding etale cohomology …

If a projective bundle $Y$ over a variety $X$ is obtained from a vector bundle $E/X$ then the cohomology and the motif of $Y$ is known to be closely related to that of $X$. …

, if $Y$ is singular then you surely don't have transversality and probably don't have the isomorphism in question (as far as I remember, the intersection homology of $X$ is isomorphic to its "usual" cohomology …

Étale cohomology with support and functoriality). … Still, does there exist a method for "computing" this relative cohomology (preferably by relating to the cohomology of some $\mathbb{C}$-varieties)? …

If $h_i:A_i\to A_{i+1}$ is a countable chain of morphisms in an abelian category $A$ that is AB3 then one can consider the (Bökstedt-Neeman) homotopy colimit of $A_i$ in $D^b(A)$. This is a two-term c …

Certainly, the usual way to switch the numeration is to consider $H^i=H_{-i}$; yet in topology one never calls singular homology a cohomology theory! See Homology or cohomology? …

Inside the 'usual' categories of motives (Chow, Voevodsky ones) there are the categories of effective motives. Similarly, there are effective Hodge structures; they seem to be closely related with eff …

subvarieties of $X$ of dimension $m$ (here $H^\ast$ is singular or etale cohomology)? Did anybody study this filtration (for fixed $X,i$, when $m$ varies) before? …

Unfortunately, I don't know how to compute the cohomology of my $S$; I only know that it has 'many holes' in it (i.e. it is 'very far from being projective'), and it is finite over a variety $X$ with $ …

One says that a triple is split over $U$ if a certain line bundle is trivial (see Definition 11.11); this has certain consequences for cohomology of varieties with coefficients in a homotopy invariant …

Let $A^\ast$ be an algebraic oriented cohomology theory (i.e. it is equipped with certain push-forwards for projective morphisms of smooth varieties over the base field $k$; see section 2 of http://www.math.uiuc.edu … It seems sufficient to prove the latter for $A^\ast$ being the algebraic cobordism (as defined by Levine and Morel), since this is the universal algebraic oriented cohomology theory. …

Let $V$ be a variety (or a Whitney stratified space); $C$ is a constant etale ('topological') sheaf on it. Let $t$ denote the middle perverse t-structure for the corresponding derived category (of she …

Note that one can compute the cohomology of $U$ as the hypercohomology $H^*(Z,Rpr'_\ast\mathbb{Z}/l^n\mathbb{Z}_U)$, where $pr': U\to Z$ is the corresponding prinicple $G_m$-bundle. … Cf. https://mathoverflow.net/questions/89171/on-the-cohomology-of-g-m-bundles-and-purity-for-singular-varieties …

Certain examples and http://ncatlab.org/nlab/show/fiber+sequence#LongSequCoh seem to suggest that the cohomology of $D$ should be something like $H^*(B)\otimes_{H^*(A)}H^\ast(C)$ (in lower cohomological …