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Register a new userBlochs cycle complex of zero cycles with $mathbb Z/p^n$ coefficients from Grothendiecks coherent duality point of view
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Fei Ren
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Let $X$ be a separated scheme of finite type over $k$ with $k$ being a perfect field of positive characteristic $p$. In this work we define a complex $K_{n,X,log}$ via Grothendiecks duality theory of coherent sheaves following Kato and build up a quasi-isomorphism from the Kato-Moser complex of logarithmic de Rham-Witt sheaves $tilde u_{n,X}$ to $K_{n,X,log}$ for the etale topology, and also for the Zariski topology under the extra assumption $k=bar k$. Combined with Zhongs quasi-isomorphism from Blochs cycle complex $mathbb Z^c_{X}$ to $tilde u_{n,X}$, we deduce certain vanishing, etale descent properties as well as invariance under rational resolutions for higher Chow groups of $0$-cycles with $mathbb Z/p^n$-coefficients.
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This note contains a generalization to $p>2$ of the authors previous calculations of the coefficients of $(mathbb{Z}/2)^n$-equivariant ordinary cohomology with coefficients in the constant $mathbb{Z}/2$-Mackey functor. The algberaic results by S.Kriz allow us to calculate the coefficients of the geometric fixed point spectrum $Phi^{(mathbb{Z}/p)^n}Hmathbb{Z}/p$, and more generally, the $mathbb{Z}$-graded coefficients of the localization of $Hmathbb{Z}/p_{(mathbb{Z}/p)^n}$ by inverting any chosen set of embeddings $S^0rightarrow S^{alpha_i}$ where $alpha_i$ are non-trivial irreducible representations. We also calculate the $RO(G)^+$-graded coefficients of $Hmathbb{Z}/p_{(mathbb{Z}/p)^n}$, which means the cohomology of a point indexed by an actual (not virtual) representation. (This is the non-derived part, which has a nice algebraic description.)
We propose a new theory of (non-split) P^n-functors. These are F: A -> B for which the adjunction monad RF is a repeated extension of Id_A by powers of an autoequivalence H and three conditions are satisfied: the monad condition, the adjoints condition, and the highest degree term condition. This unifies and extends the two earlier notions of spherical functors and split P^n-functors. We construct the P-twist of such F and prove it to be an autoequivalence. We then give a criterion for F to be a P^n-functor which is stronger than the definition but much easier to check in practice. It involves only two conditions: the strong monad condition and the weak adjoints condition. For split P^n-functors, we prove Segals conjecture on their relation to spherical functors. Finally, we give four examples of non-split P^n-functors: spherical functors, extensions by zero, cyclic covers, and family P-twists. For the latter, we show the P-twist to be the derived monodromy of associated Mukai flop, the so-called `flop-flop = twist formula.
In order to obtain existence criteria for orthogonal instanton bundles on $mathbb{P}^n$, we provide a bijection between equivalence classes of orthogonal instanton bundles with no global sections and symmetric forms. Using such correspondence we are able to provide explicit examples of orthogonal instanton bundles with no global sections on $mathbb{P}^n$ and prove that every orthogonal instanton bundle with no global sections on $mathbb{P}^n$ and charge $cgeq 3$ has rank $rleq (n-1)c$. We also prove that when the rank $r$ of the bundles reaches the upper bound, $mathcal{M}_{mathbb{P}^n}^{mathcal{O}}(c,r)$, the coarse moduli space of orthogonal instanton bundles with no global sections on $mathbb{P}^n$, with charge $cgeq 3$ and rank $r$, is affine, reduced and irreducible. Last, we construct Kronecker modules to determine the splitting type of the bundles in $mathcal{M}_{mathbb{P}^n}^{mathcal{O}}(c,r)$, whenever is non-empty.
We reformulate the problem of bounding the total rank of the homology of perfect chain complexes over the group ring $mathbb{F}_p[G]$ of an elementary abelian $p$-group $G$ in terms of commutative algebra. This extends results of Carlsson for $p=2$ to all primes. As an intermediate step, we construct an embedding of the derived category of perfect chain complexes over $mathbb{F}_p[G]$ into the derived category of $p$-DG modules over a polynomial ring.
We compute the $GL_{r+1}$-equivariant Chow class of the $GL_{r+1}$-orbit closure of any point $(x_1, ldots, x_n) in (mathbb{P}^r)^n$ in terms of the rank polytope of the matroid represented by $x_1, ldots, x_n in mathbb{P}^r$. Using these classes and generalizations involving point configurations in higher dimensional projective spaces, we define for each $dtimes n$ matrix $M$ an $n$-ary operation $[M]_hbar$ on the small equivariant quantum cohomology ring of $mathbb{P}^r$, which is the $n$-ary quantum product when $M$ is an invertible matrix. We prove that $M mapsto [M]_hbar$ is a valuative matroid polytope association. Like the quantum product, these operations satisfy recursive properties encoding solutions to enumerative problems involving point configurations of given moduli in a relative setting. As an application, we compute the number of line sections with given moduli of a general degree $2r+1$ hypersurface in $mathbb{P}^r$, generalizing the known case of quintic plane curves.
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