diff --git a/Makefile b/Makefile
index 1363245..14ca555 100644
--- a/Makefile
+++ b/Makefile
@@ -168,6 +168,8 @@ QEMUOPTS += -device virtio-blk-device,drive=x0,bus=virtio-mmio-bus.0
 qemu: $K/kernel fs.img
 	$(QEMU) $(QEMUOPTS)
 
+kernel: $K/kernel
+
 .gdbinit: .gdbinit.tmpl-riscv
 	sed "s/:1234/:$(GDBPORT)/" < $^ > $@
 
diff --git a/kernel/riscv.h b/kernel/riscv.h
index c934309..47d5397 100644
--- a/kernel/riscv.h
+++ b/kernel/riscv.h
@@ -429,6 +429,16 @@ typedef u64 *pagetable_t; // 512 PTEs
 /**
  * Page Table Entry Flags
  */
+/** Valid */
+#define PTE_V (1L << 0)
+/** Readable */
+#define PTE_R (1L << 1)
+/** Writable */
+#define PTE_W (1L << 2)
+/** Executable */
+#define PTE_X (1L << 3)
+/** User-accessible */
+#define PTE_U (1L << 4)
 
 #define PTE_V (1L << 0) /** PTE Valid */
 #define PTE_R (1L << 1) /** PTE Readable */
diff --git a/kernel/vm.c b/kernel/vm.c
index af3ff45..dc33f30 100644
--- a/kernel/vm.c
+++ b/kernel/vm.c
@@ -12,7 +12,7 @@ extern char etext[]; // kernel.ld sets this to end of kernel code.
 
 extern char trampoline[]; // trampoline.S
 
-// Make a direct-map page table for the kernel.
+/** Make a direct-map page table for the kernel. */
 pagetable_t
 kvmmake(void)
 {
@@ -49,15 +49,17 @@ kvmmake(void)
   return kpgtbl;
 }
 
-// Initialize the one kernel_pagetable
+/** Initialize the one kernel_pagetable */
 void
 kvminit(void)
 {
   kernel_pagetable = kvmmake();
 }
 
-// Switch h/w page table register to the kernel's page table,
-// and enable paging.
+/**
+ * Switch h/w page table register to the kernel's page table,
+ * and enable paging.
+ */
 void
 kvminithart()
 {